Import Champernowne normality in every base

LeanPool.lean

@@ -1435,6 +1435,15 @@ public import LeanPool.CenteredMaximal.Lattice.Witness
 public import LeanPool.CenteredMaximal.Numerics
 public import LeanPool.CenteredMaximal.Statement
 public import LeanPool.CenteredMaximal.UpperBound
+public import LeanPool.Champernowne
+public import LeanPool.Champernowne.Asymptotics
+public import LeanPool.Champernowne.Count
+public import LeanPool.Champernowne.CountExtras
+public import LeanPool.Champernowne.Defs
+public import LeanPool.Champernowne.DigitCount
+public import LeanPool.Champernowne.Main
+public import LeanPool.Champernowne.Positions
+public import LeanPool.Champernowne.Prefix
 public import LeanPool.ChannelCapacity
 public import LeanPool.ChannelCapacity.Basic
 public import LeanPool.ChannelCapacity.Capacity

LeanPool/Champernowne.lean

@@ -0,0 +1,20 @@
+/-
+Copyright (c) 2026 Arthur Champernowne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Arthur Champernowne
+-/
+module
+
+public import LeanPool.Champernowne.Main
+public import LeanPool.Champernowne.CountExtras
+
+/-!
+# Champernowne normality in every base
+
+Source: url:https://github.com/OldMathematician/ChampernowneNormality
+Authors: Arthur Champernowne
+Status: verified
+Main declarations: `Champernowne.champernowne_normal`
+Tags: normal-numbers, digit-expansions, combinatorics-on-words, asymptotic-frequency
+MSC: 11K16, 68R15
+-/

LeanPool/Champernowne/Asymptotics.lean

@@ -0,0 +1,268 @@
+/-
+Copyright (c) 2026 Arthur Champernowne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Arthur Champernowne
+-/
+module
+
+public import LeanPool.Champernowne.Positions
+public import Mathlib.Analysis.Asymptotics.Lemmas
+public import Mathlib.Analysis.SpecificLimits.Basic
+
+/-!
+# Asymptotic frequency of digit blocks
+
+`champM b n` is the digit length of the number straddling position `n`.
+The natural-number prefix bounds give an error that is little-o of `n`, and
+therefore the frequency of each nonempty length-`k` word tends to `b⁻ᵏ`.
+-/
+
+@[expose] public section
+
+namespace Champernowne
+
+open Filter Asymptotics Topology
+
+/-- Digit length of the straddling number `champIndex b n + 1`. -/
+def champM (b n : ℕ) : ℕ := Nat.log b (champIndex b n + 1) + 1
+
+theorem champM_pos (b n : ℕ) : 0 < champM b n := Nat.succ_pos _
+
+theorem pow_champM_le (b n : ℕ) :
+    b ^ (champM b n - 1) ≤ champIndex b n + 1 := by
+  rw [champM, Nat.add_sub_cancel]
+  exact Nat.pow_log_le_self b (Nat.succ_ne_zero _)
+
+theorem le_pow_champM {b : ℕ} (hb : 1 < b) (n : ℕ) :
+    champIndex b n + 1 ≤ b ^ champM b n :=
+  le_of_lt (Nat.lt_pow_succ_log_self hb _)
+
+/-- Length lower bound: `n` dominates the second-to-top cohort. -/
+theorem cohort_le_n {b : ℕ} (hb : 1 < b) (n : ℕ) :
+    (b - 1) * b ^ (champM b n - 2) * (champM b n - 1) ≤ n := by
+  rcases Nat.lt_or_ge (champM b n) 2 with h2 | h2
+  · rw [show champM b n - 1 = 0 by omega, Nat.mul_zero]
+    exact Nat.zero_le n
+  · have hN1 := pow_champM_le b n
+    have hlen := length_champBlocks_champIndex_le b n
+    rw [length_champBlocks] at hlen
+    have hsub : Finset.Ico 1 (b ^ (champM b n - 1))
+        ⊆ Finset.Ico 1 (champIndex b n + 1) :=
+      Finset.Ico_subset_Ico_right hN1
+    have hmono := Finset.sum_le_sum_of_subset
+      (f := fun k => (bigDigits b k).length) hsub
+    rw [sum_Ico_one_pow hb] at hmono
+    have hsingle : (∑ k ∈ Finset.Ico (b ^ (champM b n - 1 - 1))
+          (b ^ (champM b n - 1)), (bigDigits b k).length)
+        ≤ ∑ m ∈ Finset.Icc 1 (champM b n - 1),
+            ∑ k ∈ Finset.Ico (b ^ (m - 1)) (b ^ m),
+              (bigDigits b k).length :=
+      Finset.single_le_sum
+        (f := fun m => ∑ k ∈ Finset.Ico (b ^ (m - 1)) (b ^ m),
+          (bigDigits b k).length)
