From 908fbba2d8d16800da3da0ab718c58f6cb1501a4 Mon Sep 17 00:00:00 2001 From: Vasily Ilin Date: Sun, 27 Sep 2026 12:54:22 +0000 Subject: [PATCH 1/6] tools: preserve reproducible Zeta5 certificate templates --- python/tests/test_zeta5_certificates.py | 55 + scripts/zeta5-certificates/README.md | 60 + .../zeta5-certificates/archive/manifest.json | 479 ++++++ .../archive/records/Growth_InnerSum.json | 261 ++++ .../archive/records/Growth_InnerTable.json | 128 ++ .../archive/records/Growth_OuterTable.json | 21 + .../archive/records/Table_Intervals.json | 1371 +++++++++++++++++ .../archive/records/Table_U00.json | 207 +++ .../archive/records/Table_U01.json | 207 +++ .../archive/records/Table_U02.json | 207 +++ .../archive/records/Table_U03.json | 207 +++ .../archive/records/Table_U04.json | 207 +++ .../archive/records/Table_U05.json | 207 +++ .../archive/records/Table_U06.json | 207 +++ .../archive/records/Table_U07.json | 207 +++ .../archive/records/Table_U08.json | 207 +++ 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scripts/zeta5-certificates/archive/templates/455.txt create mode 100644 scripts/zeta5-certificates/archive/templates/456.txt create mode 100644 scripts/zeta5-certificates/archive/templates/457.txt create mode 100644 scripts/zeta5-certificates/archive/templates/458.txt create mode 100644 scripts/zeta5-certificates/archive/templates/459.txt create mode 100644 scripts/zeta5-certificates/archive/templates/460.txt create mode 100644 scripts/zeta5-certificates/regenerate.py diff --git a/python/tests/test_zeta5_certificates.py b/python/tests/test_zeta5_certificates.py new file mode 100644 index 0000000000..7753e42b3a --- /dev/null +++ b/python/tests/test_zeta5_certificates.py @@ -0,0 +1,55 @@ +"""Check exact and constrained rendering of the Zeta5 certificate archive.""" + +import importlib.util +import json +from pathlib import Path + +import pytest + +SCRIPT = Path(__file__).parents[2] / "scripts/zeta5-certificates/regenerate.py" +SPECIFICATION = importlib.util.spec_from_file_location("zeta5_regenerate", SCRIPT) +assert SPECIFICATION is not None and SPECIFICATION.loader is not None +GENERATOR = importlib.util.module_from_spec(SPECIFICATION) +SPECIFICATION.loader.exec_module(GENERATOR) + + +def test_exact_large_rational_and_name() -> None: + """Preserve huge rational numerators, signs, names, and Unicode byte for byte.""" + template = ( + "lemma U_{{integer}} : (-( {{integer}} / {{integer}} : ℝ)) ≤ x := by\n" + " exact h\n" + ) + numerator = "123456789012345678901234567890123456789" + result = GENERATOR.render(template, ["12", numerator, "200000000000000000000"]) + assert ( + result + == f"lemma U_12 : (-( {numerator} / 200000000000000000000 : ℝ)) ≤ x := by\n" + " exact h\n" + ) + + +@pytest.mark.parametrize("parameters", [[], ["1", "2"], ["1e100"], ["0; run"]]) +def test_reject_invalid_parameters(parameters: list[str]) -> None: + """Reject malformed or incomplete certificate data before rendering.""" + with pytest.raises(ValueError): + GENERATOR.render("{{integer}}", parameters) + + +@pytest.mark.parametrize("path", ["../bad.lean", "/bad.lean", "LeanPool/Other/A.lean"]) +def test_reject_output_outside_project(path: str) -> None: + """Keep accidental manifest edits from writing unrelated files.""" + with pytest.raises(ValueError): + GENERATOR.certificate_path(path) + + +def test_complete_archive(tmp_path: Path) -> None: + """Regenerate every output and independently check its pinned hash.""" + archive = SCRIPT.parent / "archive" + GENERATOR.verify_archive(archive) + assert GENERATOR.regenerate(archive, tmp_path, False) == 118 + assert GENERATOR.regenerate(archive, tmp_path, True) == 118 + manifest = json.loads((archive / "manifest.json").read_text()) + first = tmp_path / next(iter(manifest["files"])) + first.write_text(first.read_text() + "-- damaged\n") + with pytest.raises(ValueError, match="Certificates differ"): + GENERATOR.regenerate(archive, tmp_path, True) diff --git a/scripts/zeta5-certificates/README.md b/scripts/zeta5-certificates/README.md new file mode 100644 index 0000000000..8b59c880fb --- /dev/null +++ b/scripts/zeta5-certificates/README.md @@ -0,0 +1,60 @@ +# Reproducing the Zeta5 numerical certificates + +This archive reproduces all 118 generated modules in Lean Pool PR #540, including +114 U/V endpoint tables, the 684-interval assembly, and the inner/outer growth +tables and inner sum. The manifest pins source commit +`154983b2d21f27207c078cccf2646715c58e0d3c`, upstream +[`mo271/Zeta5`](https://github.com/mo271/Zeta5/tree/f19a1960609f7d38e7b63fd2acb05e6f60a7b741), +and each output's SHA-256. The later removal of the unused paper normalizer does +not change any archived file. + +Use Python 3.13 or later. No third-party packages, network, Lean installation, +floating-point arithmetic, or execution of archived code is needed for rendering: + +```sh +python3 scripts/zeta5-certificates/regenerate.py --verify-archive +python3 scripts/zeta5-certificates/regenerate.py --output "$CODEX_SCRATCH_ROOT/zeta5-regenerated" +python3 scripts/zeta5-certificates/regenerate.py --check /path/to/import-checkout +``` + +`--output` writes only the recorded project modules underneath its argument; +use an empty scratch directory when inspecting a regeneration. `--check` compares +bytes without writing. The archive hash verification is separate so a deliberate +template edit can be rendered before its baseline hashes are updated. + +## Data and shared proof templates + +The archive contains 14,147 blocks represented by 461 text templates. A record +is `[template_identifier, decimal_integer_strings]`. Replace the consecutive +`{{integer}}` markers in the named template with those strings, preserving all +other bytes, and concatenate a file's records in order. Signs, rational division, +Lean syntax, tactic calls, and layout live in the templates. Parameters include +numeric suffixes in declaration names as well as mathematical integers. Integer +strings avoid any JSON floating-point precision loss. + +The templates were extracted from the canonical, compiled Lean port, splitting +before top-level `lemma`, `theorem`, and `def` declarations (including private +ones), and replacing each ASCII decimal digit run with a marker. Templates with +identical resulting bytes are shared. Layout variants remain separate to preserve +byte-for-byte output. The complete round trip and all output hashes were checked. + +This is a generator for the **preserved certificate set**, not a search algorithm +for stronger bounds or a claim to reconstruct the original numerical search. +The upstream scripts document that search, but do not reproduce this port. Here +the exact certified rational parameters are explicit inputs. Changes to a common +proof idiom can be made once in its template rather than across its users; the +most-used template occurs 4,500 times. Grepping `records/` for its identifier finds +all affected modules. A change to the number of markers requires corresponding +parameter changes and is rejected if their counts disagree. + +After editing, render to scratch, inspect the diff, copy the intended outputs to +a content branch, and run the project build, style checks, linters, and trust +audit. Every emitted theorem must still pass the Lean kernel. New interval +partitions or new rational bounds additionally need mathematical validation; +this tool does not certify them on its own. Keep tooling edits in a separate PR +from Lean content, as required by the repository's separation policy. + +The certificate text and numeric data derive from Moritz Firsching's Apache-2.0 +formalization, with AI-assisted proof refactoring by the Lean Pool acquisition +agent. Original copyright/author headers are reproduced in the outputs. This +tool and archive are distributed under the repository's Apache-2.0 license. diff --git a/scripts/zeta5-certificates/archive/manifest.json b/scripts/zeta5-certificates/archive/manifest.json new file mode 100644 index 0000000000..2e3e3fda6a --- /dev/null +++ b/scripts/zeta5-certificates/archive/manifest.json @@ -0,0 +1,479 @@ +{ + "source_commit": "154983b2d21f27207c078cccf2646715c58e0d3c", + "upstream_commit": "f19a1960609f7d38e7b63fd2acb05e6f60a7b741", + "license": "Apache-2.0", + "files": { + "LeanPool/Zeta5Irrational/Table/U00.lean": { + "records": "Table_U00.json", + "sha256": "6082d945dfd872a62dc8d5264d61f381c9224d97bfe9abb5aa52849daf9fb090" + }, + "LeanPool/Zeta5Irrational/Table/U01.lean": { + "records": "Table_U01.json", + "sha256": "12ad80545461e9f513ed4fb2d195f01d97bdac51806c5cb3e0e80b3f9463c388" + }, + "LeanPool/Zeta5Irrational/Table/U02.lean": { + "records": "Table_U02.json", + "sha256": "171477f5cf121bbbc8f13196d515753fa5f0939de12b22c0ed18aad1ffbb3658" + }, + 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+] diff --git a/scripts/zeta5-certificates/archive/templates/000.txt b/scripts/zeta5-certificates/archive/templates/000.txt new file mode 100644 index 0000000000..1d65d7a89a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/000.txt @@ -0,0 +1,18 @@ +/- +Copyright (c) {{integer}} Moritz Firsching. All rights reserved. +Released under Apache {{integer}}.{{integer}} license as described in the file LICENSE. +Authors: Moritz Firsching +-/ + +module + +public import LeanPool.Zeta{{integer}}Irrational.Table.UpperCertificate + +/-! # Certified arcsine potential bounds (U{{integer}}) -/ + +@[expose] public section + +open Finset + +namespace Zeta{{integer}}Irrational + diff --git a/scripts/zeta5-certificates/archive/templates/001.txt b/scripts/zeta5-certificates/archive/templates/001.txt new file mode 100644 index 0000000000..52c3c98e6e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/001.txt @@ -0,0 +1,22 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_left_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/002.txt b/scripts/zeta5-certificates/archive/templates/002.txt new file mode 100644 index 0000000000..2586952eef --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/002.txt @@ -0,0 +1,21 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_left_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/003.txt b/scripts/zeta5-certificates/archive/templates/003.txt new file mode 100644 index 0000000000..8c45fffd59 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/003.txt @@ -0,0 +1,23 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_left_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) + (by norm_num [aρ_{{integer}}]) (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) + (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/004.txt b/scripts/zeta5-certificates/archive/templates/004.txt new file mode 100644 index 0000000000..f5d6bf1738 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/004.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : Uρ ({{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/005.txt b/scripts/zeta5-certificates/archive/templates/005.txt new file mode 100644 index 0000000000..f45852ef01 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/005.txt @@ -0,0 +1,22 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_left_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/006.txt b/scripts/zeta5-certificates/archive/templates/006.txt new file mode 100644 index 0000000000..fe4159ea04 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/006.txt @@ -0,0 +1,23 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_left_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/007.txt b/scripts/zeta5-certificates/archive/templates/007.txt new file mode 100644 index 0000000000..6afac2fddb --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/007.txt @@ -0,0 +1,24 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_left_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) + (by norm_num [aρ_{{integer}}]) (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) + (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/008.txt b/scripts/zeta5-certificates/archive/templates/008.txt new file mode 100644 index 0000000000..c158463da6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/008.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/009.txt b/scripts/zeta5-certificates/archive/templates/009.txt new file mode 100644 index 0000000000..d117eab27a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/009.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/010.txt b/scripts/zeta5-certificates/archive/templates/010.txt new file mode 100644 index 0000000000..fba40d3439 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/010.txt @@ -0,0 +1,5 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := + by rw [Uω_of_mem (by norm_num [aρ_{{integer}}]) (by norm_num [bρ_{{integer}}])]; exact Lmid_{{integer}} + diff --git a/scripts/zeta5-certificates/archive/templates/011.txt b/scripts/zeta5-certificates/archive/templates/011.txt new file mode 100644 index 0000000000..1910cbd62e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/011.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/012.txt b/scripts/zeta5-certificates/archive/templates/012.txt new file mode 100644 index 0000000000..9cb0484080 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/012.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/013.txt b/scripts/zeta5-certificates/archive/templates/013.txt new file mode 100644 index 0000000000..212e0b2178 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/013.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/014.txt b/scripts/zeta5-certificates/archive/templates/014.txt new file mode 100644 index 0000000000..9cb0bc39b8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/014.txt @@ -0,0 +1,21 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_left_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/015.txt b/scripts/zeta5-certificates/archive/templates/015.txt new file mode 100644 index 0000000000..42cec92ea6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/015.txt @@ -0,0 +1,22 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_left_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/016.txt b/scripts/zeta5-certificates/archive/templates/016.txt new file mode 100644 index 0000000000..15587330f6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/016.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/017.txt b/scripts/zeta5-certificates/archive/templates/017.txt new file mode 100644 index 0000000000..19aad37f89 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/017.txt @@ -0,0 +1,81 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := + by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/018.txt b/scripts/zeta5-certificates/archive/templates/018.txt new file mode 100644 index 0000000000..a485440f05 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/018.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := + by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/019.txt b/scripts/zeta5-certificates/archive/templates/019.txt new file mode 100644 index 0000000000..e7189aa3fa --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/019.txt @@ -0,0 +1,21 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_left_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/020.txt b/scripts/zeta5-certificates/archive/templates/020.txt new file mode 100644 index 0000000000..cc3e77d2ee --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/020.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := + by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/021.txt b/scripts/zeta5-certificates/archive/templates/021.txt new file mode 100644 index 0000000000..9ad5f4f038 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/021.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/022.txt b/scripts/zeta5-certificates/archive/templates/022.txt new file mode 100644 index 0000000000..3102608001 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/022.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/023.txt b/scripts/zeta5-certificates/archive/templates/023.txt new file mode 100644 index 0000000000..8bb3888676 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/023.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/024.txt b/scripts/zeta5-certificates/archive/templates/024.txt new file mode 100644 index 0000000000..18fcfd998a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/024.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/025.txt b/scripts/zeta5-certificates/archive/templates/025.txt new file mode 100644 index 0000000000..44ed9823b3 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/025.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/026.txt b/scripts/zeta5-certificates/archive/templates/026.txt new file mode 100644 index 0000000000..0523b10ccb --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/026.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/027.txt b/scripts/zeta5-certificates/archive/templates/027.txt new file mode 100644 index 0000000000..93d30d1b7e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/027.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/028.txt b/scripts/zeta5-certificates/archive/templates/028.txt new file mode 100644 index 0000000000..330adb3f34 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/028.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/029.txt b/scripts/zeta5-certificates/archive/templates/029.txt new file mode 100644 index 0000000000..65e8edb52f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/029.txt @@ -0,0 +1,22 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [bρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/030.txt b/scripts/zeta5-certificates/archive/templates/030.txt new file mode 100644 index 0000000000..50bc184cf5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/030.txt @@ -0,0 +1,86 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/031.txt b/scripts/zeta5-certificates/archive/templates/031.txt new file mode 100644 index 0000000000..ac4e4e55d4 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/031.txt @@ -0,0 +1,23 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [bρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/032.txt b/scripts/zeta5-certificates/archive/templates/032.txt new file mode 100644 index 0000000000..921aa9deba --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/032.txt @@ -0,0 +1,21 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [bρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/033.txt b/scripts/zeta5-certificates/archive/templates/033.txt new file mode 100644 index 0000000000..b3e4c52cd2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/033.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/034.txt b/scripts/zeta5-certificates/archive/templates/034.txt new file mode 100644 index 0000000000..a3a8f7939d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/034.txt @@ -0,0 +1,21 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [bρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/035.txt b/scripts/zeta5-certificates/archive/templates/035.txt new file mode 100644 index 0000000000..5a27076f87 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/035.txt @@ -0,0 +1,22 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [bρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/036.txt b/scripts/zeta5-certificates/archive/templates/036.txt new file mode 100644 index 0000000000..72aea90771 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/036.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/037.txt b/scripts/zeta5-certificates/archive/templates/037.txt new file mode 100644 index 0000000000..e8fdccbd31 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/037.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/038.txt b/scripts/zeta5-certificates/archive/templates/038.txt new file mode 100644 index 0000000000..4c6c1a90c9 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/038.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/039.txt b/scripts/zeta5-certificates/archive/templates/039.txt new file mode 100644 index 0000000000..2f89589b31 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/039.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/040.txt b/scripts/zeta5-certificates/archive/templates/040.txt new file mode 100644 index 0000000000..2096156a59 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/040.txt @@ -0,0 +1,23 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) + (by norm_num [bρ_{{integer}}]) (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/041.txt b/scripts/zeta5-certificates/archive/templates/041.txt new file mode 100644 index 0000000000..4da8581d69 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/041.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/042.txt b/scripts/zeta5-certificates/archive/templates/042.txt new file mode 100644 index 0000000000..895c01ed70 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/042.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/043.txt b/scripts/zeta5-certificates/archive/templates/043.txt new file mode 100644 index 0000000000..1051909453 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/043.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/044.txt b/scripts/zeta5-certificates/archive/templates/044.txt new file mode 100644 index 0000000000..0d9f86227c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/044.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/045.txt b/scripts/zeta5-certificates/archive/templates/045.txt new file mode 100644 index 0000000000..ebea4e399f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/045.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/046.txt b/scripts/zeta5-certificates/archive/templates/046.txt new file mode 100644 index 0000000000..cde413d17c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/046.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/047.txt b/scripts/zeta5-certificates/archive/templates/047.txt new file mode 100644 index 0000000000..372ad02f94 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/047.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/048.txt b/scripts/zeta5-certificates/archive/templates/048.txt new file mode 100644 index 0000000000..4997254f86 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/048.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/049.txt b/scripts/zeta5-certificates/archive/templates/049.txt new file mode 100644 index 0000000000..18ac9944bf --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/049.txt @@ -0,0 +1,86 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/050.txt b/scripts/zeta5-certificates/archive/templates/050.txt new file mode 100644 index 0000000000..49c7d792b0 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/050.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/051.txt b/scripts/zeta5-certificates/archive/templates/051.txt new file mode 100644 index 0000000000..8d34ccb980 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/051.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/052.txt b/scripts/zeta5-certificates/archive/templates/052.txt new file mode 100644 index 0000000000..00f54e0a3a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/052.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/053.txt b/scripts/zeta5-certificates/archive/templates/053.txt new file mode 100644 index 0000000000..190b44f5b4 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/053.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/054.txt b/scripts/zeta5-certificates/archive/templates/054.txt new file mode 100644 index 0000000000..5b57ed6e79 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/054.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/055.txt b/scripts/zeta5-certificates/archive/templates/055.txt new file mode 100644 index 0000000000..0e432d58aa --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/055.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/056.txt b/scripts/zeta5-certificates/archive/templates/056.txt new file mode 100644 index 0000000000..df5a62b860 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/056.