+        (fun _ _ => Nat.zero_le _)
+        (Finset.mem_Icc.mpr ⟨by omega, le_rfl⟩)
+    rw [sum_length_cohort hb (champM b n - 1) (by omega),
+      show champM b n - 1 - 1 = champM b n - 2 by omega] at hsingle
+    omega
+
+theorem tendsto_champIndex_atTop (b : ℕ) :
+    Tendsto (champIndex b) atTop atTop := by
+  rw [Filter.tendsto_atTop]
+  intro K
+  filter_upwards [Filter.eventually_ge_atTop ((champBlocks b K).length)]
+    with n hn
+  exact Nat.le_findGreatest (le_trans (le_length_champBlocks b K) hn) hn
+
+theorem tendsto_champM_atTop {b : ℕ} (hb : 1 < b) :
+    Tendsto (champM b) atTop atTop := by
+  rw [Filter.tendsto_atTop]
+  intro K
+  have h10 : Tendsto (fun n => champIndex b n + 1) atTop atTop :=
+    tendsto_atTop_mono (fun n => Nat.le_succ (champIndex b n))
+      (tendsto_champIndex_atTop b)
+  filter_upwards [h10.eventually_ge_atTop (b ^ K)] with n hn
+  have hn' : b ^ K ≤ champIndex b n + 1 := hn
+  have hlog := (Nat.le_log_iff_pow_le hb
+    (show champIndex b n + 1 ≠ 0 by omega)).mpr hn'
+  change K ≤ Nat.log b (champIndex b n + 1) + 1
+  omega
+
+/-! ### General-`w` packaging
+
+Instantiates the `Positions.lean` straddle transfer at `M := champM b n`.
+`w.length ≤ champM b n` holds only eventually, since `champM b n → ∞`
+(`tendsto_champM_atTop`), so the transfer itself is wrapped in `∀ᶠ`. -/
+
+theorem base_pow_count_le {b : ℕ} (hb : 1 < b) {w : List ℕ}
+    (hw : ∀ d ∈ w, d < b) (hwne : w ≠ []) :
+    ∀ᶠ n in atTop, b ^ w.length * countOccurrences w (champPrefix b n)
+      ≤ n + 7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n := by
+  filter_upwards [(tendsto_champM_atTop hb).eventually_ge_atTop w.length] with n hn
+  exact base_pow_countOccurrences_champPrefix_le hb n (champM b n) hw hwne
+    (champM_pos b n) hn (pow_champM_le b n) (le_pow_champM hb n)
+
+theorem le_base_pow_count {b : ℕ} (hb : 1 < b) {w : List ℕ}
+    (hw : ∀ d ∈ w, d < b) (hwne : w ≠ []) :
+    ∀ᶠ n in atTop, n ≤ b ^ w.length * countOccurrences w (champPrefix b n)
+      + 7 * (w.length + 1) * b ^ w.length * b ^ champM b n := by
+  filter_upwards [(tendsto_champM_atTop hb).eventually_ge_atTop w.length] with n hn
+  exact le_base_pow_countOccurrences_champPrefix hb n (champM b n) hw hwne
+    (champM_pos b n) hn (pow_champM_le b n) (le_pow_champM hb n)
+
+/-- The error dominates its own budget: `E(n)·(b-1)(M−1) ≤ E'·n`. -/
+theorem err_mul_le_pow {b : ℕ} (hb : 1 < b) (n : ℕ) (w : List ℕ) :
+    7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n * ((b - 1) * (champM b n - 1))
+      ≤ (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ 2) * n := by
+  rcases Nat.lt_or_ge (champM b n) 2 with h2 | h2
+  · rw [show champM b n - 1 = 0 by omega]
+    simp
+  · have hD := cohort_le_n hb n
+    have hpow : b ^ champM b n = b ^ 2 * b ^ (champM b n - 2) := by
+      rw [← pow_add]; congr 1; omega
+    calc 7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n
+        * ((b - 1) * (champM b n - 1))
+        = (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ 2)
+          * ((b - 1) * b ^ (champM b n - 2) * (champM b n - 1)) := by
+          rw [hpow]; ring
+    _ ≤ (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ 2) * n := Nat.mul_le_mul_left _ hD
+
+/-- Subtraction-free form for `M ≥ 2`. -/
+theorem err_mul_le_pow' {b : ℕ} (hb : 1 < b) (n : ℕ) (w : List ℕ) (h2 : 2 ≤ champM b n) :
+    7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n * ((b - 1) * champM b n)
+      ≤ 2 * (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ 2) * n := by
+  have hF := err_mul_le_pow hb n w
+  have h9 : (b - 1) * champM b n ≤ 2 * ((b - 1) * (champM b n - 1)) := by
+    calc (b - 1) * champM b n ≤ (b - 1) * (2 * (champM b n - 1)) :=
+        Nat.mul_le_mul_left _ (by omega)
+    _ = 2 * ((b - 1) * (champM b n - 1)) := by ring
+  calc 7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n * ((b - 1) * champM b n)
+      ≤ 7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n
+        * (2 * ((b - 1) * (champM b n - 1))) := Nat.mul_le_mul_left _ h9
+  _ = 2 * (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n
+        * ((b - 1) * (champM b n - 1))) := by ring
+  _ ≤ 2 * ((7 * (w.length + 1) * b ^ (2 * w.length) * b ^ 2) * n) :=
+      Nat.mul_le_mul_left _ hF
+  _ = 2 * (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ 2) * n := by ring
+
+/-- The error sequence is negligible relative to `n`. -/
+theorem tendsto_err_div_pow {b : ℕ} (hb : 1 < b) (w : List ℕ) :
+    Tendsto (fun n => (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n : ℝ) / n)
+      atTop (𝓝 0) := by
+  have hbR : (1 : ℝ) ≤ (b : ℝ) - 1 := by
+    have : (2 : ℝ) ≤ b := by exact_mod_cast hb
+    linarith
+  have hden : Tendsto (fun n => ((b : ℝ) - 1) * (champM b n : ℝ))
+      atTop atTop :=
+    (tendsto_const_mul_atTop_of_pos (by linarith)).mpr
+      (tendsto_natCast_atTop_atTop.comp (tendsto_champM_atTop hb))
+  have hg : Tendsto (fun n =>
+      (2 * (7 * ((w.length : ℝ) + 1) * (b : ℝ) ^ (2 * w.length) * (b : ℝ) ^ 2))
+      / (((b : ℝ) - 1) * (champM b n : ℝ))) atTop (𝓝 0) :=
+    Filter.Tendsto.div_atTop tendsto_const_nhds hden
+  refine squeeze_zero' ?_ ?_ hg
+  · filter_upwards [Filter.eventually_ge_atTop 1] with n hn
+    have h0 : (0 : ℝ) < n := by exact_mod_cast hn
+    positivity
+  · filter_upwards [(tendsto_champM_atTop hb).eventually_ge_atTop 2,
+      Filter.eventually_ge_atTop 1] with n hM2 hn1
+    have h0 : (0 : ℝ) < n := by exact_mod_cast hn1
+    have hM0 : (0 : ℝ) < ((b : ℝ) - 1) * (champM b n : ℝ) := by
+      have hMpos : (0 : ℝ) < (champM b n : ℝ) := by
+        exact_mod_cast champM_pos b n
+      nlinarith
+    rw [div_le_div_iff₀ h0 hM0]
+    have := err_mul_le_pow' hb n w hM2
+    have hcast2 : ((7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n
+        * ((b - 1) * champM b n) : ℕ) : ℝ)
+        = (7 * ((w.length : ℝ) + 1) * (b : ℝ) ^ (2 * w.length) * (b : ℝ) ^ champM b n)
+          * (((b : ℝ) - 1) * (champM b n : ℝ)) := by
+      have hb1 : ((b - 1 : ℕ) : ℝ) = (b : ℝ) - 1 := by
+        rw [Nat.cast_sub (by omega : 1 ≤ b), Nat.cast_one]
+      push_cast [hb1]
+      ring
+    have hcast3 : ((2 * (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ 2) * n : ℕ) : ℝ)
+        = 2 * (7 * ((w.length : ℝ) + 1) * (b : ℝ) ^ (2 * w.length) * (b : ℝ) ^ 2) * n := by
+      push_cast; ring
+    rw [← hcast2, ← hcast3]
+    exact_mod_cast this
+
+/-- `countOccurrences w (champPrefix b n) −
+n/b^k = o(n)`. -/
+theorem countOccurrences_champPrefix_sub_isLittleO {b : ℕ} (hb : 1 < b) {w : List ℕ}
+    (hw : ∀ d ∈ w, d < b) (hwne : w ≠ []) :
+    (fun n => (countOccurrences w (champPrefix b n) : ℝ) - n / (b : ℝ) ^ w.length)
+      =o[atTop] (fun n => (n : ℝ)) := by
+  have hbR : (0 : ℝ) < (b : ℝ) ^ w.length := by positivity
+  have hb1R : (1 : ℝ) ≤ (b : ℝ) ^ w.length := by
+    have : (1 : ℝ) ≤ (b : ℝ) := by exact_mod_cast le_of_lt hb
+    exact one_le_pow₀ this
+  have hEo : (fun n => (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n : ℝ))
+      =o[atTop] (fun n => (n : ℝ)) := by
+    rw [Asymptotics.isLittleO_iff_tendsto']
+    · exact tendsto_err_div_pow hb w
+    · filter_upwards [Filter.eventually_ge_atTop 1] with n hn h0
+      have : (0 : ℝ) < n := by exact_mod_cast hn
+      exact absurd h0 (ne_of_gt this)
+  refine Asymptotics.IsBigO.trans_isLittleO ?_ hEo
+  rw [Asymptotics.isBigO_iff]
+  refine ⟨1, ?_⟩
+  filter_upwards [base_pow_count_le hb hw hwne, le_base_pow_count hb hw hwne]
+    with n h1 h2
+  -- Widen the lower transfer's error b^k·… to the upper's b^(2k)·… (ℕ level).