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/057.txt b/scripts/zeta5-certificates/archive/templates/057.txt new file mode 100644 index 0000000000..077dd16cf6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/057.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/058.txt b/scripts/zeta5-certificates/archive/templates/058.txt new file mode 100644 index 0000000000..1b437a154b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/058.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/059.txt b/scripts/zeta5-certificates/archive/templates/059.txt new file mode 100644 index 0000000000..9096462475 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/059.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/060.txt b/scripts/zeta5-certificates/archive/templates/060.txt new file mode 100644 index 0000000000..0e1451abb8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/060.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/061.txt b/scripts/zeta5-certificates/archive/templates/061.txt new file mode 100644 index 0000000000..f85f66bbb0 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/061.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/062.txt b/scripts/zeta5-certificates/archive/templates/062.txt new file mode 100644 index 0000000000..34d4b53b12 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/062.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/063.txt b/scripts/zeta5-certificates/archive/templates/063.txt new file mode 100644 index 0000000000..5747f9345f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/063.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/064.txt b/scripts/zeta5-certificates/archive/templates/064.txt new file mode 100644 index 0000000000..c522872b2c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/064.txt @@ -0,0 +1,24 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) + (by norm_num [bρ_{{integer}}]) (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) + (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/065.txt b/scripts/zeta5-certificates/archive/templates/065.txt new file mode 100644 index 0000000000..75307b73b8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/065.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/066.txt b/scripts/zeta5-certificates/archive/templates/066.txt new file mode 100644 index 0000000000..38f305b9f5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/066.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/067.txt b/scripts/zeta5-certificates/archive/templates/067.txt new file mode 100644 index 0000000000..58c59ffc1a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/067.txt @@ -0,0 +1,86 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/068.txt b/scripts/zeta5-certificates/archive/templates/068.txt new file mode 100644 index 0000000000..c36cb9aa73 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/068.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/069.txt b/scripts/zeta5-certificates/archive/templates/069.txt new file mode 100644 index 0000000000..b15c4acf00 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/069.txt @@ -0,0 +1,86 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/070.txt b/scripts/zeta5-certificates/archive/templates/070.txt new file mode 100644 index 0000000000..4ec013d69d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/070.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/071.txt b/scripts/zeta5-certificates/archive/templates/071.txt new file mode 100644 index 0000000000..f4ffa5ee67 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/071.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/072.txt b/scripts/zeta5-certificates/archive/templates/072.txt new file mode 100644 index 0000000000..319025e894 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/072.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/073.txt b/scripts/zeta5-certificates/archive/templates/073.txt new file mode 100644 index 0000000000..16881bf67c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/073.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/074.txt b/scripts/zeta5-certificates/archive/templates/074.txt new file mode 100644 index 0000000000..28ce103dad --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/074.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := + by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/075.txt b/scripts/zeta5-certificates/archive/templates/075.txt new file mode 100644 index 0000000000..e25a2d996f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/075.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/076.txt b/scripts/zeta5-certificates/archive/templates/076.txt new file mode 100644 index 0000000000..09e32f979e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/076.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/077.txt b/scripts/zeta5-certificates/archive/templates/077.txt new file mode 100644 index 0000000000..7561931dd7 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/077.txt @@ -0,0 +1,81 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/078.txt b/scripts/zeta5-certificates/archive/templates/078.txt new file mode 100644 index 0000000000..b5b81bf61f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/078.txt @@ -0,0 +1,81 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := + by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/079.txt b/scripts/zeta5-certificates/archive/templates/079.txt new file mode 100644 index 0000000000..b854d8aa8f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/079.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/080.txt b/scripts/zeta5-certificates/archive/templates/080.txt new file mode 100644 index 0000000000..21fe3b728f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/080.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/081.txt b/scripts/zeta5-certificates/archive/templates/081.txt new file mode 100644 index 0000000000..5595c4b04a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/081.txt @@ -0,0 +1,81 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/082.txt b/scripts/zeta5-certificates/archive/templates/082.txt new file mode 100644 index 0000000000..bdb8db92fe --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/082.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/083.txt b/scripts/zeta5-certificates/archive/templates/083.txt new file mode 100644 index 0000000000..5d2c2386b8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/083.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/084.txt b/scripts/zeta5-certificates/archive/templates/084.txt new file mode 100644 index 0000000000..5857ce1d18 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/084.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/085.txt b/scripts/zeta5-certificates/archive/templates/085.txt new file mode 100644 index 0000000000..6553f52311 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/085.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/086.txt b/scripts/zeta5-certificates/archive/templates/086.txt new file mode 100644 index 0000000000..2b217e626f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/086.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/087.txt b/scripts/zeta5-certificates/archive/templates/087.txt new file mode 100644 index 0000000000..4ac2a31227 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/087.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/088.txt b/scripts/zeta5-certificates/archive/templates/088.txt new file mode 100644 index 0000000000..35fd333f54 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/088.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/089.txt b/scripts/zeta5-certificates/archive/templates/089.txt new file mode 100644 index 0000000000..b87383a310 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/089.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/090.txt b/scripts/zeta5-certificates/archive/templates/090.txt new file mode 100644 index 0000000000..54bffa8167 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/090.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/091.txt b/scripts/zeta5-certificates/archive/templates/091.txt new file mode 100644 index 0000000000..222c45c265 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/091.txt @@ -0,0 +1,87 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/092.txt b/scripts/zeta5-certificates/archive/templates/092.txt new file mode 100644 index 0000000000..65a8e20b7a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/092.txt @@ -0,0 +1,86 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/093.txt b/scripts/zeta5-certificates/archive/templates/093.txt new file mode 100644 index 0000000000..5b228bff29 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/093.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/094.txt b/scripts/zeta5-certificates/archive/templates/094.txt new file mode 100644 index 0000000000..3d2c5350a3 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/094.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/095.txt b/scripts/zeta5-certificates/archive/templates/095.txt new file mode 100644 index 0000000000..c4a4187301 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/095.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/096.txt b/scripts/zeta5-certificates/archive/templates/096.txt new file mode 100644 index 0000000000..b187480633 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/096.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/097.txt b/scripts/zeta5-certificates/archive/templates/097.txt new file mode 100644 index 0000000000..af81c832c2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/097.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/098.txt b/scripts/zeta5-certificates/archive/templates/098.txt new file mode 100644 index 0000000000..80a4d21f06 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/098.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/099.txt b/scripts/zeta5-certificates/archive/templates/099.txt new file mode 100644 index 0000000000..de3fcf78e7 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/099.txt @@ -0,0 +1,86 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/100.txt b/scripts/zeta5-certificates/archive/templates/100.txt new file mode 100644 index 0000000000..5dbb1c8af8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/100.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/101.txt b/scripts/zeta5-certificates/archive/templates/101.txt new file mode 100644 index 0000000000..6757636b14 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/101.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/102.txt b/scripts/zeta5-certificates/archive/templates/102.txt new file mode 100644 index 0000000000..ea7820fcff --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/102.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/103.txt b/scripts/zeta5-certificates/archive/templates/103.txt new file mode 100644 index 0000000000..d89ab417b5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/103.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/104.txt b/scripts/zeta5-certificates/archive/templates/104.txt new file mode 100644 index 0000000000..f34652388a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/104.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/105.txt b/scripts/zeta5-certificates/archive/templates/105.txt new file mode 100644 index 0000000000..57bdb6a3bb --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/105.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/106.txt b/scripts/zeta5-certificates/archive/templates/106.txt new file mode 100644 index 0000000000..c199a091ef --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/106.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/107.txt b/scripts/zeta5-certificates/archive/templates/107.txt new file mode 100644 index 0000000000..1d372afcdb --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/107.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/108.txt b/scripts/zeta5-certificates/archive/templates/108.txt new file mode 100644 index 0000000000..4752ccc702 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/108.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/109.txt b/scripts/zeta5-certificates/archive/templates/109.txt new file mode 100644 index 0000000000..05c463eed9 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/109.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/110.txt b/scripts/zeta5-certificates/archive/templates/110.txt new file mode 100644 index 0000000000..902e8b00ae --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/110.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/111.txt b/scripts/zeta5-certificates/archive/templates/111.txt new file mode 100644 index 0000000000..c929ff3575 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/111.txt @@ -0,0 +1,81 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/112.txt b/scripts/zeta5-certificates/archive/templates/112.txt new file mode 100644 index 0000000000..1b5d1417b9 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/112.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/113.txt b/scripts/zeta5-certificates/archive/templates/113.txt new file mode 100644 index 0000000000..ac7e4e2242 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/113.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/114.txt b/scripts/zeta5-certificates/archive/templates/114.txt new file mode 100644 index 0000000000..c9c1e0c2e1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/114.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/115.txt b/scripts/zeta5-certificates/archive/templates/115.txt new file mode 100644 index 0000000000..9ad895a48d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/115.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/116.txt b/scripts/zeta5-certificates/archive/templates/116.txt new file mode 100644 index 0000000000..1433f92e84 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/116.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/117.txt b/scripts/zeta5-certificates/archive/templates/117.txt new file mode 100644 index 0000000000..c3468a4719 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/117.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/118.txt b/scripts/zeta5-certificates/archive/templates/118.txt new file mode 100644 index 0000000000..6da2e267ed --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/118.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/119.txt b/scripts/zeta5-certificates/archive/templates/119.txt new file mode 100644 index 0000000000..deacf9de4a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/119.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/120.txt b/scripts/zeta5-certificates/archive/templates/120.txt new file mode 100644 index 0000000000..8c87bcab00 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/120.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/121.txt b/scripts/zeta5-certificates/archive/templates/121.txt new file mode 100644 index 0000000000..de41c583c5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/121.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/122.txt b/scripts/zeta5-certificates/archive/templates/122.txt new file mode 100644 index 0000000000..f85cf0cc54 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/122.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/123.txt b/scripts/zeta5-certificates/archive/templates/123.txt new file mode 100644 index 0000000000..8037c78088 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/123.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/124.txt b/scripts/zeta5-certificates/archive/templates/124.txt new file mode 100644 index 0000000000..f7d03c2e21 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/124.txt @@ -0,0 +1,81 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/125.txt b/scripts/zeta5-certificates/archive/templates/125.txt new file mode 100644 index 0000000000..1b18379602 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/125.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/126.txt b/scripts/zeta5-certificates/archive/templates/126.txt new file mode 100644 index 0000000000..679885d5b7 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/126.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/127.txt b/scripts/zeta5-certificates/archive/templates/127.txt new file mode 100644 index 0000000000..77668366e9 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/127.txt @@ -0,0 +1,84 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/128.txt b/scripts/zeta5-certificates/archive/templates/128.txt new file mode 100644 index 0000000000..466f2f5ede --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/128.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/129.txt b/scripts/zeta5-certificates/archive/templates/129.txt new file mode 100644 index 0000000000..64b2eff00e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/129.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/130.txt b/scripts/zeta5-certificates/archive/templates/130.txt new file mode 100644 index 0000000000..92aade6a8f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/130.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/131.txt b/scripts/zeta5-certificates/archive/templates/131.txt new file mode 100644 index 0000000000..192d07da18 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/131.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/132.txt b/scripts/zeta5-certificates/archive/templates/132.txt new file mode 100644 index 0000000000..5a9eaaead0 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/132.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/133.txt b/scripts/zeta5-certificates/archive/templates/133.txt new file mode 100644 index 0000000000..1c632f6ed2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/133.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/134.txt b/scripts/zeta5-certificates/archive/templates/134.txt new file mode 100644 index 0000000000..f4606b8b1a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/134.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/135.txt b/scripts/zeta5-certificates/archive/templates/135.txt new file mode 100644 index 0000000000..fb90283a6d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/135.txt @@ -0,0 +1,82 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/136.txt b/scripts/zeta5-certificates/archive/templates/136.txt new file mode 100644 index 0000000000..09fbbd7e16 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/136.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/137.txt b/scripts/zeta5-certificates/archive/templates/137.txt new file mode 100644 index 0000000000..8bf22d89ba --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/137.txt @@ -0,0 +1,98 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/138.txt b/scripts/zeta5-certificates/archive/templates/138.txt new file mode 100644 index 0000000000..d1eab5a41f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/138.txt @@ -0,0 +1,92 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/139.txt b/scripts/zeta5-certificates/archive/templates/139.txt new file mode 100644 index 0000000000..dd451b4a22 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/139.txt @@ -0,0 +1,97 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/140.txt b/scripts/zeta5-certificates/archive/templates/140.txt new file mode 100644 index 0000000000..6b1304c69c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/140.txt @@ -0,0 +1,97 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/141.txt b/scripts/zeta5-certificates/archive/templates/141.txt new file mode 100644 index 0000000000..d58efb967b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/141.txt @@ -0,0 +1,93 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/142.txt b/scripts/zeta5-certificates/archive/templates/142.txt new file mode 100644 index 0000000000..db362805a6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/142.txt @@ -0,0 +1,86 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/143.txt b/scripts/zeta5-certificates/archive/templates/143.txt new file mode 100644 index 0000000000..78c79b6cf3 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/143.txt @@ -0,0 +1,92 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/144.txt b/scripts/zeta5-certificates/archive/templates/144.txt new file mode 100644 index 0000000000..e4fef5bdcd --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/144.txt @@ -0,0 +1,91 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/145.txt b/scripts/zeta5-certificates/archive/templates/145.txt new file mode 100644 index 0000000000..6398ed21f1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/145.txt @@ -0,0 +1,90 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/146.txt b/scripts/zeta5-certificates/archive/templates/146.txt new file mode 100644 index 0000000000..6d9c80821c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/146.txt @@ -0,0 +1,86 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/147.txt b/scripts/zeta5-certificates/archive/templates/147.txt new file mode 100644 index 0000000000..c7c9230a32 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/147.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/148.txt b/scripts/zeta5-certificates/archive/templates/148.txt new file mode 100644 index 0000000000..1a2ceee8f2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/148.txt @@ -0,0 +1,92 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/149.txt b/scripts/zeta5-certificates/archive/templates/149.txt new file mode 100644 index 0000000000..aaba999508 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/149.txt @@ -0,0 +1,91 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/150.txt b/scripts/zeta5-certificates/archive/templates/150.txt new file mode 100644 index 0000000000..58b1a0fe92 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/150.txt @@ -0,0 +1,90 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/151.txt b/scripts/zeta5-certificates/archive/templates/151.txt new file mode 100644 index 0000000000..1448d251cc --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/151.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/152.txt b/scripts/zeta5-certificates/archive/templates/152.txt new file mode 100644 index 0000000000..7ad2757299 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/152.txt @@ -0,0 +1,93 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/153.txt b/scripts/zeta5-certificates/archive/templates/153.txt new file mode 100644 index 0000000000..bd6943ae0e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/153.txt @@ -0,0 +1,92 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/154.txt b/scripts/zeta5-certificates/archive/templates/154.txt new file mode 100644 index 0000000000..fa0cf22254 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/154.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/155.txt b/scripts/zeta5-certificates/archive/templates/155.txt new file mode 100644 index 0000000000..89f5e8b02c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/155.txt @@ -0,0 +1,89 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/156.txt b/scripts/zeta5-certificates/archive/templates/156.txt new file mode 100644 index 0000000000..4f421ad259 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/156.txt @@ -0,0 +1,93 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/157.txt b/scripts/zeta5-certificates/archive/templates/157.txt new file mode 100644 index 0000000000..56aa6953d9 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/157.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/158.txt b/scripts/zeta5-certificates/archive/templates/158.txt new file mode 100644 index 0000000000..69483feff9 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/158.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/159.txt b/scripts/zeta5-certificates/archive/templates/159.txt new file mode 100644 index 0000000000..cd868abbe4 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/159.txt @@ -0,0 +1,92 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/160.txt b/scripts/zeta5-certificates/archive/templates/160.txt new file mode 100644 index 0000000000..61cd9cfac0 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/160.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/161.txt b/scripts/zeta5-certificates/archive/templates/161.txt new file mode 100644 index 0000000000..2261fa53e8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/161.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/162.txt b/scripts/zeta5-certificates/archive/templates/162.txt new file mode 100644 index 0000000000..f0dd9fd224 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/162.txt @@ -0,0 +1,85 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/163.txt b/scripts/zeta5-certificates/archive/templates/163.txt new file mode 100644 index 0000000000..bc0bab180e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/163.txt @@ -0,0 +1,83 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/164.txt b/scripts/zeta5-certificates/archive/templates/164.txt new file mode 100644 index 0000000000..2ad59b9dce --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/164.txt @@ -0,0 +1,13 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [bρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/165.txt b/scripts/zeta5-certificates/archive/templates/165.txt new file mode 100644 index 0000000000..6dc3cbb64f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/165.txt @@ -0,0 +1,79 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/166.txt b/scripts/zeta5-certificates/archive/templates/166.txt new file mode 100644 index 0000000000..701f03e49b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/166.txt @@ -0,0 +1,79 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/167.txt b/scripts/zeta5-certificates/archive/templates/167.txt new file mode 100644 index 0000000000..2d9747f289 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/167.txt @@ -0,0 +1,14 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) + (by norm_num [bρ_{{integer}}]) (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) + (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/168.txt b/scripts/zeta5-certificates/archive/templates/168.txt new file mode 100644 index 0000000000..0fb0eb92cc --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/168.txt @@ -0,0 +1,79 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/169.txt b/scripts/zeta5-certificates/archive/templates/169.txt new file mode 100644 index 0000000000..5ebb181c98 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/169.txt @@ -0,0 +1,75 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + (-({{integer}} / {{integer}} : ℝ)) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/170.txt b/scripts/zeta5-certificates/archive/templates/170.txt new file mode 100644 index 0000000000..d6dffcd56b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/170.txt @@ -0,0 +1,74 @@ +lemma U_{{integer}} : + Uρ ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + diff --git a/scripts/zeta5-certificates/archive/templates/171.txt b/scripts/zeta5-certificates/archive/templates/171.txt new file mode 100644 index 0000000000..856bacf777 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/171.txt @@ -0,0 +1,11 @@ +lemma U_{{integer}}_{{integer}} : Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [bρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/172.txt b/scripts/zeta5-certificates/archive/templates/172.txt new file mode 100644 index 0000000000..3b0aee6ec2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/172.txt @@ -0,0 +1,12 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [bρ_{{integer}}]) + (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/173.txt b/scripts/zeta5-certificates/archive/templates/173.txt new file mode 100644 index 0000000000..e1fdfefe3e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/173.txt @@ -0,0 +1,13 @@ +lemma U_{{integer}}_{{integer}} : + Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := by + have hlg : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + exact + Uω_le_right_certificate (s := ({{integer}} / {{integer}} : ℝ)) (q := + ({{integer}} / {{integer}} : ℝ)) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) + (by norm_num [bρ_{{integer}}]) (by norm_num) (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) + (by norm_num [aρ_{{integer}}, bρ_{{integer}}]) hlg + diff --git a/scripts/zeta5-certificates/archive/templates/174.txt b/scripts/zeta5-certificates/archive/templates/174.txt new file mode 100644 index 0000000000..48155309fe --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/174.txt @@ -0,0 +1,72 @@ +lemma U_{{integer}} : Uρ ({{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := by + rw [Uρ_expand, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}, + cρ_{{integer}}, cρ_{{integer}}, cρ_{{integer}}] + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * + ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + have hm{{integer}} : + ({{integer}} / {{integer}} : ℝ) * Uω (aρ {{integer}}) (bρ {{integer}}) ({{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) * ({{integer}} / {{integer}} : ℝ) := + mul_le_mul_of_nonneg_left U_{{integer}}_{{integer}} (by norm_num) + linarith only [hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, hm{{integer}}, + hm{{integer}}] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/175.txt b/scripts/zeta5-certificates/archive/templates/175.txt new file mode 100644 index 0000000000..0cd5b82197 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/175.txt @@ -0,0 +1,26 @@ +/- +Copyright (c) {{integer}} Moritz Firsching. All rights reserved. +Released under Apache {{integer}}.{{integer}} license as described in the file LICENSE. +Authors: Moritz Firsching +-/ + +module + +public import LeanPool.Zeta{{integer}}Irrational.Energy +import LeanPool.Zeta{{integer}}Irrational.Certificates +import LeanPool.Zeta{{integer}}Irrational.PotentialV +import LeanPool.Zeta{{integer}}Irrational.TableCheck +import Mathlib.Analysis.Complex.ExponentialBounds +import Mathlib.Analysis.Real.Pi.Bounds +import Mathlib.NumberTheory.Chebyshev +import Mathlib.Tactic.NormNum.Prime +import Mathlib.Tactic.NormNum.RealSqrt + +/-! # V{{integer}}: certified bounds for the zeta({{integer}}) proof -/ + +@[expose] public section + +open Finset + +namespace Zeta{{integer}}Irrational + diff --git a/scripts/zeta5-certificates/archive/templates/176.txt b/scripts/zeta5-certificates/archive/templates/176.txt new file mode 100644 index 0000000000..36104045c4 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/176.txt @@ -0,0 +1,111 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := by + norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/177.txt b/scripts/zeta5-certificates/archive/templates/177.txt new file mode 100644 index 0000000000..aafa2e9be3 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/177.txt @@ -0,0 +1,112 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/178.txt b/scripts/zeta5-certificates/archive/templates/178.txt new file mode 100644 index 0000000000..f7ce583583 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/178.txt @@ -0,0 +1,111 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := by + norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/179.txt b/scripts/zeta5-certificates/archive/templates/179.txt new file mode 100644 index 0000000000..48956bddec --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/179.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/180.txt b/scripts/zeta5-certificates/archive/templates/180.txt new file mode 100644 index 0000000000..59a3a76c93 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/180.txt @@ -0,0 +1,113 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/181.txt b/scripts/zeta5-certificates/archive/templates/181.txt new file mode 100644 index 0000000000..3ac82f396d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/181.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/182.txt b/scripts/zeta5-certificates/archive/templates/182.txt new file mode 100644 index 0000000000..f3542d9fd6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/182.txt @@ -0,0 +1,111 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/183.txt b/scripts/zeta5-certificates/archive/templates/183.txt new file mode 100644 index 0000000000..feaac22c7c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/183.txt @@ -0,0 +1,109 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/184.txt b/scripts/zeta5-certificates/archive/templates/184.txt new file mode 100644 index 0000000000..2d769702d8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/184.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/185.txt b/scripts/zeta5-certificates/archive/templates/185.txt new file mode 100644 index 0000000000..8155f15470 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/185.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/186.txt b/scripts/zeta5-certificates/archive/templates/186.txt new file mode 100644 index 0000000000..13b9e1da8e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/186.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/187.txt b/scripts/zeta5-certificates/archive/templates/187.txt new file mode 100644 index 0000000000..8c62a28571 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/187.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/188.txt b/scripts/zeta5-certificates/archive/templates/188.txt new file mode 100644 index 0000000000..08e878553e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/188.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/189.txt b/scripts/zeta5-certificates/archive/templates/189.txt new file mode 100644 index 0000000000..7c785f7768 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/189.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/190.txt b/scripts/zeta5-certificates/archive/templates/190.txt new file mode 100644 index 0000000000..534c83d950 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/190.txt @@ -0,0 +1,113 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/191.txt b/scripts/zeta5-certificates/archive/templates/191.txt new file mode 100644 index 0000000000..c168563213 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/191.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/192.txt b/scripts/zeta5-certificates/archive/templates/192.txt new file mode 100644 index 0000000000..cfb78a9788 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/192.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/193.txt b/scripts/zeta5-certificates/archive/templates/193.txt new file mode 100644 index 0000000000..5b7cf209d8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/193.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/194.txt b/scripts/zeta5-certificates/archive/templates/194.txt new file mode 100644 index 0000000000..450b6e1b0a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/194.txt @@ -0,0 +1,26 @@ +/- +Copyright (c) {{integer}} Moritz Firsching. All rights reserved. +Released under Apache {{integer}}.{{integer}} license as described in the file LICENSE. +Authors: Moritz Firsching +-/ + +module + +public import LeanPool.Zeta{{integer}}Irrational.Potential +import LeanPool.Zeta{{integer}}Irrational.Certificates +import LeanPool.Zeta{{integer}}Irrational.PotentialV +import LeanPool.Zeta{{integer}}Irrational.TableCheck +import Mathlib.Analysis.Complex.ExponentialBounds +import Mathlib.Analysis.Real.Pi.Bounds +import Mathlib.NumberTheory.Chebyshev +import Mathlib.Tactic.NormNum.Prime +import Mathlib.Tactic.NormNum.RealSqrt + +/-! # V{{integer}}: certified bounds for the zeta({{integer}}) proof -/ + +@[expose] public section + +open Finset + +namespace Zeta{{integer}}Irrational + diff --git a/scripts/zeta5-certificates/archive/templates/195.txt b/scripts/zeta5-certificates/archive/templates/195.txt new file mode 100644 index 0000000000..a0400f495d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/195.txt @@ -0,0 +1,131 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hal{{integer}} : + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_half (by norm_num) + (sqrt_le_of_sq_le (by norm_num) (by norm_num) : + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}}) ≤ + ({{integer}} / {{integer}} : ℝ)) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/196.txt b/scripts/zeta5-certificates/archive/templates/196.txt new file mode 100644 index 0000000000..93fc7bc65b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/196.txt @@ -0,0 +1,132 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hal{{integer}} : + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_half (by norm_num) + (sqrt_le_of_sq_le (by norm_num) (by norm_num) : + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}}) ≤ + ({{integer}} / {{integer}} : ℝ)) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/197.txt b/scripts/zeta5-certificates/archive/templates/197.txt new file mode 100644 index 0000000000..bbcfd0ab90 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/197.txt @@ -0,0 +1,128 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hal{{integer}} : + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_half (by norm_num) + (sqrt_le_of_sq_le (by norm_num) (by norm_num) : + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}}) ≤ + ({{integer}} / {{integer}} : ℝ)) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/198.txt b/scripts/zeta5-certificates/archive/templates/198.txt new file mode 100644 index 0000000000..71f950d873 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/198.txt @@ -0,0 +1,130 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hal{{integer}} : + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_half (by norm_num) + (sqrt_le_of_sq_le (by norm_num) (by norm_num) : + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}}) ≤ + ({{integer}} / {{integer}} : ℝ)) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/199.txt b/scripts/zeta5-certificates/archive/templates/199.txt new file mode 100644 index 0000000000..365a32e198 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/199.txt @@ -0,0 +1,129 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := by + norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hal{{integer}} : + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_half (by norm_num) + (sqrt_le_of_sq_le (by norm_num) (by norm_num) : + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}}) ≤ + ({{integer}} / {{integer}} : ℝ)) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/200.txt b/scripts/zeta5-certificates/archive/templates/200.txt new file mode 100644 index 0000000000..9d628d3da9 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/200.txt @@ -0,0 +1,131 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hal{{integer}} : + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_half (by norm_num) + (sqrt_le_of_sq_le (by norm_num) (by norm_num) : + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}}) ≤ + ({{integer}} / {{integer}} : ℝ)) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/201.txt b/scripts/zeta5-certificates/archive/templates/201.txt new file mode 100644 index 0000000000..4effeaa6bf --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/201.txt @@ -0,0 +1,130 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hal{{integer}} : + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_half (by norm_num) + (sqrt_le_of_sq_le (by norm_num) (by norm_num) : + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}}) ≤ + ({{integer}} / {{integer}} : ℝ)) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) := + arctan_le_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/202.txt b/scripts/zeta5-certificates/archive/templates/202.txt new file mode 100644 index 0000000000..ea3ff160ac --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/202.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield qm := by + rw [Vfield_eq (by norm_num)] + rw [show qm = ({{integer}} / {{integer}} : ℝ) by norm_num] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := by + norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := by + have h{{integer}} := + mul_le_mul_of_nonpos_right hs_hi + (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}] : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + {{integer}}) + have h{{integer}} := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg ({{integer}} / {{integer}} : ℝ)) + linarith only [h{{integer}}, h{{integer}}] + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/203.txt b/scripts/zeta5-certificates/archive/templates/203.txt new file mode 100644 index 0000000000..b4687d84f2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/203.txt @@ -0,0 +1,113 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield qp := by + rw [Vfield_eq (by norm_num)] + rw [show qp = ({{integer}} / {{integer}} : ℝ) by norm_num] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := by + norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/204.txt b/scripts/zeta5-certificates/archive/templates/204.txt new file mode 100644 index 0000000000..f37968f3c9 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/204.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/205.txt b/scripts/zeta5-certificates/archive/templates/205.txt new file mode 100644 index 0000000000..ad44c614f8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/205.txt @@ -0,0 +1,113 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/206.txt b/scripts/zeta5-certificates/archive/templates/206.txt new file mode 100644 index 0000000000..2a9eb4b07e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/206.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/207.txt b/scripts/zeta5-certificates/archive/templates/207.txt new file mode 100644 index 0000000000..12b5af3ea5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/207.txt @@ -0,0 +1,112 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/208.txt b/scripts/zeta5-certificates/archive/templates/208.txt new file mode 100644 index 0000000000..b5ad157bf2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/208.txt @@ -0,0 +1,111 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/209.txt b/scripts/zeta5-certificates/archive/templates/209.txt new file mode 100644 index 0000000000..4676d7d3ae --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/209.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/210.txt b/scripts/zeta5-certificates/archive/templates/210.txt new file mode 100644 index 0000000000..0270f0c81c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/210.txt @@ -0,0 +1,113 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/211.txt b/scripts/zeta5-certificates/archive/templates/211.txt new file mode 100644 index 0000000000..eb9b4b7337 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/211.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/212.txt b/scripts/zeta5-certificates/archive/templates/212.txt new file mode 100644 index 0000000000..0826e43209 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/212.txt @@ -0,0 +1,113 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/213.txt b/scripts/zeta5-certificates/archive/templates/213.txt new file mode 100644 index 0000000000..fcdaf653e8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/213.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/214.txt b/scripts/zeta5-certificates/archive/templates/214.txt new file mode 100644 index 0000000000..fd2a39d8b5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/214.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/215.txt b/scripts/zeta5-certificates/archive/templates/215.txt new file mode 100644 index 0000000000..153a9affa0 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/215.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/216.txt b/scripts/zeta5-certificates/archive/templates/216.txt new file mode 100644 index 0000000000..e3c045d919 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/216.txt @@ -0,0 +1,98 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/217.txt b/scripts/zeta5-certificates/archive/templates/217.txt new file mode 100644 index 0000000000..a39b4de1b5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/217.txt @@ -0,0 +1,97 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/218.txt b/scripts/zeta5-certificates/archive/templates/218.txt new file mode 100644 index 0000000000..9ff757670e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/218.txt @@ -0,0 +1,100 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/219.txt b/scripts/zeta5-certificates/archive/templates/219.txt new file mode 100644 index 0000000000..af9a310fc4 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/219.txt @@ -0,0 +1,100 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/220.txt b/scripts/zeta5-certificates/archive/templates/220.txt new file mode 100644 index 0000000000..68b2d7a395 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/220.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/221.txt b/scripts/zeta5-certificates/archive/templates/221.txt new file mode 100644 index 0000000000..952d5dbbdb --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/221.txt @@ -0,0 +1,103 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/222.txt b/scripts/zeta5-certificates/archive/templates/222.txt new file mode 100644 index 0000000000..c348c5d653 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/222.txt @@ -0,0 +1,97 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/223.txt b/scripts/zeta5-certificates/archive/templates/223.txt new file mode 100644 index 0000000000..0431382dfc --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/223.txt @@ -0,0 +1,102 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/224.txt b/scripts/zeta5-certificates/archive/templates/224.txt new file mode 100644 index 0000000000..067a494573 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/224.txt @@ -0,0 +1,100 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/225.txt b/scripts/zeta5-certificates/archive/templates/225.txt new file mode 100644 index 0000000000..8a3a783269 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/225.txt @@ -0,0 +1,98 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/226.txt b/scripts/zeta5-certificates/archive/templates/226.txt new file mode 100644 index 0000000000..7118b31380 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/226.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/227.txt b/scripts/zeta5-certificates/archive/templates/227.txt new file mode 100644 index 0000000000..0e10e027d6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/227.txt @@ -0,0 +1,97 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/228.txt b/scripts/zeta5-certificates/archive/templates/228.txt new file mode 100644 index 0000000000..7fca075a04 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/228.txt @@ -0,0 +1,99 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/229.txt b/scripts/zeta5-certificates/archive/templates/229.txt new file mode 100644 index 0000000000..6326bf2d2e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/229.txt @@ -0,0 +1,99 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/230.txt b/scripts/zeta5-certificates/archive/templates/230.txt new file mode 100644 index 0000000000..8962f3369a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/230.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/231.txt b/scripts/zeta5-certificates/archive/templates/231.txt new file mode 100644 index 0000000000..741ea6b2ab --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/231.txt @@ -0,0 +1,96 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/232.txt b/scripts/zeta5-certificates/archive/templates/232.txt new file mode 100644 index 0000000000..ec758e40e7 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/232.txt @@ -0,0 +1,98 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/233.txt b/scripts/zeta5-certificates/archive/templates/233.txt new file mode 100644 index 0000000000..a874c3c6e1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/233.txt @@ -0,0 +1,100 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/234.txt b/scripts/zeta5-certificates/archive/templates/234.txt new file mode 100644 index 0000000000..ca4cf2f7b1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/234.