+  have h2' : n ≤ b ^ w.length * countOccurrences w (champPrefix b n)
+      + 7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n := by
+    have hpow : b ^ w.length ≤ b ^ (2 * w.length) :=
+      Nat.pow_le_pow_right (by omega) (by omega)
+    have hmul := Nat.mul_le_mul_right (b ^ champM b n)
+      (Nat.mul_le_mul_left (7 * (w.length + 1)) hpow)
+    omega
+  have e1 : (b : ℝ) ^ w.length * countOccurrences w (champPrefix b n)
+      ≤ n + 7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n := by exact_mod_cast h1
+  have e2 : (n : ℝ) ≤ (b : ℝ) ^ w.length * countOccurrences w (champPrefix b n)
+      + 7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n := by exact_mod_cast h2'
+  have hEpos : (0 : ℝ) ≤ (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n : ℝ) :=
+    by positivity
+  have key : |(b : ℝ) ^ w.length * countOccurrences w (champPrefix b n) - n|
+      ≤ (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n : ℝ) := by
+    rw [abs_le]
+    constructor
+    · linarith
+    · linarith
+  rw [Real.norm_eq_abs, Real.norm_eq_abs, one_mul, abs_of_nonneg hEpos,
+    show (countOccurrences w (champPrefix b n) : ℝ) - n / (b : ℝ) ^ w.length
+      = ((b : ℝ) ^ w.length * countOccurrences w (champPrefix b n) - n) / (b : ℝ) ^ w.length by
+        field_simp,
+    abs_div, abs_of_pos hbR]
+  calc |(b : ℝ) ^ w.length * countOccurrences w (champPrefix b n) - n| / (b : ℝ) ^ w.length
+      ≤ (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n : ℝ) / (b : ℝ) ^ w.length := by
+        exact div_le_div_of_nonneg_right key hbR.le
+  _ ≤ (7 * (w.length + 1) * b ^ (2 * w.length) * b ^ champM b n : ℝ) := by
+        rw [div_le_iff₀ hbR]
+        exact le_mul_of_one_le_right hEpos hb1R
+
+/-- Every nonempty block `w` (digits `< b`)
+occurs in the base-`b` Champernowne stream with asymptotic frequency
+`b⁻ᵏ`. -/
+theorem tendsto_countOccurrences_champPrefix_div {b : ℕ} (hb : 1 < b) {w : List ℕ}
+    (hw : ∀ d ∈ w, d < b) (hwne : w ≠ []) :
+    Tendsto (fun n => (countOccurrences w (champPrefix b n) : ℝ) / n)
+      atTop (𝓝 (((b : ℝ) ^ w.length)⁻¹)) := by
+  have hb0 : ((b : ℝ) ^ w.length) ≠ 0 := by positivity
+  have ho := countOccurrences_champPrefix_sub_isLittleO hb hw hwne
+  rw [Asymptotics.isLittleO_iff_tendsto'
+    (by
+      filter_upwards [Filter.eventually_ge_atTop 1] with n hn h0
+      have : (0 : ℝ) < n := by exact_mod_cast hn
+      exact absurd h0 (ne_of_gt this))] at ho
+  have hsum := ho.add_const ((1 : ℝ) / (b : ℝ) ^ w.length)
+  rw [zero_add] at hsum
+  rw [← one_div]
+  refine Tendsto.congr' ?_ hsum
+  filter_upwards [Filter.eventually_ge_atTop 1] with n hn
+  have h0 : (0 : ℝ) < n := by exact_mod_cast hn
+  rw [one_div]
+  field_simp
+  ring
+
+end Champernowne