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/235.txt b/scripts/zeta5-certificates/archive/templates/235.txt new file mode 100644 index 0000000000..75f334cd72 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/235.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/236.txt b/scripts/zeta5-certificates/archive/templates/236.txt new file mode 100644 index 0000000000..c0e644c9fc --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/236.txt @@ -0,0 +1,99 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/237.txt b/scripts/zeta5-certificates/archive/templates/237.txt new file mode 100644 index 0000000000..6650c59ff5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/237.txt @@ -0,0 +1,100 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/238.txt b/scripts/zeta5-certificates/archive/templates/238.txt new file mode 100644 index 0000000000..7ba7feef53 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/238.txt @@ -0,0 +1,100 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/239.txt b/scripts/zeta5-certificates/archive/templates/239.txt new file mode 100644 index 0000000000..5874fe555b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/239.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/240.txt b/scripts/zeta5-certificates/archive/templates/240.txt new file mode 100644 index 0000000000..55f32e8f59 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/240.txt @@ -0,0 +1,99 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/241.txt b/scripts/zeta5-certificates/archive/templates/241.txt new file mode 100644 index 0000000000..cedf264757 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/241.txt @@ -0,0 +1,99 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/242.txt b/scripts/zeta5-certificates/archive/templates/242.txt new file mode 100644 index 0000000000..f46fa9b07f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/242.txt @@ -0,0 +1,100 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/243.txt b/scripts/zeta5-certificates/archive/templates/243.txt new file mode 100644 index 0000000000..8549284104 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/243.txt @@ -0,0 +1,100 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/244.txt b/scripts/zeta5-certificates/archive/templates/244.txt new file mode 100644 index 0000000000..ccbe5f25ca --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/244.txt @@ -0,0 +1,98 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/245.txt b/scripts/zeta5-certificates/archive/templates/245.txt new file mode 100644 index 0000000000..85a4977816 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/245.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/246.txt b/scripts/zeta5-certificates/archive/templates/246.txt new file mode 100644 index 0000000000..d1375f8aaf --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/246.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/247.txt b/scripts/zeta5-certificates/archive/templates/247.txt new file mode 100644 index 0000000000..e91fa85f2c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/247.txt @@ -0,0 +1,102 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/248.txt b/scripts/zeta5-certificates/archive/templates/248.txt new file mode 100644 index 0000000000..d611da8684 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/248.txt @@ -0,0 +1,100 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/249.txt b/scripts/zeta5-certificates/archive/templates/249.txt new file mode 100644 index 0000000000..a7d08d47f4 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/249.txt @@ -0,0 +1,102 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/250.txt b/scripts/zeta5-certificates/archive/templates/250.txt new file mode 100644 index 0000000000..7d0e55d4e3 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/250.txt @@ -0,0 +1,102 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/251.txt b/scripts/zeta5-certificates/archive/templates/251.txt new file mode 100644 index 0000000000..57e1960e17 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/251.txt @@ -0,0 +1,97 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/252.txt b/scripts/zeta5-certificates/archive/templates/252.txt new file mode 100644 index 0000000000..3597fe84bd --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/252.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/253.txt b/scripts/zeta5-certificates/archive/templates/253.txt new file mode 100644 index 0000000000..dd85196eb0 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/253.txt @@ -0,0 +1,102 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/254.txt b/scripts/zeta5-certificates/archive/templates/254.txt new file mode 100644 index 0000000000..6ada3e775b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/254.txt @@ -0,0 +1,97 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/255.txt b/scripts/zeta5-certificates/archive/templates/255.txt new file mode 100644 index 0000000000..64ade149e2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/255.txt @@ -0,0 +1,99 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/256.txt b/scripts/zeta5-certificates/archive/templates/256.txt new file mode 100644 index 0000000000..4d43b6edcd --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/256.txt @@ -0,0 +1,104 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/257.txt b/scripts/zeta5-certificates/archive/templates/257.txt new file mode 100644 index 0000000000..3fc86cfa6b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/257.txt @@ -0,0 +1,105 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/258.txt b/scripts/zeta5-certificates/archive/templates/258.txt new file mode 100644 index 0000000000..ebd6a68a99 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/258.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/259.txt b/scripts/zeta5-certificates/archive/templates/259.txt new file mode 100644 index 0000000000..1ff9ce2113 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/259.txt @@ -0,0 +1,104 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/260.txt b/scripts/zeta5-certificates/archive/templates/260.txt new file mode 100644 index 0000000000..bb4d197461 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/260.txt @@ -0,0 +1,103 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/261.txt b/scripts/zeta5-certificates/archive/templates/261.txt new file mode 100644 index 0000000000..a5cfee9943 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/261.txt @@ -0,0 +1,103 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/262.txt b/scripts/zeta5-certificates/archive/templates/262.txt new file mode 100644 index 0000000000..09992bad26 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/262.txt @@ -0,0 +1,102 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/263.txt b/scripts/zeta5-certificates/archive/templates/263.txt new file mode 100644 index 0000000000..53ad82d819 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/263.txt @@ -0,0 +1,102 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/264.txt b/scripts/zeta5-certificates/archive/templates/264.txt new file mode 100644 index 0000000000..c15819d0fe --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/264.txt @@ -0,0 +1,99 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/265.txt b/scripts/zeta5-certificates/archive/templates/265.txt new file mode 100644 index 0000000000..e769113293 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/265.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/266.txt b/scripts/zeta5-certificates/archive/templates/266.txt new file mode 100644 index 0000000000..091c8578e9 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/266.txt @@ -0,0 +1,101 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/267.txt b/scripts/zeta5-certificates/archive/templates/267.txt new file mode 100644 index 0000000000..b46de6fc19 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/267.txt @@ -0,0 +1,98 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := by + norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/268.txt b/scripts/zeta5-certificates/archive/templates/268.txt new file mode 100644 index 0000000000..dbdb3679f8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/268.txt @@ -0,0 +1,102 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/269.txt b/scripts/zeta5-certificates/archive/templates/269.txt new file mode 100644 index 0000000000..04f9893634 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/269.txt @@ -0,0 +1,100 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/270.txt b/scripts/zeta5-certificates/archive/templates/270.txt new file mode 100644 index 0000000000..df5cb8b3ab --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/270.txt @@ -0,0 +1,100 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/271.txt b/scripts/zeta5-certificates/archive/templates/271.txt new file mode 100644 index 0000000000..5f59ba03d0 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/271.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/272.txt b/scripts/zeta5-certificates/archive/templates/272.txt new file mode 100644 index 0000000000..f26084cc1a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/272.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/273.txt b/scripts/zeta5-certificates/archive/templates/273.txt new file mode 100644 index 0000000000..0265102695 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/273.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/274.txt b/scripts/zeta5-certificates/archive/templates/274.txt new file mode 100644 index 0000000000..0a5be2bfd1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/274.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/275.txt b/scripts/zeta5-certificates/archive/templates/275.txt new file mode 100644 index 0000000000..8099fbb7cf --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/275.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/276.txt b/scripts/zeta5-certificates/archive/templates/276.txt new file mode 100644 index 0000000000..58b04d928a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/276.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/277.txt b/scripts/zeta5-certificates/archive/templates/277.txt new file mode 100644 index 0000000000..5eadc68548 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/277.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/278.txt b/scripts/zeta5-certificates/archive/templates/278.txt new file mode 100644 index 0000000000..ffebd8abd2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/278.txt @@ -0,0 +1,120 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/279.txt b/scripts/zeta5-certificates/archive/templates/279.txt new file mode 100644 index 0000000000..1899fd8953 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/279.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/280.txt b/scripts/zeta5-certificates/archive/templates/280.txt new file mode 100644 index 0000000000..57307cbc52 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/280.txt @@ -0,0 +1,120 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/281.txt b/scripts/zeta5-certificates/archive/templates/281.txt new file mode 100644 index 0000000000..bf2b0aae25 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/281.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/282.txt b/scripts/zeta5-certificates/archive/templates/282.txt new file mode 100644 index 0000000000..3c121f740d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/282.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/283.txt b/scripts/zeta5-certificates/archive/templates/283.txt new file mode 100644 index 0000000000..a09bdf4c7b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/283.txt @@ -0,0 +1,120 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/284.txt b/scripts/zeta5-certificates/archive/templates/284.txt new file mode 100644 index 0000000000..bad492107b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/284.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/285.txt b/scripts/zeta5-certificates/archive/templates/285.txt new file mode 100644 index 0000000000..658202aaaa --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/285.txt @@ -0,0 +1,119 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/286.txt b/scripts/zeta5-certificates/archive/templates/286.txt new file mode 100644 index 0000000000..f0687154bf --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/286.txt @@ -0,0 +1,121 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/287.txt b/scripts/zeta5-certificates/archive/templates/287.txt new file mode 100644 index 0000000000..110553f489 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/287.txt @@ -0,0 +1,119 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/288.txt b/scripts/zeta5-certificates/archive/templates/288.txt new file mode 100644 index 0000000000..51734645cd --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/288.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/289.txt b/scripts/zeta5-certificates/archive/templates/289.txt new file mode 100644 index 0000000000..16e73c1938 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/289.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/290.txt b/scripts/zeta5-certificates/archive/templates/290.txt new file mode 100644 index 0000000000..79576aa37d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/290.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/291.txt b/scripts/zeta5-certificates/archive/templates/291.txt new file mode 100644 index 0000000000..9968ceb29b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/291.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/292.txt b/scripts/zeta5-certificates/archive/templates/292.txt new file mode 100644 index 0000000000..7985b95964 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/292.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/293.txt b/scripts/zeta5-certificates/archive/templates/293.txt new file mode 100644 index 0000000000..50e34d1654 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/293.txt @@ -0,0 +1,121 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/294.txt b/scripts/zeta5-certificates/archive/templates/294.txt new file mode 100644 index 0000000000..0e3e6b923a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/294.txt @@ -0,0 +1,122 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/295.txt b/scripts/zeta5-certificates/archive/templates/295.txt new file mode 100644 index 0000000000..c80681c710 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/295.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/296.txt b/scripts/zeta5-certificates/archive/templates/296.txt new file mode 100644 index 0000000000..e26a2a484f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/296.txt @@ -0,0 +1,119 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/297.txt b/scripts/zeta5-certificates/archive/templates/297.txt new file mode 100644 index 0000000000..9e7333d092 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/297.txt @@ -0,0 +1,119 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/298.txt b/scripts/zeta5-certificates/archive/templates/298.txt new file mode 100644 index 0000000000..5feba15fce --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/298.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/299.txt b/scripts/zeta5-certificates/archive/templates/299.txt new file mode 100644 index 0000000000..f25822644a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/299.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/300.txt b/scripts/zeta5-certificates/archive/templates/300.txt new file mode 100644 index 0000000000..f788deab84 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/300.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/301.txt b/scripts/zeta5-certificates/archive/templates/301.txt new file mode 100644 index 0000000000..0a0807b063 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/301.txt @@ -0,0 +1,119 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/302.txt b/scripts/zeta5-certificates/archive/templates/302.txt new file mode 100644 index 0000000000..a4b8cda6a3 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/302.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/303.txt b/scripts/zeta5-certificates/archive/templates/303.txt new file mode 100644 index 0000000000..dc08721c33 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/303.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/304.txt b/scripts/zeta5-certificates/archive/templates/304.txt new file mode 100644 index 0000000000..7ec3e80c64 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/304.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/305.txt b/scripts/zeta5-certificates/archive/templates/305.txt new file mode 100644 index 0000000000..4c6daa24e4 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/305.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/306.txt b/scripts/zeta5-certificates/archive/templates/306.txt new file mode 100644 index 0000000000..0f00fc43c6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/306.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/307.txt b/scripts/zeta5-certificates/archive/templates/307.txt new file mode 100644 index 0000000000..ae0563efa1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/307.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/308.txt b/scripts/zeta5-certificates/archive/templates/308.txt new file mode 100644 index 0000000000..2fbfdbbf96 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/308.txt @@ -0,0 +1,120 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/309.txt b/scripts/zeta5-certificates/archive/templates/309.txt new file mode 100644 index 0000000000..cd76df4688 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/309.txt @@ -0,0 +1,119 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/310.txt b/scripts/zeta5-certificates/archive/templates/310.txt new file mode 100644 index 0000000000..d33d59aa7e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/310.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/311.txt b/scripts/zeta5-certificates/archive/templates/311.txt new file mode 100644 index 0000000000..28e38653bb --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/311.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/312.txt b/scripts/zeta5-certificates/archive/templates/312.txt new file mode 100644 index 0000000000..850d21c39e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/312.txt @@ -0,0 +1,119 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/313.txt b/scripts/zeta5-certificates/archive/templates/313.txt new file mode 100644 index 0000000000..493c2cf800 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/313.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/314.txt b/scripts/zeta5-certificates/archive/templates/314.txt new file mode 100644 index 0000000000..b118ec36a0 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/314.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/315.txt b/scripts/zeta5-certificates/archive/templates/315.txt new file mode 100644 index 0000000000..aa01b96d9f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/315.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/316.txt b/scripts/zeta5-certificates/archive/templates/316.txt new file mode 100644 index 0000000000..6255156aa9 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/316.txt @@ -0,0 +1,120 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/317.txt b/scripts/zeta5-certificates/archive/templates/317.txt new file mode 100644 index 0000000000..e8f7ff7ee5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/317.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/318.txt b/scripts/zeta5-certificates/archive/templates/318.txt new file mode 100644 index 0000000000..a4fae6da0a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/318.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/319.txt b/scripts/zeta5-certificates/archive/templates/319.txt new file mode 100644 index 0000000000..d67ee9d0c3 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/319.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/320.txt b/scripts/zeta5-certificates/archive/templates/320.txt new file mode 100644 index 0000000000..e94ff03aa1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/320.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/321.txt b/scripts/zeta5-certificates/archive/templates/321.txt new file mode 100644 index 0000000000..cb841ff97f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/321.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/322.txt b/scripts/zeta5-certificates/archive/templates/322.txt new file mode 100644 index 0000000000..3a1fe5a8c2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/322.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/323.txt b/scripts/zeta5-certificates/archive/templates/323.txt new file mode 100644 index 0000000000..93556e6a2f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/323.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/324.txt b/scripts/zeta5-certificates/archive/templates/324.txt new file mode 100644 index 0000000000..13d6d5c787 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/324.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/325.txt b/scripts/zeta5-certificates/archive/templates/325.txt new file mode 100644 index 0000000000..5d9730cac2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/325.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/326.txt b/scripts/zeta5-certificates/archive/templates/326.txt new file mode 100644 index 0000000000..53251702a4 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/326.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/327.txt b/scripts/zeta5-certificates/archive/templates/327.txt new file mode 100644 index 0000000000..de2cb0ee73 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/327.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/328.txt b/scripts/zeta5-certificates/archive/templates/328.txt new file mode 100644 index 0000000000..37f445cf31 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/328.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := by + norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/329.txt b/scripts/zeta5-certificates/archive/templates/329.txt new file mode 100644 index 0000000000..55793d4e3d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/329.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/330.txt b/scripts/zeta5-certificates/archive/templates/330.txt new file mode 100644 index 0000000000..75f60d0757 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/330.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/331.txt b/scripts/zeta5-certificates/archive/templates/331.txt new file mode 100644 index 0000000000..3d33af9390 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/331.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/332.txt b/scripts/zeta5-certificates/archive/templates/332.txt new file mode 100644 index 0000000000..28be71ab9f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/332.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/333.txt b/scripts/zeta5-certificates/archive/templates/333.txt new file mode 100644 index 0000000000..f97e3468d6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/333.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/334.txt b/scripts/zeta5-certificates/archive/templates/334.txt new file mode 100644 index 0000000000..32b2c5f72f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/334.txt @@ -0,0 +1,110 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = Real.log ({{integer}} / {{integer}} : ℝ) := by + norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/335.txt b/scripts/zeta5-certificates/archive/templates/335.txt new file mode 100644 index 0000000000..31ba225793 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/335.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/336.txt b/scripts/zeta5-certificates/archive/templates/336.txt new file mode 100644 index 0000000000..6a3fb5062f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/336.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/337.txt b/scripts/zeta5-certificates/archive/templates/337.txt new file mode 100644 index 0000000000..b0f6ed762d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/337.txt @@ -0,0 +1,119 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/338.txt b/scripts/zeta5-certificates/archive/templates/338.txt new file mode 100644 index 0000000000..485a187ab1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/338.txt @@ -0,0 +1,119 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/339.txt b/scripts/zeta5-certificates/archive/templates/339.txt new file mode 100644 index 0000000000..ef8fe98b2d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/339.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/340.txt b/scripts/zeta5-certificates/archive/templates/340.txt new file mode 100644 index 0000000000..e90156b43f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/340.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/341.txt b/scripts/zeta5-certificates/archive/templates/341.txt new file mode 100644 index 0000000000..e3fbdeeb6f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/341.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/342.txt b/scripts/zeta5-certificates/archive/templates/342.txt new file mode 100644 index 0000000000..d788d52b5e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/342.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/343.txt b/scripts/zeta5-certificates/archive/templates/343.txt new file mode 100644 index 0000000000..391b4ec86b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/343.txt @@ -0,0 +1,121 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/344.txt b/scripts/zeta5-certificates/archive/templates/344.txt new file mode 100644 index 0000000000..09c8ff5a2d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/344.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/345.txt b/scripts/zeta5-certificates/archive/templates/345.txt new file mode 100644 index 0000000000..ad1d10f9c1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/345.txt @@ -0,0 +1,113 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/346.txt b/scripts/zeta5-certificates/archive/templates/346.txt new file mode 100644 index 0000000000..42609b1553 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/346.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/347.txt b/scripts/zeta5-certificates/archive/templates/347.txt new file mode 100644 index 0000000000..3925687950 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/347.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/348.txt b/scripts/zeta5-certificates/archive/templates/348.txt new file mode 100644 index 0000000000..f1cf6d747a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/348.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/349.txt b/scripts/zeta5-certificates/archive/templates/349.txt new file mode 100644 index 0000000000..9d0aa6944c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/349.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/350.txt b/scripts/zeta5-certificates/archive/templates/350.txt new file mode 100644 index 0000000000..15fcbe6239 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/350.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/351.txt b/scripts/zeta5-certificates/archive/templates/351.txt new file mode 100644 index 0000000000..18dc17c636 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/351.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/352.txt b/scripts/zeta5-certificates/archive/templates/352.txt new file mode 100644 index 0000000000..3c721d40ac --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/352.txt @@ -0,0 +1,113 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/353.txt b/scripts/zeta5-certificates/archive/templates/353.txt new file mode 100644 index 0000000000..221633e4cb --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/353.txt @@ -0,0 +1,114 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/354.txt b/scripts/zeta5-certificates/archive/templates/354.txt new file mode 100644 index 0000000000..ab519d9bc5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/354.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/355.txt b/scripts/zeta5-certificates/archive/templates/355.txt new file mode 100644 index 0000000000..ac3f8b33b0 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/355.txt @@ -0,0 +1,119 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/356.txt b/scripts/zeta5-certificates/archive/templates/356.txt new file mode 100644 index 0000000000..42326f8b55 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/356.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/357.txt b/scripts/zeta5-certificates/archive/templates/357.txt new file mode 100644 index 0000000000..47f6ee0382 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/357.txt @@ -0,0 +1,115 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := + by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/358.txt b/scripts/zeta5-certificates/archive/templates/358.txt new file mode 100644 index 0000000000..23482a03b3 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/358.txt @@ -0,0 +1,120 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/359.txt b/scripts/zeta5-certificates/archive/templates/359.txt new file mode 100644 index 0000000000..3c706e888e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/359.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/360.txt b/scripts/zeta5-certificates/archive/templates/360.txt new file mode 100644 index 0000000000..6055a7ef29 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/360.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/361.txt b/scripts/zeta5-certificates/archive/templates/361.txt new file mode 100644 index 0000000000..9b43d70a8f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/361.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/362.txt b/scripts/zeta5-certificates/archive/templates/362.txt new file mode 100644 index 0000000000..1bbcbf27ef --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/362.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/363.txt b/scripts/zeta5-certificates/archive/templates/363.txt new file mode 100644 index 0000000000..699ee21164 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/363.txt @@ -0,0 +1,117 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/364.txt b/scripts/zeta5-certificates/archive/templates/364.txt new file mode 100644 index 0000000000..51253d8889 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/364.txt @@ -0,0 +1,118 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/365.txt b/scripts/zeta5-certificates/archive/templates/365.txt new file mode 100644 index 0000000000..bfacd68690 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/365.txt @@ -0,0 +1,116 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} by + norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/366.txt b/scripts/zeta5-certificates/archive/templates/366.txt new file mode 100644 index 0000000000..887e0e097f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/366.txt @@ -0,0 +1,120 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/367.txt b/scripts/zeta5-certificates/archive/templates/367.txt new file mode 100644 index 0000000000..7529a8ff73 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/367.txt @@ -0,0 +1,121 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/368.txt b/scripts/zeta5-certificates/archive/templates/368.txt new file mode 100644 index 0000000000..d8c963effa --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/368.txt @@ -0,0 +1,122 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/369.txt b/scripts/zeta5-certificates/archive/templates/369.txt new file mode 100644 index 0000000000..7ff73eb6ab --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/369.txt @@ -0,0 +1,122 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/370.txt b/scripts/zeta5-certificates/archive/templates/370.txt new file mode 100644 index 0000000000..d091bc774b --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/370.txt @@ -0,0 +1,121 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/371.txt b/scripts/zeta5-certificates/archive/templates/371.txt new file mode 100644 index 0000000000..32269d82e5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/371.txt @@ -0,0 +1,123 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/372.txt b/scripts/zeta5-certificates/archive/templates/372.txt new file mode 100644 index 0000000000..fbb0cc1205 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/372.txt @@ -0,0 +1,119 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * (-({{integer}} / {{integer}} : ℝ)) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := + log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + (-({{integer}} / {{integer}} : ℝ)) := by + have h := + log_le_of_split_neg {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = ({{integer}} / {{integer}} : ℝ) / {{integer}} ^ {{integer}} + by norm_num) + (by norm_num) + (log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hau{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) := + arctan_le_half (by norm_num) (by norm_num) + (le_sqrt_of_sq_le (by norm_num) (by norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}})) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hat{{integer}} : + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_inv (by norm_num) + (by + rw [show + ({{integer}} : ℝ) / ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hau{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + (Real.pi / {{integer}} - ({{integer}} * ({{integer}} / {{integer}} : ℝ))) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/373.txt b/scripts/zeta5-certificates/archive/templates/373.txt new file mode 100644 index 0000000000..235558bb1f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/373.txt @@ -0,0 +1,106 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * ({{integer}} / {{integer}} : ℝ) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := by + have h := + log_ge_of_split_pos {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = {{integer}} ^ {{integer}} * ({{integer}} / {{integer}} : ℝ) by + norm_num) + (by norm_num) + (log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_half (by norm_num) + (sqrt_le_of_sq_le (by norm_num) (by norm_num) : + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}}) ≤ + ({{integer}} / {{integer}} : ℝ)) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/374.txt b/scripts/zeta5-certificates/archive/templates/374.txt new file mode 100644 index 0000000000..72f21fd925 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/374.txt @@ -0,0 +1,108 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * ({{integer}} / {{integer}} : ℝ) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := by + have h := + log_ge_of_split_pos {{integer}} + (show + ({{integer}} / {{integer}} : ℝ) = {{integer}} ^ {{integer}} * ({{integer}} / {{integer}} : ℝ) by + norm_num) + (by norm_num) + (log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_half (by norm_num) + (sqrt_le_of_sq_le (by norm_num) (by norm_num) : + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}}) ≤ + ({{integer}} / {{integer}} : ℝ)) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/375.txt b/scripts/zeta5-certificates/archive/templates/375.txt new file mode 100644 index 0000000000..59f020810d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/375.txt @@ -0,0 +1,105 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * ({{integer}} / {{integer}} : ℝ) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := by + have h := + log_ge_of_split_pos {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = {{integer}} ^ {{integer}} * ({{integer}} / {{integer}} : ℝ) by + norm_num) + (by norm_num) + (log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_half (by norm_num) + (sqrt_le_of_sq_le (by norm_num) (by norm_num) : + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}}) ≤ + ({{integer}} / {{integer}} : ℝ)) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + diff --git a/scripts/zeta5-certificates/archive/templates/376.txt b/scripts/zeta5-certificates/archive/templates/376.txt new file mode 100644 index 0000000000..d77d354ddf --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/376.txt @@ -0,0 +1,106 @@ +lemma V_{{integer}} : + (({{integer}} / {{integer}} : ℝ) - + {{integer}} * ({{integer}} / {{integer}}) * ({{integer}} / {{integer}} : ℝ) - + {{integer}} + + {{integer}} * ({{integer}} / {{integer}}) + + {{integer}} * + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)))) ≤ + Vfield ({{integer}} / {{integer}} : ℝ) := by + rw [Vfield_eq (by norm_num)] + have e{{integer}} : + Real.log ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have e{{integer}} : + Real.log (({{integer}} / {{integer}} : ℝ) + ({{integer}} / {{integer}}) ^ {{integer}}) = + Real.log ({{integer}} / {{integer}} : ℝ) := + by norm_num + have hL{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ) := by + have h := + log_ge_of_split_pos {{integer}} + (show ({{integer}} / {{integer}} : ℝ) = {{integer}} ^ {{integer}} * ({{integer}} / {{integer}} : ℝ) + by norm_num) + (by norm_num) + (log_ge_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.log ({{integer}} / {{integer}} : ℝ)) + linarith only [h, Real.log_two_gt_d{{integer}}] + have hU{{integer}} : + Real.log ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + log_le_of_ge_one (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hs_lo : + ({{integer}} / {{integer}} : ℝ) ≤ Real.sqrt ({{integer}} / {{integer}} : ℝ) := + le_sqrt_of_sq_le (by norm_num) (by norm_num) + have hs_hi : + Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ ({{integer}} / {{integer}} : ℝ) := + sqrt_le_of_sq_le (by norm_num) (by norm_num) + have hspos : {{integer}} < Real.sqrt ({{integer}} / {{integer}} : ℝ) := + Real.sqrt_pos.mpr (by norm_num) + have hal{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hat{{integer}} : + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + Real.arctan ({{integer}} / {{integer}} : ℝ) := + arctan_ge_half (by norm_num) + (sqrt_le_of_sq_le (by norm_num) (by norm_num) : + Real.sqrt ({{integer}} + ({{integer}} / {{integer}} : ℝ) ^ {{integer}}) ≤ + ({{integer}} / {{integer}} : ℝ)) + (by + rw [show + ({{integer}} / {{integer}} : ℝ) / + ({{integer}} + ({{integer}} / {{integer}} : ℝ)) = + ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact hal{{integer}}) + have hz{{integer}} : + ({{integer}} / {{integer}} : ℝ) ≤ + {{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = {{integer}} / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact one_div_le_one_div_of_le hspos hs_hi + have hA{{integer}} := arctan_ge_of_le hz{{integer}} hat{{integer}} + have hat{{integer}} : + Real.arctan ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := + arctan_le_of_series (by norm_num) (by norm_num) {{integer}} + (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) + have hz{{integer}} : + ({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ) ≤ + ({{integer}} / {{integer}} : ℝ) := by + rw [show + ({{integer}} / {{integer}} : ℝ) = + ({{integer}} / {{integer}}) / ({{integer}} / {{integer}} : ℝ) + by + norm_num]; + exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo + have hA{{integer}} := arctan_le_of_ge hz{{integer}} hat{{integer}} + have hP : + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ)) ≤ + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + by linarith only [hA{{integer}}, hA{{integer}}] + have hprod : + (({{integer}} / {{integer}} : ℝ) * + (Real.pi + ({{integer}} * ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * ({{integer}} / {{integer}} : ℝ))) ≤ + Real.sqrt ({{integer}} / {{integer}} : ℝ) * + (Real.pi + Real.arctan ({{integer}} / Real.sqrt ({{integer}} / {{integer}} : ℝ)) - + {{integer}} * Real.arctan (({{integer}} / {{integer}}) / Real.sqrt ({{integer}} / {{integer}} : ℝ))) := + mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) (Real.sqrt_nonneg _) + linarith only [e{{integer}}, e{{integer}}, hL{{integer}}, hU{{integer}}, hprod] + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/377.txt b/scripts/zeta5-certificates/archive/templates/377.txt new file mode 100644 index 0000000000..5a9d58c790 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/377.txt @@ -0,0 +1,135 @@ +/- +Copyright (c) {{integer}} Moritz Firsching. All rights reserved. +Released under Apache {{integer}}.{{integer}} license as described in the file LICENSE. +Authors: Moritz Firsching +-/ + +module + +import Mathlib.Analysis.Real.Pi.Bounds + +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.U{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.V{{integer}} +public import LeanPool.Zeta{{integer}}Irrational.Table.Bracket +public import LeanPool.Zeta{{integer}}Irrational.Table.Tail + +/-! # Intervals: certified bounds for the zeta({{integer}}) proof -/ + +@[expose] public section + +open Finset + +namespace Zeta{{integer}}Irrational + diff --git a/scripts/zeta5-certificates/archive/templates/378.txt b/scripts/zeta5-certificates/archive/templates/378.txt new file mode 100644 index 0000000000..b3b7eac174 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/378.txt @@ -0,0 +1,5 @@ +lemma tc_{{integer}} : + ∀ t, ({{integer}} : ℝ) ≤ t → t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + check_left (by norm_num) (by norm_num [aρ_{{integer}}]) U_{{integer}} V_{{integer}} + (by unfold M{{integer}}; linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) + diff --git a/scripts/zeta5-certificates/archive/templates/379.txt b/scripts/zeta5-certificates/archive/templates/379.txt new file mode 100644 index 0000000000..1e61b9369c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/379.txt @@ -0,0 +1,7 @@ +lemma tc_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → + t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + check_left (by norm_num) (by norm_num [aρ_{{integer}}]) U_{{integer}} V_{{integer}} + (by unfold M{{integer}}; linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) + diff --git a/scripts/zeta5-certificates/archive/templates/380.txt b/scripts/zeta5-certificates/archive/templates/380.txt new file mode 100644 index 0000000000..717a40b7e8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/380.txt @@ -0,0 +1,10 @@ +lemma tc_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → + t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := by + intro t h{{integer}} h{{integer}} h{{integer}} + have eq{{integer}} : qm = ({{integer}} / {{integer}} : ℝ) := by norm_num [qm] + exact + check_mid_left Umid_le V_{{integer}} (by unfold M{{integer}}; linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) t + (by rw [aρ_{{integer}}]; exact h{{integer}}) (by rw [eq{{integer}}]; exact h{{integer}}) h{{integer}} + diff --git a/scripts/zeta5-certificates/archive/templates/381.txt b/scripts/zeta5-certificates/archive/templates/381.txt new file mode 100644 index 0000000000..164c6a1794 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/381.txt @@ -0,0 +1,11 @@ +lemma tc_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → + t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := by + intro t h{{integer}} h{{integer}} h{{integer}} + have eq{{integer}} : qm = ({{integer}} / {{integer}} : ℝ) := by norm_num [qm] + have eq{{integer}} : qp = ({{integer}} / {{integer}} : ℝ) := by norm_num [qp] + exact + check_mid Umid_le V_br (by unfold M{{integer}}; linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) t + (by rw [eq{{integer}}]; exact h{{integer}}) (by rw [eq{{integer}}]; exact h{{integer}}) h{{integer}} + diff --git a/scripts/zeta5-certificates/archive/templates/382.txt b/scripts/zeta5-certificates/archive/templates/382.txt new file mode 100644 index 0000000000..216b1ce05a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/382.txt @@ -0,0 +1,10 @@ +lemma tc_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → + t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := by + intro t h{{integer}} h{{integer}} h{{integer}} + have eq{{integer}} : qp = ({{integer}} / {{integer}} : ℝ) := by norm_num [qp] + exact + check_mid_right Umid_le V_{{integer}} (by unfold M{{integer}}; linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) t + (by rw [eq{{integer}}]; exact h{{integer}}) (by rw [bρ_{{integer}}]; exact h{{integer}}) h{{integer}} + diff --git a/scripts/zeta5-certificates/archive/templates/383.txt b/scripts/zeta5-certificates/archive/templates/383.txt new file mode 100644 index 0000000000..cfafca74a2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/383.txt @@ -0,0 +1,7 @@ +lemma tc_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → + t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + check_right (by norm_num [bρ_{{integer}}]) (by norm_num) U_{{integer}} V_{{integer}} + (by unfold M{{integer}}; linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) + diff --git a/scripts/zeta5-certificates/archive/templates/384.txt b/scripts/zeta5-certificates/archive/templates/384.txt new file mode 100644 index 0000000000..cb14823bf4 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/384.txt @@ -0,0 +1,6 @@ +lemma tc_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → t ≤ ({{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + check_right (by norm_num [bρ_{{integer}}]) (by norm_num) U_{{integer}} V_{{integer}} + (by unfold M{{integer}}; linarith only [Real.pi_gt_d{{integer}}, Real.pi_lt_d{{integer}}]) + diff --git a/scripts/zeta5-certificates/archive/templates/385.txt b/scripts/zeta5-certificates/archive/templates/385.txt new file mode 100644 index 0000000000..175eb44777 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/385.txt @@ -0,0 +1,6 @@ +lemma cov_{{integer}}_{{integer}} : + ∀ t, ({{integer}} : ℝ) ≤ t → t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + fun t h{{integer}} h{{integer}} h{{integer}} => + (le_or_gt t ({{integer}} / {{integer}} : ℝ)).elim (fun h => tc_{{integer}} t h{{integer}} h h{{integer}}) + (fun h => tc_{{integer}} t h.le h{{integer}} h{{integer}}) + diff --git a/scripts/zeta5-certificates/archive/templates/386.txt b/scripts/zeta5-certificates/archive/templates/386.txt new file mode 100644 index 0000000000..a7b7717e16 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/386.txt @@ -0,0 +1,8 @@ +lemma cov_{{integer}}_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → + t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + fun t h{{integer}} h{{integer}} h{{integer}} => + (le_or_gt t ({{integer}} / {{integer}} : ℝ)).elim (fun h => tc_{{integer}} t h{{integer}} h h{{integer}}) + (fun h => tc_{{integer}} t h.le h{{integer}} h{{integer}}) + diff --git a/scripts/zeta5-certificates/archive/templates/387.txt b/scripts/zeta5-certificates/archive/templates/387.txt new file mode 100644 index 0000000000..474f57d257 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/387.txt @@ -0,0 +1,8 @@ +lemma cov_{{integer}}_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → + t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + fun t h{{integer}} h{{integer}} h{{integer}} => + (le_or_gt t ({{integer}} / {{integer}} : ℝ)).elim (fun h => tc_{{integer}} t h{{integer}} h h{{integer}}) + (fun h => cov_{{integer}}_{{integer}} t h.le h{{integer}} h{{integer}}) + diff --git a/scripts/zeta5-certificates/archive/templates/388.txt b/scripts/zeta5-certificates/archive/templates/388.txt new file mode 100644 index 0000000000..a71458c636 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/388.txt @@ -0,0 +1,6 @@ +lemma cov_{{integer}}_{{integer}} : + ∀ t, ({{integer}} : ℝ) ≤ t → t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + fun t h{{integer}} h{{integer}} h{{integer}} => + (le_or_gt t ({{integer}} / {{integer}} : ℝ)).elim (fun h => cov_{{integer}}_{{integer}} t h{{integer}} h h{{integer}}) + (fun h => cov_{{integer}}_{{integer}} t h.le h{{integer}} h{{integer}}) + diff --git a/scripts/zeta5-certificates/archive/templates/389.txt b/scripts/zeta5-certificates/archive/templates/389.txt new file mode 100644 index 0000000000..8393342b95 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/389.txt @@ -0,0 +1,8 @@ +lemma cov_{{integer}}_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → + t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + fun t h{{integer}} h{{integer}} h{{integer}} => + (le_or_gt t ({{integer}} / {{integer}} : ℝ)).elim (fun h => cov_{{integer}}_{{integer}} t h{{integer}} h h{{integer}}) + (fun h => cov_{{integer}}_{{integer}} t h.le h{{integer}} h{{integer}}) + diff --git a/scripts/zeta5-certificates/archive/templates/390.txt b/scripts/zeta5-certificates/archive/templates/390.txt new file mode 100644 index 0000000000..66ace8c527 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/390.txt @@ -0,0 +1,7 @@ +lemma cov_{{integer}}_{{integer}} : + ∀ t, + ({{integer}} : ℝ) ≤ t → t ≤ ({{integer}} / {{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + fun t h{{integer}} h{{integer}} h{{integer}} => + (le_or_gt t ({{integer}} / {{integer}} : ℝ)).elim (fun h => cov_{{integer}}_{{integer}} t h{{integer}} h h{{integer}}) + (fun h => cov_{{integer}}_{{integer}} t h.le h{{integer}} h{{integer}}) + diff --git a/scripts/zeta5-certificates/archive/templates/391.txt b/scripts/zeta5-certificates/archive/templates/391.txt new file mode 100644 index 0000000000..04656531ac --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/391.txt @@ -0,0 +1,7 @@ +lemma cov_{{integer}}_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → t ≤ ({{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + fun t h{{integer}} h{{integer}} h{{integer}} => + (le_or_gt t ({{integer}} / {{integer}} : ℝ)).elim (fun h => tc_{{integer}} t h{{integer}} h h{{integer}}) + (fun h => tc_{{integer}} t h.le h{{integer}} h{{integer}}) + diff --git a/scripts/zeta5-certificates/archive/templates/392.txt b/scripts/zeta5-certificates/archive/templates/392.txt new file mode 100644 index 0000000000..ac3518f203 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/392.txt @@ -0,0 +1,7 @@ +lemma cov_{{integer}}_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → t ≤ ({{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + fun t h{{integer}} h{{integer}} h{{integer}} => + (le_or_gt t ({{integer}} / {{integer}} : ℝ)).elim (fun h => tc_{{integer}} t h{{integer}} h h{{integer}}) + (fun h => cov_{{integer}}_{{integer}} t h.le h{{integer}} h{{integer}}) + diff --git a/scripts/zeta5-certificates/archive/templates/393.txt b/scripts/zeta5-certificates/archive/templates/393.txt new file mode 100644 index 0000000000..511564fb0f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/393.txt @@ -0,0 +1,8 @@ +lemma cov_{{integer}}_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → + t ≤ ({{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + fun t h{{integer}} h{{integer}} h{{integer}} => + (le_or_gt t ({{integer}} / {{integer}} : ℝ)).elim (fun h => cov_{{integer}}_{{integer}} t h{{integer}} h h{{integer}}) + (fun h => cov_{{integer}}_{{integer}} t h.le h{{integer}} h{{integer}}) + diff --git a/scripts/zeta5-certificates/archive/templates/394.txt b/scripts/zeta5-certificates/archive/templates/394.txt new file mode 100644 index 0000000000..8a258d64f1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/394.txt @@ -0,0 +1,7 @@ +lemma cov_{{integer}}_{{integer}} : + ∀ t, + ({{integer}} / {{integer}} : ℝ) ≤ t → t ≤ ({{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + fun t h{{integer}} h{{integer}} h{{integer}} => + (le_or_gt t ({{integer}} / {{integer}} : ℝ)).elim (fun h => cov_{{integer}}_{{integer}} t h{{integer}} h h{{integer}}) + (fun h => cov_{{integer}}_{{integer}} t h.le h{{integer}} h{{integer}}) + diff --git a/scripts/zeta5-certificates/archive/templates/395.txt b/scripts/zeta5-certificates/archive/templates/395.txt new file mode 100644 index 0000000000..041c5d6698 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/395.txt @@ -0,0 +1,6 @@ +lemma cov_{{integer}}_{{integer}} : ∀ t, ({{integer}} : ℝ) ≤ t → t ≤ ({{integer}} : ℝ) → {{integer}} < t → {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := + fun t h{{integer}} h{{integer}} h{{integer}} => + (le_or_gt t ({{integer}} / {{integer}} : ℝ)).elim (fun h => cov_{{integer}}_{{integer}} t h{{integer}} h h{{integer}}) + (fun h => cov_{{integer}}_{{integer}} t h.le h{{integer}} h{{integer}}) + +/-- Lemma {{integer}}.{{integer}} of the paper: the potential inequality ({{integer}}.{{integer}}) on the whole half-line. -/ diff --git a/scripts/zeta5-certificates/archive/templates/396.txt b/scripts/zeta5-certificates/archive/templates/396.txt new file mode 100644 index 0000000000..d49f0646a6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/396.txt @@ -0,0 +1,6 @@ +theorem potential_ineq (t : ℝ) (ht : {{integer}} < t) : {{integer}} * Uρ t - Vfield t ≤ M{{integer}} := by + rcases le_or_gt t {{integer}} with h | h + · exact cov_{{integer}}_{{integer}} t ht.le h ht + · exact potential_ineq_tail h.le + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/397.txt b/scripts/zeta5-certificates/archive/templates/397.txt new file mode 100644 index 0000000000..56de61fea6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/397.txt @@ -0,0 +1,23 @@ +/- +Copyright (c) {{integer}} Moritz Firsching. All rights reserved. +Released under Apache {{integer}}.{{integer}} license as described in the file LICENSE. +Authors: Moritz Firsching +-/ + +module + +import LeanPool.Zeta{{integer}}Irrational.Growth.TableGen +public import LeanPool.Zeta{{integer}}Irrational.Growth.TablePrime +import Mathlib.Tactic.ReduceModChar +import Mathlib.Tactic.Polynomial.Basic +import Mathlib.Tactic.NormNum.RealSqrt +import Mathlib.NumberTheory.Chebyshev +import Mathlib.Tactic.NormNum.Prime +import Mathlib.Analysis.SpecialFunctions.Gamma.Basic + +/-!Generated by `gen_lean_inner.py`: the pieces of `Ê_in` on `[{{integer}}, {{integer}}]`. -/ + +@[expose] public section + +namespace Zeta{{integer}}Irrational + diff --git a/scripts/zeta5-certificates/archive/templates/398.txt b/scripts/zeta5-certificates/archive/templates/398.txt new file mode 100644 index 0000000000..b9b64df28f --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/398.txt @@ -0,0 +1,63 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : ({{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/399.txt b/scripts/zeta5-certificates/archive/templates/399.txt new file mode 100644 index 0000000000..b5012cc129 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/399.txt @@ -0,0 +1,63 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/400.txt b/scripts/zeta5-certificates/archive/templates/400.txt new file mode 100644 index 0000000000..94b45e6906 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/400.txt @@ -0,0 +1,64 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/401.txt b/scripts/zeta5-certificates/archive/templates/401.txt new file mode 100644 index 0000000000..b18230a8b1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/401.txt @@ -0,0 +1,64 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/402.txt b/scripts/zeta5-certificates/archive/templates/402.txt new file mode 100644 index 0000000000..6ee6de57ea --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/402.txt @@ -0,0 +1,65 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/403.txt b/scripts/zeta5-certificates/archive/templates/403.txt new file mode 100644 index 0000000000..dee510a8be --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/403.txt @@ -0,0 +1,66 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/404.txt b/scripts/zeta5-certificates/archive/templates/404.txt new file mode 100644 index 0000000000..c2fbe71008 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/404.txt @@ -0,0 +1,66 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < ({{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/405.txt b/scripts/zeta5-certificates/archive/templates/405.txt new file mode 100644 index 0000000000..05881df911 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/405.txt @@ -0,0 +1,66 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/406.txt b/scripts/zeta5-certificates/archive/templates/406.txt new file mode 100644 index 0000000000..07ad62dfc5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/406.txt @@ -0,0 +1,63 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/407.txt b/scripts/zeta5-certificates/archive/templates/407.txt new file mode 100644 index 0000000000..d0acd334be --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/407.txt @@ -0,0 +1,67 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/408.txt b/scripts/zeta5-certificates/archive/templates/408.txt new file mode 100644 index 0000000000..0f1b01dde8 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/408.txt @@ -0,0 +1,67 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < ({{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/409.txt b/scripts/zeta5-certificates/archive/templates/409.txt new file mode 100644 index 0000000000..4d72ca4bbb --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/409.txt @@ -0,0 +1,63 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : ({{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/410.txt b/scripts/zeta5-certificates/archive/templates/410.txt new file mode 100644 index 0000000000..4495b7b3eb --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/410.txt @@ -0,0 +1,63 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/411.txt b/scripts/zeta5-certificates/archive/templates/411.txt new file mode 100644 index 0000000000..e393d33da2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/411.txt @@ -0,0 +1,63 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/412.txt b/scripts/zeta5-certificates/archive/templates/412.txt new file mode 100644 index 0000000000..8378518784 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/412.txt @@ -0,0 +1,64 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/413.txt b/scripts/zeta5-certificates/archive/templates/413.txt new file mode 100644 index 0000000000..6d5be2c6e7 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/413.txt @@ -0,0 +1,67 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/414.txt b/scripts/zeta5-certificates/archive/templates/414.txt new file mode 100644 index 0000000000..227a6d1add --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/414.txt @@ -0,0 +1,67 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = ({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/415.txt b/scripts/zeta5-certificates/archive/templates/415.txt new file mode 100644 index 0000000000..e39de18a1a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/415.txt @@ -0,0 +1,63 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : ({{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = ({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/416.txt b/scripts/zeta5-certificates/archive/templates/416.txt new file mode 100644 index 0000000000..9b5dd88348 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/416.txt @@ -0,0 +1,65 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : ({{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/417.txt b/scripts/zeta5-certificates/archive/templates/417.txt new file mode 100644 index 0000000000..596819d102 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/417.txt @@ -0,0 +1,66 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/418.txt b/scripts/zeta5-certificates/archive/templates/418.txt new file mode 100644 index 0000000000..5c5f9b379a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/418.txt @@ -0,0 +1,71 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/419.txt b/scripts/zeta5-certificates/archive/templates/419.txt new file mode 100644 index 0000000000..34e401a984 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/419.txt @@ -0,0 +1,71 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = ({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/420.txt b/scripts/zeta5-certificates/archive/templates/420.txt new file mode 100644 index 0000000000..88c803b68c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/420.txt @@ -0,0 +1,71 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < ({{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/421.txt b/scripts/zeta5-certificates/archive/templates/421.txt new file mode 100644 index 0000000000..6059679baa --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/421.txt @@ -0,0 +1,65 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : ({{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = ({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/422.txt b/scripts/zeta5-certificates/archive/templates/422.txt new file mode 100644 index 0000000000..f13e636660 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/422.txt @@ -0,0 +1,71 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/423.txt b/scripts/zeta5-certificates/archive/templates/423.txt new file mode 100644 index 0000000000..fc8ea08e5d --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/423.txt @@ -0,0 +1,65 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/424.txt b/scripts/zeta5-certificates/archive/templates/424.txt new file mode 100644 index 0000000000..bb94e7f967 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/424.txt @@ -0,0 +1,65 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/425.txt b/scripts/zeta5-certificates/archive/templates/425.txt new file mode 100644 index 0000000000..486a975f84 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/425.txt @@ -0,0 +1,67 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/426.txt b/scripts/zeta5-certificates/archive/templates/426.txt new file mode 100644 index 0000000000..088508c85c --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/426.txt @@ -0,0 +1,68 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < ({{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/427.txt b/scripts/zeta5-certificates/archive/templates/427.txt new file mode 100644 index 0000000000..dc0fa6d717 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/427.txt @@ -0,0 +1,64 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : ({{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/428.txt b/scripts/zeta5-certificates/archive/templates/428.txt new file mode 100644 index 0000000000..bebce49c73 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/428.txt @@ -0,0 +1,64 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/429.txt b/scripts/zeta5-certificates/archive/templates/429.txt new file mode 100644 index 0000000000..223979f7be --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/429.txt @@ -0,0 +1,67 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/430.txt b/scripts/zeta5-certificates/archive/templates/430.txt new file mode 100644 index 0000000000..a208a76693 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/430.txt @@ -0,0 +1,68 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/431.txt b/scripts/zeta5-certificates/archive/templates/431.txt new file mode 100644 index 0000000000..2eb7dbb58e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/431.txt @@ -0,0 +1,68 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/432.txt b/scripts/zeta5-certificates/archive/templates/432.txt new file mode 100644 index 0000000000..7d108029f0 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/432.txt @@ -0,0 +1,64 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + diff --git a/scripts/zeta5-certificates/archive/templates/433.txt b/scripts/zeta5-certificates/archive/templates/433.txt new file mode 100644 index 0000000000..b7c3b74a82 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/433.txt @@ -0,0 +1,69 @@ +theorem inPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < ({{integer}} : ℝ)) : + Ein x ({{integer}}) = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Ein_expand ({{integer}}) (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}] + simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q] + push_cast + norm_num + ring + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/434.txt b/scripts/zeta5-certificates/archive/templates/434.txt new file mode 100644 index 0000000000..6a72424835 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/434.txt @@ -0,0 +1,24 @@ +/- +Copyright (c) {{integer}} Moritz Firsching. All rights reserved. +Released under Apache {{integer}}.{{integer}} license as described in the file LICENSE. +Authors: Moritz Firsching +-/ + +module + +import LeanPool.Zeta{{integer}}Irrational.Growth.TableGen +public import LeanPool.Zeta{{integer}}Irrational.Growth.OuterPrime +import Mathlib.Tactic.ReduceModChar +import Mathlib.Tactic.Polynomial.Basic +import Mathlib.Tactic.NormNum.RealSqrt +import Mathlib.NumberTheory.Chebyshev +import Mathlib.Data.Rat.Star +import Mathlib.Tactic.NormNum.Prime +import Mathlib.Analysis.SpecialFunctions.Gamma.Basic + +/-!Generated by `gen_lean_outer.py`: the table of `Ê_out` on `[{{integer}}/{{integer}}, {{integer}}]`. -/ + +@[expose] public section + +namespace Zeta{{integer}}Irrational + diff --git a/scripts/zeta5-certificates/archive/templates/435.txt b/scripts/zeta5-certificates/archive/templates/435.txt new file mode 100644 index 0000000000..48097115f5 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/435.txt @@ -0,0 +1,67 @@ +theorem outPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Eout x = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Eout_expand (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have em : max {{integer}} ({{integer}} / {{integer}} * x - {{integer}}) = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + simp only [max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, em] + simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, GO] + norm_num + ring_nf + diff --git a/scripts/zeta5-certificates/archive/templates/436.txt b/scripts/zeta5-certificates/archive/templates/436.txt new file mode 100644 index 0000000000..056c11e7f6 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/436.txt @@ -0,0 +1,68 @@ +theorem outPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < ({{integer}} : ℝ)) : + Eout x = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Eout_expand (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have em : max {{integer}} ({{integer}} / {{integer}} * x - {{integer}}) = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + simp only [max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, em] + simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, GO] + norm_num + ring_nf + diff --git a/scripts/zeta5-certificates/archive/templates/437.txt b/scripts/zeta5-certificates/archive/templates/437.txt new file mode 100644 index 0000000000..c526179b14 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/437.txt @@ -0,0 +1,65 @@ +theorem outPiece_{{integer}} (x : ℝ) (h{{integer}} : ({{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Eout x = ({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Eout_expand (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have em : max {{integer}} ({{integer}} / {{integer}} * x - {{integer}}) = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + simp only [max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, em] + simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero] + norm_num + ring_nf + diff --git a/scripts/zeta5-certificates/archive/templates/438.txt b/scripts/zeta5-certificates/archive/templates/438.txt new file mode 100644 index 0000000000..ffdac629d2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/438.txt @@ -0,0 +1,66 @@ +theorem outPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Eout x = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Eout_expand (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have em : max {{integer}} ({{integer}} / {{integer}} * x - {{integer}}) = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + simp only [max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, em] + simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero] + norm_num + ring_nf + diff --git a/scripts/zeta5-certificates/archive/templates/439.txt b/scripts/zeta5-certificates/archive/templates/439.txt new file mode 100644 index 0000000000..3d06721aa2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/439.txt @@ -0,0 +1,67 @@ +theorem outPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Eout x = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Eout_expand (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have em : max {{integer}} ({{integer}} / {{integer}} * x - {{integer}}) = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + simp only [max_def] + split_ifs <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, em] + simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero] + norm_num + ring_nf + diff --git a/scripts/zeta5-certificates/archive/templates/440.txt b/scripts/zeta5-certificates/archive/templates/440.txt new file mode 100644 index 0000000000..3b6ed80e30 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/440.txt @@ -0,0 +1,66 @@ +theorem outPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Eout x = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Eout_expand (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have em : max {{integer}} ({{integer}} / {{integer}} * x - {{integer}}) = ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + (try simp only [max_def]); (try split_ifs) <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, em] + simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero] + norm_num + ring_nf + diff --git a/scripts/zeta5-certificates/archive/templates/441.txt b/scripts/zeta5-certificates/archive/templates/441.txt new file mode 100644 index 0000000000..7fc6198619 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/441.txt @@ -0,0 +1,67 @@ +theorem outPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < ({{integer}} : ℝ)) : + Eout x = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Eout_expand (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have em : max {{integer}} ({{integer}} / {{integer}} * x - {{integer}}) = ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + (try simp only [max_def]); (try split_ifs) <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, em] + simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero] + norm_num + ring_nf + diff --git a/scripts/zeta5-certificates/archive/templates/442.txt b/scripts/zeta5-certificates/archive/templates/442.txt new file mode 100644 index 0000000000..d12eaa341e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/442.txt @@ -0,0 +1,64 @@ +theorem outPiece_{{integer}} (x : ℝ) (h{{integer}} : ({{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Eout x = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Eout_expand (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have em : max {{integer}} ({{integer}} / {{integer}} * x - {{integer}}) = ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + (try simp only [max_def]); (try split_ifs) <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, em] + simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero] + norm_num + ring_nf + diff --git a/scripts/zeta5-certificates/archive/templates/443.txt b/scripts/zeta5-certificates/archive/templates/443.txt new file mode 100644 index 0000000000..54456f228a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/443.txt @@ -0,0 +1,65 @@ +theorem outPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < (({{integer}}) / {{integer}} : ℝ)) : + Eout x = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Eout_expand (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have em : max {{integer}} ({{integer}} / {{integer}} * x - {{integer}}) = ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + (try simp only [max_def]); (try split_ifs) <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, em] + simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero] + norm_num + ring_nf + diff --git a/scripts/zeta5-certificates/archive/templates/444.txt b/scripts/zeta5-certificates/archive/templates/444.txt new file mode 100644 index 0000000000..82acc61150 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/444.txt @@ -0,0 +1,68 @@ +theorem outPiece_{{integer}} (x : ℝ) (h{{integer}} : (({{integer}}) / {{integer}} : ℝ) ≤ x) (h{{integer}} : x < ({{integer}} : ℝ)) : + Eout x = (({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ) := by + rw [Eout_expand (q := {{integer}}) (q' := {{integer}}) (μ := {{integer}}) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith) + (by norm_num; linarith) (by linarith)] + have hf{{integer}} : ({{integer}} : ℝ) ≤ x - ({{integer}} : ℝ) := by linarith + have hf{{integer}} : x - ({{integer}} : ℝ) < {{integer}} := by linarith + have hg{{integer}} : ({{integer}} : ℝ) ≤ {{integer}} / {{integer}} * x - ({{integer}} : ℝ) := by linarith + have hg{{integer}} : {{integer}} / {{integer}} * x - ({{integer}} : ℝ) < {{integer}} := by linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}}] + ring + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + (({{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}} hf{{integer}} hg{{integer}}] + simp only [min_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + (({{integer}} : ℝ) * x + ((-{{integer}}) / {{integer}} : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = + ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + rw [cc{{integer}} hf{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have e{{integer}} : + ccount (x - ({{integer}} : ℝ)) ({{integer}} / {{integer}} * x - ({{integer}} : ℝ)) {{integer}} {{integer}} = (({{integer}} : ℝ) * x + ({{integer}} : ℝ)) := by + rw [cc{{integer}} hg{{integer}}] + simp only [min_def, max_def] + split_ifs <;> linarith + have em : max {{integer}} ({{integer}} / {{integer}} * x - {{integer}}) = ((({{integer}}) / {{integer}} : ℝ) * x + ((-{{integer}}) : ℝ)) := by + (try simp only [max_def]); (try split_ifs) <;> linarith + push_cast + simp only [Fin.sum_univ_three] + rw [e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, e{{integer}}, em] + simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two, + Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero] + norm_num + ring_nf + +/-- The {{integer}} rational endpoints of the {{integer}} outer-range linear pieces. -/ diff --git a/scripts/zeta5-certificates/archive/templates/445.txt b/scripts/zeta5-certificates/archive/templates/445.txt new file mode 100644 index 0000000000..39d2a30d89 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/445.txt @@ -0,0 +1,5 @@ +def tOutL : List ℚ := + [({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}})] + +/-- Rational slopes of the {{integer}} outer-range linear pieces. -/ diff --git a/scripts/zeta5-certificates/archive/templates/446.txt b/scripts/zeta5-certificates/archive/templates/446.txt new file mode 100644 index 0000000000..ede66c2233 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/446.txt @@ -0,0 +1,5 @@ +def aOutL : List ℚ := + [({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}})] + +/-- Rational intercepts of the {{integer}} outer-range linear pieces. -/ diff --git a/scripts/zeta5-certificates/archive/templates/447.txt b/scripts/zeta5-certificates/archive/templates/447.txt new file mode 100644 index 0000000000..ff84504cb4 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/447.txt @@ -0,0 +1,11 @@ +def bOutL : List ℚ := + [(-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}})] + +/-- The `i`th outer-range endpoint as a real number; zero outside the table. -/ +noncomputable def tOut (i : ℕ) : ℝ := + ((tOutL.getD i {{integer}} : ℚ) : ℝ) + +/-- The affine bound on outer piece `i`, with coefficients from `aOutL` and `bOutL`. -/ +noncomputable def gOut (i : ℕ) (x : ℝ) : ℝ := + ((aOutL.getD i {{integer}} : ℚ) : ℝ) * x + ((bOutL.getD i {{integer}} : ℚ) : ℝ) + diff --git a/scripts/zeta5-certificates/archive/templates/448.txt b/scripts/zeta5-certificates/archive/templates/448.txt new file mode 100644 index 0000000000..a8e0fe63f1 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/448.txt @@ -0,0 +1,5 @@ +theorem tOut_mono : ∀ i < {{integer}}, tOut i < tOut (i + {{integer}}) := by + intro i hi + interval_cases i <;> simp only [tOut, tOutL, List.getD_cons_succ, List.getD_cons_zero] <;> + norm_num + diff --git a/scripts/zeta5-certificates/archive/templates/449.txt b/scripts/zeta5-certificates/archive/templates/449.txt new file mode 100644 index 0000000000..d8342398c2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/449.txt @@ -0,0 +1,70 @@ +theorem outer_table : ∀ i < {{integer}}, ∀ x ∈ Set.Ico (tOut i) (tOut (i + {{integer}})), Eout x = gOut i x := by + intro i hi x hx + interval_cases i + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := outPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/archive/templates/450.txt b/scripts/zeta5-certificates/archive/templates/450.txt new file mode 100644 index 0000000000..37afb2c7e9 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/450.txt @@ -0,0 +1,25 @@ +/- +Copyright (c) {{integer}} Moritz Firsching. All rights reserved. +Released under Apache {{integer}}.{{integer}} license as described in the file LICENSE. +Authors: Moritz Firsching +-/ + +module + +import LeanPool.Zeta{{integer}}Irrational.Growth.InnerTable +import LeanPool.Zeta{{integer}}Irrational.Growth.TableGen +public import LeanPool.Zeta{{integer}}Irrational.Growth.TablePrime +import Mathlib.Tactic.NormNum.RealSqrt +import Mathlib.NumberTheory.Chebyshev +import Mathlib.Tactic.NormNum.Prime +import Mathlib.Analysis.SpecialFunctions.Gamma.Basic +import Mathlib.Data.Int.Star + +/-!Generated by `gen_inner_sum.py`: the table of `Ê_in` on `[{{integer}}, {{integer}}]`. -/ + + +@[expose] public section + +namespace Zeta{{integer}}Irrational + +/-- The {{integer}} rational endpoints of the {{integer}} inner-range linear pieces. -/ diff --git a/scripts/zeta5-certificates/archive/templates/451.txt b/scripts/zeta5-certificates/archive/templates/451.txt new file mode 100644 index 0000000000..8ba97a7011 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/451.txt @@ -0,0 +1,19 @@ +def tInL : List ℚ := + [({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}})] + +/-- Integer allocation levels chosen for the {{integer}} inner-range pieces. -/ diff --git a/scripts/zeta5-certificates/archive/templates/452.txt b/scripts/zeta5-certificates/archive/templates/452.txt new file mode 100644 index 0000000000..f291686336 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/452.txt @@ -0,0 +1,9 @@ +def kInL : List ℤ := + [{{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, + {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, + {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, + {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, + {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, {{integer}}, + {{integer}}, {{integer}}, {{integer}}, {{integer}}] + +/-- Rational slopes of the {{integer}} inner-range linear pieces. -/ diff --git a/scripts/zeta5-certificates/archive/templates/453.txt b/scripts/zeta5-certificates/archive/templates/453.txt new file mode 100644 index 0000000000..35ea60bbe7 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/453.txt @@ -0,0 +1,19 @@ +def aInL : List ℚ := + [({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}}), + ({{integer}} / {{integer}}), ({{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), + ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}}), ({{integer}} / {{integer}})] + +/-- Rational intercepts of the {{integer}} inner-range linear pieces. -/ diff --git a/scripts/zeta5-certificates/archive/templates/454.txt b/scripts/zeta5-certificates/archive/templates/454.txt new file mode 100644 index 0000000000..9851b44f6e --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/454.txt @@ -0,0 +1,19 @@ +def bInL : List ℚ := + [(-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), + (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), + (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), + (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), + (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), + (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), + (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), + (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), + (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), + (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), + (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), + (-{{integer}} / {{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}}), (-{{integer}} / {{integer}}), (-{{integer}} / {{integer}}), (-{{integer}})] + +/-- The `i`th inner-range endpoint as a real number; zero outside the table. -/ +noncomputable def tIn (i : ℕ) : ℝ := + ((tInL.getD i {{integer}} : ℚ) : ℝ) + +/-- The allocation level for inner piece `i`; zero outside the table. -/ diff --git a/scripts/zeta5-certificates/archive/templates/455.txt b/scripts/zeta5-certificates/archive/templates/455.txt new file mode 100644 index 0000000000..8bf6a4f3fc --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/455.txt @@ -0,0 +1,7 @@ +def kIn (i : ℕ) : ℤ := + kInL.getD i {{integer}} + +/-- The affine bound on inner piece `i`, with coefficients from `aInL` and `bInL`. -/ +noncomputable def gIn (i : ℕ) (x : ℝ) : ℝ := + ((aInL.getD i {{integer}} : ℚ) : ℝ) * x + ((bInL.getD i {{integer}} : ℚ) : ℝ) + diff --git a/scripts/zeta5-certificates/archive/templates/456.txt b/scripts/zeta5-certificates/archive/templates/456.txt new file mode 100644 index 0000000000..26c35ebcc2 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/456.txt @@ -0,0 +1,7 @@ +theorem tIn_mono : ∀ i < {{integer}}, tIn i < tIn (i + {{integer}}) := by + have certificate : ∀ i : Fin {{integer}}, + tInL.getD i.val {{integer}} < tInL.getD (i.val + {{integer}}) {{integer}} := by decide +kernel + intro i hi + simp only [tIn] + exact_mod_cast certificate ⟨i, hi⟩ + diff --git a/scripts/zeta5-certificates/archive/templates/457.txt b/scripts/zeta5-certificates/archive/templates/457.txt new file mode 100644 index 0000000000..ede24ea46a --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/457.txt @@ -0,0 +1,8 @@ +private lemma inner_table_{{integer}} (x : ℝ) (hx : x ∈ Set.Ico (tIn {{integer}}) (tIn ({{integer}} + {{integer}}))) : + Ein x (kIn {{integer}}) = gIn {{integer}} x := by + simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero, + Set.mem_Ico] at hx ⊢ + push_cast at hx ⊢ + have := inPiece_{{integer}} x (by linarith [hx.{{integer}}]) (by linarith [hx.{{integer}}]) + linarith + diff --git a/scripts/zeta5-certificates/archive/templates/458.txt b/scripts/zeta5-certificates/archive/templates/458.txt new file mode 100644 index 0000000000..5fdcec75f3 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/458.txt @@ -0,0 +1,129 @@ +theorem inner_table : ∀ i < {{integer}}, ∀ x ∈ Set.Ico (tIn i) (tIn (i + {{integer}})), Ein x (kIn i) = gIn i x := by + intro i hi x hx + interval_cases i + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + · exact inner_table_{{integer}} x hx + diff --git a/scripts/zeta5-certificates/archive/templates/459.txt b/scripts/zeta5-certificates/archive/templates/459.txt new file mode 100644 index 0000000000..903043b822 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/459.txt @@ -0,0 +1,7 @@ +private lemma inner_table_k_{{integer}} (x : ℝ) (hx : x ∈ Set.Ico (tIn {{integer}}) (tIn ({{integer}} + {{integer}}))) : + -{{integer}} * (⌊x⌋₊ : ℤ) - {{integer}} ≤ kIn {{integer}} ∧ kIn {{integer}} ≤ {{integer}} * (⌊x⌋₊ : ℤ) + {{integer}} := by + simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢ + push_cast at hx + rw [nat_floor_of_bounds (q := {{integer}}) (by norm_num; linarith [hx.{{integer}}]) (by norm_num; linarith [hx.{{integer}}])] + norm_num + diff --git a/scripts/zeta5-certificates/archive/templates/460.txt b/scripts/zeta5-certificates/archive/templates/460.txt new file mode 100644 index 0000000000..2337926219 --- /dev/null +++ b/scripts/zeta5-certificates/archive/templates/460.txt @@ -0,0 +1,133 @@ +theorem inner_table_k : + ∀ i < {{integer}}, + ∀ x ∈ Set.Ico (tIn i) (tIn (i + {{integer}})), + -{{integer}} * (⌊x⌋₊ : ℤ) - {{integer}} ≤ kIn i ∧ kIn i ≤ {{integer}} * (⌊x⌋₊ : ℤ) + {{integer}} := by + intro i hi x hx + interval_cases i + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + · exact inner_table_k_{{integer}} x hx + +end Zeta{{integer}}Irrational diff --git a/scripts/zeta5-certificates/regenerate.py b/scripts/zeta5-certificates/regenerate.py new file mode 100644 index 0000000000..1734b8d01f --- /dev/null +++ b/scripts/zeta5-certificates/regenerate.py @@ -0,0 +1,109 @@ +#!/usr/bin/env python3 +"""Render the archived Zeta5 certificate templates using exact integer parameters.""" + +import argparse +import hashlib +import json +import re +from pathlib import Path + +PLACEHOLDER = "{{integer}}" +PROJECT = Path("LeanPool/Zeta5Irrational") + + +def render(template: str, parameters: list[str]) -> str: + """Instantiate one proof block, rejecting incomplete or nonnumeric records.""" + if any(re.fullmatch(r"[0-9]+", value) is None for value in parameters): + raise ValueError("Certificate parameters must be decimal integer strings") + pieces = template.split(PLACEHOLDER) + if len(pieces) != len(parameters) + 1: + raise ValueError("Template and parameter counts differ") + return ( + "".join(part + number for part, number in zip(pieces, parameters)) + pieces[-1] + ) + + +def certificate_path(name: str) -> Path: + """Restrict output to the archived project's Lean modules.""" + path = Path(name) + if path.is_absolute() or ".." in path.parts or path.suffix != ".lean": + raise ValueError(f"Invalid certificate path: {name}") + if not path.is_relative_to(PROJECT): + raise ValueError(f"Certificate outside the project: {name}") + return path + + +def regenerate(archive: Path, destination: Path, check: bool) -> int: + """Render every module and either write it or compare it with a checkout.""" + manifest = json.loads((archive / "manifest.json").read_text()) + mismatches = [] + for name, specification in manifest["files"].items(): + relative = certificate_path(name) + records = json.loads( + (archive / "records" / specification["records"]).read_text() + ) + blocks = [] + for template, parameters in records: + if re.fullmatch(r"[0-9]{3}", template) is None: + raise ValueError("Invalid template identifier") + source = (archive / "templates" / f"{template}.txt").read_text() + blocks.append(render(source, parameters)) + content = "".join(blocks).encode() + target = destination / relative + if check: + if not target.is_file() or target.read_bytes() != content: + mismatches.append(name) + else: + target.parent.mkdir(parents=True, exist_ok=True) + target.write_bytes(content) + if mismatches: + raise ValueError("Certificates differ: " + ", ".join(mismatches)) + return len(manifest["files"]) + + +def verify_archive(archive: Path) -> None: + """Check the immutable baseline hashes independently of a target checkout.""" + manifest = json.loads((archive / "manifest.json").read_text()) + for specification in manifest["files"].values(): + records = json.loads( + (archive / "records" / specification["records"]).read_text() + ) + content = "".join( + render((archive / "templates" / f"{name}.txt").read_text(), parameters) + for name, parameters in records + ).encode() + if hashlib.sha256(content).hexdigest() != specification["sha256"]: + raise ValueError("Archived certificate differs from its pinned source hash") + + +def main() -> None: + """Run generation, checkout comparison, or baseline archive verification.""" + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument( + "--archive", type=Path, default=Path(__file__).parent / "archive" + ) + mode = parser.add_mutually_exclusive_group(required=True) + mode.add_argument( + "--output", type=Path, help="Write generated modules beneath this directory" + ) + mode.add_argument( + "--check", type=Path, help="Compare generated modules with this checkout" + ) + mode.add_argument("--verify-archive", action="store_true") + arguments = parser.parse_args() + if arguments.verify_archive: + verify_archive(arguments.archive) + print("Archive matches every pinned certificate hash") + else: + count = regenerate( + arguments.archive, + arguments.check or arguments.output, + arguments.check is not None, + ) + print( + f"{'Checked' if arguments.check else 'Generated'} {count} certificate modules" + ) + + +if __name__ == "__main__": + main() From 6a9863bb2f17169fb8107bdbf8131f20ca7aa071 Mon Sep 17 00:00:00 2001 From: Vasily Ilin Date: Sun, 27 Sep 2026 12:59:36 +0000 Subject: [PATCH 2/6] fix: publish regenerated certificates only into a fresh tree --- python/tests/test_zeta5_certificates.py | 39 +++++++++++++++++++++--- scripts/zeta5-certificates/README.md | 5 ++- scripts/zeta5-certificates/regenerate.py | 18 ++++++++++- 3 files changed, 56 insertions(+), 6 deletions(-) diff --git a/python/tests/test_zeta5_certificates.py b/python/tests/test_zeta5_certificates.py index 7753e42b3a..ecaedcc576 100644 --- a/python/tests/test_zeta5_certificates.py +++ b/python/tests/test_zeta5_certificates.py @@ -2,6 +2,7 @@ import importlib.util import json +import shutil from pathlib import Path import pytest @@ -46,10 +47,40 @@ def test_complete_archive(tmp_path: Path) -> None: """Regenerate every output and independently check its pinned hash.""" archive = SCRIPT.parent / "archive" GENERATOR.verify_archive(archive) - assert GENERATOR.regenerate(archive, tmp_path, False) == 118 - assert GENERATOR.regenerate(archive, tmp_path, True) == 118 + assert GENERATOR.regenerate(archive, tmp_path / "generated", False) == 118 + assert GENERATOR.regenerate(archive, tmp_path / "generated", True) == 118 manifest = json.loads((archive / "manifest.json").read_text()) - first = tmp_path / next(iter(manifest["files"])) + first = tmp_path / "generated" / next(iter(manifest["files"])) first.write_text(first.read_text() + "-- damaged\n") with pytest.raises(ValueError, match="Certificates differ"): - GENERATOR.regenerate(archive, tmp_path, True) + GENERATOR.regenerate(archive, tmp_path / "generated", True) + + +def test_reject_reused_or_symlinked_destination(tmp_path: Path) -> None: + """Protect pre-existing output trees and symlink targets from writes.""" + outside = tmp_path / "outside" + outside.mkdir() + sentinel = outside / "sentinel" + sentinel.write_text("preserve me") + symlink = tmp_path / "link" + symlink.symlink_to(outside, target_is_directory=True) + for destination in (outside, symlink): + with pytest.raises(FileExistsError): + GENERATOR.regenerate(SCRIPT.parent / "archive", destination, False) + assert sentinel.read_text() == "preserve me" + assert list(outside.iterdir()) == [sentinel] + + +def test_failed_later_record_publishes_nothing(tmp_path: Path) -> None: + """Remove staging output when a later template is absent.""" + archive = tmp_path / "archive" + shutil.copytree(SCRIPT.parent / "archive", archive) + manifest = json.loads((archive / "manifest.json").read_text()) + last = list(manifest["files"].values())[-1] + records = archive / "records" / last["records"] + records.write_text('[["999", []]]') + output = tmp_path / "generated" + with pytest.raises(FileNotFoundError): + GENERATOR.regenerate(archive, output, False) + assert not output.exists() + assert list(tmp_path.glob(".zeta5-certificates-*")) == [] diff --git a/scripts/zeta5-certificates/README.md b/scripts/zeta5-certificates/README.md index 8b59c880fb..7b4789fc13 100644 --- a/scripts/zeta5-certificates/README.md +++ b/scripts/zeta5-certificates/README.md @@ -18,7 +18,10 @@ python3 scripts/zeta5-certificates/regenerate.py --check /path/to/import-checkou ``` `--output` writes only the recorded project modules underneath its argument; -use an empty scratch directory when inspecting a regeneration. `--check` compares +the destination must not already exist, even as an empty directory or symlink. +Generation stages all files in a new sibling directory and publishes the complete +tree by rename only after every record renders successfully. A failed generation +removes its staging tree and publishes nothing. `--check` compares bytes without writing. The archive hash verification is separate so a deliberate template edit can be rendered before its baseline hashes are updated. diff --git a/scripts/zeta5-certificates/regenerate.py b/scripts/zeta5-certificates/regenerate.py index 1734b8d01f..bf2854e7ff 100644 --- a/scripts/zeta5-certificates/regenerate.py +++ b/scripts/zeta5-certificates/regenerate.py @@ -5,6 +5,7 @@ import hashlib import json import re +import tempfile from pathlib import Path PLACEHOLDER = "{{integer}}" @@ -33,7 +34,7 @@ def certificate_path(name: str) -> Path: return path -def regenerate(archive: Path, destination: Path, check: bool) -> int: +def _render_into(archive: Path, destination: Path, check: bool) -> int: """Render every module and either write it or compare it with a checkout.""" manifest = json.loads((archive / "manifest.json").read_text()) mismatches = [] @@ -61,6 +62,21 @@ def regenerate(archive: Path, destination: Path, check: bool) -> int: return len(manifest["files"]) +def regenerate(archive: Path, destination: Path, check: bool) -> int: + """Check a checkout or publish a complete generation to a new directory.""" + if check: + return _render_into(archive, destination, True) + if destination.exists() or destination.is_symlink(): + raise FileExistsError("Generation requires a new output directory") + with tempfile.TemporaryDirectory( + prefix=".zeta5-certificates-", dir=destination.parent + ) as staging_name: + staging = Path(staging_name) + count = _render_into(archive, staging, False) + staging.rename(destination) + return count + + def verify_archive(archive: Path) -> None: """Check the immutable baseline hashes independently of a target checkout.""" manifest = json.loads((archive / "manifest.json").read_text()) From cb25dbb6fa557ba317144aa1da05109d4235da7c Mon Sep 17 00:00:00 2001 From: Vasily Ilin Date: Sun, 27 Sep 2026 13:04:52 +0000 Subject: [PATCH 3/6] fix: validate certificates before exclusively creating output --- python/tests/test_zeta5_certificates.py | 41 +++++++++++++- scripts/zeta5-certificates/README.md | 7 ++- scripts/zeta5-certificates/regenerate.py | 70 ++++++++++++------------ 3 files changed, 77 insertions(+), 41 deletions(-) diff --git a/python/tests/test_zeta5_certificates.py b/python/tests/test_zeta5_certificates.py index ecaedcc576..a2ce6c4c6f 100644 --- a/python/tests/test_zeta5_certificates.py +++ b/python/tests/test_zeta5_certificates.py @@ -71,8 +71,8 @@ def test_reject_reused_or_symlinked_destination(tmp_path: Path) -> None: assert list(outside.iterdir()) == [sentinel] -def test_failed_later_record_publishes_nothing(tmp_path: Path) -> None: - """Remove staging output when a later template is absent.""" +def test_failed_later_record_creates_nothing(tmp_path: Path) -> None: + """Create no output when a later template is absent.""" archive = tmp_path / "archive" shutil.copytree(SCRIPT.parent / "archive", archive) manifest = json.loads((archive / "manifest.json").read_text()) @@ -84,3 +84,40 @@ def test_failed_later_record_publishes_nothing(tmp_path: Path) -> None: GENERATOR.regenerate(archive, output, False) assert not output.exists() assert list(tmp_path.glob(".zeta5-certificates-*")) == [] + + +def test_destination_created_during_render_is_preserved( + tmp_path: Path, monkeypatch: pytest.MonkeyPatch +) -> None: + """Do not replace a destination created after the operation starts.""" + output = tmp_path / "generated" + + def competing_creation(archive: Path) -> dict[Path, bytes]: + output.mkdir() + return {Path("LeanPool/Zeta5Irrational/A.lean"): b"test"} + + monkeypatch.setattr(GENERATOR, "_render_modules", competing_creation) + with pytest.raises(FileExistsError): + GENERATOR.regenerate(SCRIPT.parent / "archive", output, False) + assert output.is_dir() + assert list(output.iterdir()) == [] + + +def test_nested_output_and_write_failure_cleanup( + tmp_path: Path, monkeypatch: pytest.MonkeyPatch +) -> None: + """Support new parents and clean up output if writing fails.""" + relative = Path("LeanPool/Zeta5Irrational/A.lean") + monkeypatch.setattr(GENERATOR, "_render_modules", lambda _: {relative: b"test"}) + output = tmp_path / "new" / "parents" / "generated" + assert GENERATOR.regenerate(SCRIPT.parent / "archive", output, False) == 1 + assert (output / relative).read_bytes() == b"test" + + def fail_write(path: Path, content: bytes) -> None: + raise OSError("disk full") + + monkeypatch.setattr(Path, "write_bytes", fail_write) + failed_output = tmp_path / "failed" + with pytest.raises(OSError, match="disk full"): + GENERATOR.regenerate(SCRIPT.parent / "archive", failed_output, False) + assert not failed_output.exists() diff --git a/scripts/zeta5-certificates/README.md b/scripts/zeta5-certificates/README.md index 7b4789fc13..2949eaf9b5 100644 --- a/scripts/zeta5-certificates/README.md +++ b/scripts/zeta5-certificates/README.md @@ -19,9 +19,10 @@ python3 scripts/zeta5-certificates/regenerate.py --check /path/to/import-checkou `--output` writes only the recorded project modules underneath its argument; the destination must not already exist, even as an empty directory or symlink. -Generation stages all files in a new sibling directory and publishes the complete -tree by rename only after every record renders successfully. A failed generation -removes its staging tree and publishes nothing. `--check` compares +All records are validated and rendered in memory before the destination is +created exclusively. Missing parent directories are created automatically. A +malformed record creates no output; a write failure removes the new output tree. +Wait for successful completion before reading or copying generated files. `--check` compares bytes without writing. The archive hash verification is separate so a deliberate template edit can be rendered before its baseline hashes are updated. diff --git a/scripts/zeta5-certificates/regenerate.py b/scripts/zeta5-certificates/regenerate.py index bf2854e7ff..220c04d5c6 100644 --- a/scripts/zeta5-certificates/regenerate.py +++ b/scripts/zeta5-certificates/regenerate.py @@ -5,7 +5,7 @@ import hashlib import json import re -import tempfile +import shutil from pathlib import Path PLACEHOLDER = "{{integer}}" @@ -34,10 +34,10 @@ def certificate_path(name: str) -> Path: return path -def _render_into(archive: Path, destination: Path, check: bool) -> int: - """Render every module and either write it or compare it with a checkout.""" +def _render_modules(archive: Path) -> dict[Path, bytes]: + """Validate and render every recorded module before touching output.""" manifest = json.loads((archive / "manifest.json").read_text()) - mismatches = [] + modules = {} for name, specification in manifest["files"].items(): relative = certificate_path(name) records = json.loads( @@ -49,46 +49,44 @@ def _render_into(archive: Path, destination: Path, check: bool) -> int: raise ValueError("Invalid template identifier") source = (archive / "templates" / f"{template}.txt").read_text() blocks.append(render(source, parameters)) - content = "".join(blocks).encode() - target = destination / relative - if check: - if not target.is_file() or target.read_bytes() != content: - mismatches.append(name) - else: - target.parent.mkdir(parents=True, exist_ok=True) - target.write_bytes(content) - if mismatches: - raise ValueError("Certificates differ: " + ", ".join(mismatches)) - return len(manifest["files"]) + modules[relative] = "".join(blocks).encode() + return modules def regenerate(archive: Path, destination: Path, check: bool) -> int: - """Check a checkout or publish a complete generation to a new directory.""" + """Check a checkout or write prevalidated modules to an exclusively new tree.""" + modules = _render_modules(archive) if check: - return _render_into(archive, destination, True) - if destination.exists() or destination.is_symlink(): - raise FileExistsError("Generation requires a new output directory") - with tempfile.TemporaryDirectory( - prefix=".zeta5-certificates-", dir=destination.parent - ) as staging_name: - staging = Path(staging_name) - count = _render_into(archive, staging, False) - staging.rename(destination) - return count + mismatches = [ + str(path) + for path, content in modules.items() + if not (destination / path).is_file() + or (destination / path).read_bytes() != content + ] + if mismatches: + raise ValueError("Certificates differ: " + ", ".join(mismatches)) + return len(modules) + destination.parent.mkdir(parents=True, exist_ok=True) + destination.mkdir(mode=0o700) + try: + for path, content in modules.items(): + target = destination / path + target.parent.mkdir(parents=True, exist_ok=True) + target.write_bytes(content) + except BaseException: + shutil.rmtree(destination) + raise + return len(modules) def verify_archive(archive: Path) -> None: - """Check the immutable baseline hashes independently of a target checkout.""" + """Check baseline hashes independently of a target checkout.""" manifest = json.loads((archive / "manifest.json").read_text()) - for specification in manifest["files"].values(): - records = json.loads( - (archive / "records" / specification["records"]).read_text() - ) - content = "".join( - render((archive / "templates" / f"{name}.txt").read_text(), parameters) - for name, parameters in records - ).encode() - if hashlib.sha256(content).hexdigest() != specification["sha256"]: + for path, content in _render_modules(archive).items(): + if ( + hashlib.sha256(content).hexdigest() + != manifest["files"][str(path)]["sha256"] + ): raise ValueError("Archived certificate differs from its pinned source hash") From c05607ee81e49c4a4ae95af00d1fac7ba53e732a Mon Sep 17 00:00:00 2001 From: Vasily Ilin Date: Sun, 27 Sep 2026 13:09:45 +0000 Subject: [PATCH 4/6] fix: verify certificate hashes with portable paths and UTF-8 --- python/tests/test_zeta5_certificates.py | 27 +++++++++++++++++++----- scripts/zeta5-certificates/regenerate.py | 14 ++++++------ 2 files changed, 30 insertions(+), 11 deletions(-) diff --git a/python/tests/test_zeta5_certificates.py b/python/tests/test_zeta5_certificates.py index a2ce6c4c6f..90ff23435f 100644 --- a/python/tests/test_zeta5_certificates.py +++ b/python/tests/test_zeta5_certificates.py @@ -1,9 +1,10 @@ """Check exact and constrained rendering of the Zeta5 certificate archive.""" +import hashlib import importlib.util import json import shutil -from pathlib import Path +from pathlib import Path, PureWindowsPath import pytest @@ -49,9 +50,11 @@ def test_complete_archive(tmp_path: Path) -> None: GENERATOR.verify_archive(archive) assert GENERATOR.regenerate(archive, tmp_path / "generated", False) == 118 assert GENERATOR.regenerate(archive, tmp_path / "generated", True) == 118 - manifest = json.loads((archive / "manifest.json").read_text()) + manifest = json.loads((archive / "manifest.json").read_text(encoding="utf-8")) first = tmp_path / "generated" / next(iter(manifest["files"])) - first.write_text(first.read_text() + "-- damaged\n") + first.write_text( + first.read_text(encoding="utf-8") + "-- damaged\n", encoding="utf-8" + ) with pytest.raises(ValueError, match="Certificates differ"): GENERATOR.regenerate(archive, tmp_path / "generated", True) @@ -67,7 +70,7 @@ def test_reject_reused_or_symlinked_destination(tmp_path: Path) -> None: for destination in (outside, symlink): with pytest.raises(FileExistsError): GENERATOR.regenerate(SCRIPT.parent / "archive", destination, False) - assert sentinel.read_text() == "preserve me" + assert sentinel.read_text(encoding="utf-8") == "preserve me" assert list(outside.iterdir()) == [sentinel] @@ -75,7 +78,7 @@ def test_failed_later_record_creates_nothing(tmp_path: Path) -> None: """Create no output when a later template is absent.""" archive = tmp_path / "archive" shutil.copytree(SCRIPT.parent / "archive", archive) - manifest = json.loads((archive / "manifest.json").read_text()) + manifest = json.loads((archive / "manifest.json").read_text(encoding="utf-8")) last = list(manifest["files"].values())[-1] records = archive / "records" / last["records"] records.write_text('[["999", []]]') @@ -121,3 +124,17 @@ def fail_write(path: Path, content: bytes) -> None: with pytest.raises(OSError, match="disk full"): GENERATOR.regenerate(SCRIPT.parent / "archive", failed_output, False) assert not failed_output.exists() + + +def test_archive_manifest_uses_portable_paths( + tmp_path: Path, monkeypatch: pytest.MonkeyPatch +) -> None: + """Look up POSIX manifest keys even when rendering uses Windows paths.""" + content = b"certificate" + path = PureWindowsPath("LeanPool/Zeta5Irrational/Table/U00.lean") + manifest = { + "files": {path.as_posix(): {"sha256": hashlib.sha256(content).hexdigest()}} + } + (tmp_path / "manifest.json").write_text(json.dumps(manifest), encoding="utf-8") + monkeypatch.setattr(GENERATOR, "_render_modules", lambda _: {path: content}) + GENERATOR.verify_archive(tmp_path) diff --git a/scripts/zeta5-certificates/regenerate.py b/scripts/zeta5-certificates/regenerate.py index 220c04d5c6..3dc86cfc88 100644 --- a/scripts/zeta5-certificates/regenerate.py +++ b/scripts/zeta5-certificates/regenerate.py @@ -36,20 +36,22 @@ def certificate_path(name: str) -> Path: def _render_modules(archive: Path) -> dict[Path, bytes]: """Validate and render every recorded module before touching output.""" - manifest = json.loads((archive / "manifest.json").read_text()) + manifest = json.loads((archive / "manifest.json").read_text(encoding="utf-8")) modules = {} for name, specification in manifest["files"].items(): relative = certificate_path(name) records = json.loads( - (archive / "records" / specification["records"]).read_text() + (archive / "records" / specification["records"]).read_text(encoding="utf-8") ) blocks = [] for template, parameters in records: if re.fullmatch(r"[0-9]{3}", template) is None: raise ValueError("Invalid template identifier") - source = (archive / "templates" / f"{template}.txt").read_text() + source = (archive / "templates" / f"{template}.txt").read_text( + encoding="utf-8" + ) blocks.append(render(source, parameters)) - modules[relative] = "".join(blocks).encode() + modules[relative] = "".join(blocks).encode("utf-8") return modules @@ -81,11 +83,11 @@ def regenerate(archive: Path, destination: Path, check: bool) -> int: def verify_archive(archive: Path) -> None: """Check baseline hashes independently of a target checkout.""" - manifest = json.loads((archive / "manifest.json").read_text()) + manifest = json.loads((archive / "manifest.json").read_text(encoding="utf-8")) for path, content in _render_modules(archive).items(): if ( hashlib.sha256(content).hexdigest() - != manifest["files"][str(path)]["sha256"] + != manifest["files"][path.as_posix()]["sha256"] ): raise ValueError("Archived certificate differs from its pinned source hash") From 37c58c2b1ef349fef79b7ed053012d726ff67ce6 Mon Sep 17 00:00:00 2001 From: Vasily Ilin Date: Sun, 27 Sep 2026 14:04:52 +0000 Subject: [PATCH 5/6] Refresh preserved certificate headers after final import cleanup --- scripts/zeta5-certificates/README.md | 6 +- .../zeta5-certificates/archive/manifest.json | 122 +++++++++--------- .../archive/records/Growth_InnerSum.json | 2 +- .../archive/records/Growth_InnerTable.json | 2 +- .../archive/records/Growth_OuterTable.json | 2 +- .../archive/records/Table_V00.json | 2 +- .../archive/records/Table_V01.json | 2 +- .../archive/records/Table_V02.json | 2 +- .../archive/records/Table_V03.json | 2 +- .../archive/records/Table_V04.json | 2 +- .../archive/records/Table_V05.json | 2 +- .../archive/records/Table_V06.json | 2 +- .../archive/records/Table_V07.json | 2 +- .../archive/records/Table_V08.json | 2 +- .../archive/records/Table_V09.json | 2 +- .../archive/records/Table_V10.json | 2 +- .../archive/records/Table_V11.json | 2 +- .../archive/records/Table_V12.json | 2 +- .../archive/records/Table_V13.json | 2 +- .../archive/records/Table_V14.json | 2 +- .../archive/records/Table_V15.json | 2 +- .../archive/records/Table_V16.json | 2 +- .../archive/records/Table_V17.json | 2 +- .../archive/records/Table_V18.json | 2 +- .../archive/records/Table_V19.json | 2 +- .../archive/records/Table_V20.json | 2 +- .../archive/records/Table_V21.json | 2 +- .../archive/records/Table_V22.json | 2 +- .../archive/records/Table_V23.json | 2 +- .../archive/records/Table_V24.json | 2 +- .../archive/records/Table_V25.json | 2 +- .../archive/records/Table_V26.json | 2 +- .../archive/records/Table_V27.json | 2 +- .../archive/records/Table_V28.json | 2 +- .../archive/records/Table_V29.json | 2 +- .../archive/records/Table_V30.json | 2 +- .../archive/records/Table_V31.json | 2 +- .../archive/records/Table_V32.json | 2 +- .../archive/records/Table_V33.json | 2 +- .../archive/records/Table_V34.json | 2 +- .../archive/records/Table_V35.json | 2 +- .../archive/records/Table_V36.json | 2 +- .../archive/records/Table_V37.json | 2 +- .../archive/records/Table_V38.json | 2 +- .../archive/records/Table_V39.json | 2 +- .../archive/records/Table_V40.json | 2 +- .../archive/records/Table_V41.json | 2 +- .../archive/records/Table_V42.json | 2 +- .../archive/records/Table_V43.json | 2 +- .../archive/records/Table_V44.json | 2 +- .../archive/records/Table_V45.json | 2 +- .../archive/records/Table_V46.json | 2 +- .../archive/records/Table_V47.json | 2 +- .../archive/records/Table_V48.json | 2 +- .../archive/records/Table_V49.json | 2 +- .../archive/records/Table_V50.json | 2 +- .../archive/records/Table_V51.json | 2 +- .../archive/records/Table_V52.json | 2 +- .../archive/records/Table_V53.json | 2 +- .../archive/records/Table_V54.json | 2 +- .../archive/records/Table_V55.json | 2 +- .../archive/records/Table_V56.json | 2 +- .../archive/templates/{175.txt => 461.txt} | 7 +- .../archive/templates/{194.txt => 462.txt} | 7 +- .../archive/templates/{397.txt => 463.txt} | 9 +- .../archive/templates/{434.txt => 464.txt} | 10 +- .../archive/templates/{450.txt => 465.txt} | 8 +- 67 files changed, 147 insertions(+), 142 deletions(-) rename scripts/zeta5-certificates/archive/templates/{175.txt => 461.txt} (82%) rename scripts/zeta5-certificates/archive/templates/{194.txt => 462.txt} (82%) rename scripts/zeta5-certificates/archive/templates/{397.txt => 463.txt} (76%) rename scripts/zeta5-certificates/archive/templates/{434.txt => 464.txt} (74%) rename scripts/zeta5-certificates/archive/templates/{450.txt => 465.txt} (79%) diff --git a/scripts/zeta5-certificates/README.md b/scripts/zeta5-certificates/README.md index 2949eaf9b5..3afccae0b4 100644 --- a/scripts/zeta5-certificates/README.md +++ b/scripts/zeta5-certificates/README.md @@ -3,10 +3,10 @@ This archive reproduces all 118 generated modules in Lean Pool PR #540, including 114 U/V endpoint tables, the 684-interval assembly, and the inner/outer growth tables and inner sum. The manifest pins source commit -`154983b2d21f27207c078cccf2646715c58e0d3c`, upstream +`b00b44fc50beb974417bfdff2898aeedd83cfdde`, upstream [`mo271/Zeta5`](https://github.com/mo271/Zeta5/tree/f19a1960609f7d38e7b63fd2acb05e6f60a7b741), -and each output's SHA-256. The later removal of the unused paper normalizer does -not change any archived file. +and each output's SHA-256. The archive includes the import cleanup from the +final review; every mathematical certificate body is preserved. Use Python 3.13 or later. No third-party packages, network, Lean installation, floating-point arithmetic, or execution of archived code is needed for rendering: diff --git a/scripts/zeta5-certificates/archive/manifest.json b/scripts/zeta5-certificates/archive/manifest.json index 2e3e3fda6a..6490fa400f 100644 --- a/scripts/zeta5-certificates/archive/manifest.json +++ b/scripts/zeta5-certificates/archive/manifest.json @@ -1,5 +1,5 @@ { - "source_commit": "154983b2d21f27207c078cccf2646715c58e0d3c", + "source_commit": "b00b44fc50beb974417bfdff2898aeedd83cfdde", "upstream_commit": "f19a1960609f7d38e7b63fd2acb05e6f60a7b741", "license": "Apache-2.0", "files": { @@ -233,231 +233,231 @@ }, "LeanPool/Zeta5Irrational/Table/V00.lean": { "records": "Table_V00.json", - "sha256": "b4336809612f6ced8228af5f443b52304e26ff6a39a08af374ad883c740d3f6b" + "sha256": "4d37f2ddd1a1b932bbaf93e7243f71332ca885c4f9dd2d665ea40af78ea0b4c0" }, "LeanPool/Zeta5Irrational/Table/V01.lean": { "records": "Table_V01.json", - "sha256": "56d20df920c8f6c03c9e1f0b618bc2e37ac5920dfd7b281949c15f618487bf12" + "sha256": "74dbd11f69f7494ea7fd3032310c19e69ace1ce919972b2a89615573895d4161" }, "LeanPool/Zeta5Irrational/Table/V02.lean": { "records": "Table_V02.json", - "sha256": "e187d031dd20c510a33b5f137bc1930091b21f8eee5fb33122cfb821194a389c" + "sha256": "b3fd7317471b077808a43a6291708fa19d984f7fedf311d9056441bc1e4e25fe" }, "LeanPool/Zeta5Irrational/Table/V03.lean": { "records": "Table_V03.json", - "sha256": "b6a38cadfdeedbfc6213a72197d0d9cc2194ed0408f8c7a27bfba67f2fcba97b" + "sha256": "b513e9ba722ccab29550905254ba85167149eab529ce2654d1c0ca4ab427be7c" }, "LeanPool/Zeta5Irrational/Table/V04.lean": { "records": "Table_V04.json", - "sha256": "06bf879d3615ee3c19fbc9e15262f4073c7482a6205b8857305fe4f75fbb2a64" + "sha256": "9aa9ce0637cc0c715cb867687fe4c8346dfc63d54f61ef283c648c77aa180f0a" }, "LeanPool/Zeta5Irrational/Table/V05.lean": { "records": "Table_V05.json", - "sha256": 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diff --git a/scripts/zeta5-certificates/archive/templates/175.txt b/scripts/zeta5-certificates/archive/templates/461.txt similarity index 82% rename from scripts/zeta5-certificates/archive/templates/175.txt rename to scripts/zeta5-certificates/archive/templates/461.txt index 0cd5b82197..69477afa1a 100644 --- a/scripts/zeta5-certificates/archive/templates/175.txt +++ b/scripts/zeta5-certificates/archive/templates/461.txt @@ -12,9 +12,10 @@ import LeanPool.Zeta{{integer}}Irrational.PotentialV import LeanPool.Zeta{{integer}}Irrational.TableCheck import Mathlib.Analysis.Complex.ExponentialBounds import Mathlib.Analysis.Real.Pi.Bounds -import Mathlib.NumberTheory.Chebyshev -import Mathlib.Tactic.NormNum.Prime -import Mathlib.Tactic.NormNum.RealSqrt +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.NormNum.Ineq +import Mathlib.Tactic.NormNum.Inv +import Mathlib.Tactic.NormNum.Pow /-! # V{{integer}}: certified bounds for the zeta({{integer}}) proof -/ diff --git a/scripts/zeta5-certificates/archive/templates/194.txt b/scripts/zeta5-certificates/archive/templates/462.txt similarity index 82% rename from scripts/zeta5-certificates/archive/templates/194.txt rename to scripts/zeta5-certificates/archive/templates/462.txt index 450b6e1b0a..d10e748e62 100644 --- a/scripts/zeta5-certificates/archive/templates/194.txt +++ b/scripts/zeta5-certificates/archive/templates/462.txt @@ -12,9 +12,10 @@ import LeanPool.Zeta{{integer}}Irrational.PotentialV import LeanPool.Zeta{{integer}}Irrational.TableCheck import Mathlib.Analysis.Complex.ExponentialBounds import Mathlib.Analysis.Real.Pi.Bounds -import Mathlib.NumberTheory.Chebyshev -import Mathlib.Tactic.NormNum.Prime -import Mathlib.Tactic.NormNum.RealSqrt +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.NormNum.Ineq +import Mathlib.Tactic.NormNum.Inv +import Mathlib.Tactic.NormNum.Pow /-! # V{{integer}}: certified bounds for the zeta({{integer}}) proof -/ diff --git a/scripts/zeta5-certificates/archive/templates/397.txt b/scripts/zeta5-certificates/archive/templates/463.txt similarity index 76% rename from scripts/zeta5-certificates/archive/templates/397.txt rename to scripts/zeta5-certificates/archive/templates/463.txt index 56de61fea6..2716bb91d7 100644 --- a/scripts/zeta5-certificates/archive/templates/397.txt +++ b/scripts/zeta5-certificates/archive/templates/463.txt @@ -10,10 +10,11 @@ import LeanPool.Zeta{{integer}}Irrational.Growth.TableGen public import LeanPool.Zeta{{integer}}Irrational.Growth.TablePrime import Mathlib.Tactic.ReduceModChar import Mathlib.Tactic.Polynomial.Basic -import Mathlib.Tactic.NormNum.RealSqrt -import Mathlib.NumberTheory.Chebyshev -import Mathlib.Tactic.NormNum.Prime -import Mathlib.Analysis.SpecialFunctions.Gamma.Basic +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.NormNum.Ineq +import Mathlib.Tactic.NormNum.Inv +import Mathlib.Tactic.NormNum.Pow +import Mathlib.Tactic.Ring.Basic /-!Generated by `gen_lean_inner.py`: the pieces of `Ê_in` on `[{{integer}}, {{integer}}]`. -/ diff --git a/scripts/zeta5-certificates/archive/templates/434.txt b/scripts/zeta5-certificates/archive/templates/464.txt similarity index 74% rename from scripts/zeta5-certificates/archive/templates/434.txt rename to scripts/zeta5-certificates/archive/templates/464.txt index 6a72424835..4b31132cc1 100644 --- a/scripts/zeta5-certificates/archive/templates/434.txt +++ b/scripts/zeta5-certificates/archive/templates/464.txt @@ -10,11 +10,13 @@ import LeanPool.Zeta{{integer}}Irrational.Growth.TableGen public import LeanPool.Zeta{{integer}}Irrational.Growth.OuterPrime import Mathlib.Tactic.ReduceModChar import Mathlib.Tactic.Polynomial.Basic -import Mathlib.Tactic.NormNum.RealSqrt -import Mathlib.NumberTheory.Chebyshev import Mathlib.Data.Rat.Star -import Mathlib.Tactic.NormNum.Prime -import Mathlib.Analysis.SpecialFunctions.Gamma.Basic +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.NormNum.Ineq +import Mathlib.Tactic.NormNum.Inv +import Mathlib.Tactic.NormNum.Pow +import Mathlib.Tactic.Ring.Basic +import Mathlib.Tactic.Ring.RingNF /-!Generated by `gen_lean_outer.py`: the table of `Ê_out` on `[{{integer}}/{{integer}}, {{integer}}]`. -/ diff --git a/scripts/zeta5-certificates/archive/templates/450.txt b/scripts/zeta5-certificates/archive/templates/465.txt similarity index 79% rename from scripts/zeta5-certificates/archive/templates/450.txt rename to scripts/zeta5-certificates/archive/templates/465.txt index 37afb2c7e9..20f52be885 100644 --- a/scripts/zeta5-certificates/archive/templates/450.txt +++ b/scripts/zeta5-certificates/archive/templates/465.txt @@ -9,11 +9,11 @@ module import LeanPool.Zeta{{integer}}Irrational.Growth.InnerTable import LeanPool.Zeta{{integer}}Irrational.Growth.TableGen public import LeanPool.Zeta{{integer}}Irrational.Growth.TablePrime -import Mathlib.Tactic.NormNum.RealSqrt -import Mathlib.NumberTheory.Chebyshev -import Mathlib.Tactic.NormNum.Prime -import Mathlib.Analysis.SpecialFunctions.Gamma.Basic import Mathlib.Data.Int.Star +import Mathlib.Tactic.Linarith +import Mathlib.Tactic.NormNum.Ineq +import Mathlib.Tactic.NormNum.Inv +import Mathlib.Tactic.NormNum.Pow /-!Generated by `gen_inner_sum.py`: the table of `Ê_in` on `[{{integer}}, {{integer}}]`. -/ From ce9b3854facae6f015c33f5dcf546c932d7f7e45 Mon Sep 17 00:00:00 2001 From: Vasily Ilin Date: Sun, 27 Sep 2026 14:43:53 +0000 Subject: [PATCH 6/6] Link the certificate reproduction guide from the README --- README.md | 3 +++ 1 file changed, 3 insertions(+) diff --git a/README.md b/README.md index 77c31441d1..6505c77885 100644 --- a/README.md +++ b/README.md @@ -40,6 +40,9 @@ make setup # pull Mathlib oleans, build the whole pool (~1.5h), install Pytho To work on a single project you don't need the whole pool built — see the [fast per-project build](CONTRIBUTING.md#dev-setup) in `CONTRIBUTING.md`. +To regenerate the preserved Zeta5 numerical certificates, see the +[certificate reproduction guide](scripts/zeta5-certificates/README.md). + ### Challenge mode [`Challenge/`](Challenge/) is the other half of the pool: open *statements* rather than finished proofs. A challenge is a theorem written in Mathlib vocabulary and left as `sorry`, registered in [`Challenge/challenges.yml`](Challenge/challenges.yml) alongside the English statement it is supposed to say. It is the only place `sorry` is allowed, and only for the declarations the registry lists — everything else in the file must be closed, and every other gate still applies.