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Merged
RemyDegenne merged 1 commit into from
Oct 8, 2025
Merged

Review Traj.lean #29

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24 changes: 11 additions & 13 deletions LeanBandits/ForMathlib/Traj.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -49,32 +49,30 @@ lemma traj_map_eq_kernel {a : ℕ} : (traj κ a).map (fun x ↦ x (a + 1)) = κ
have hp : g ∘ IicProdIoc a (a + 1) = (piSingleton a).symm ∘ Prod.snd := by
ext
simp [g, _root_.IicProdIoc, piSingleton]
rw [hf, map_comp_right, traj_map_frestrictLe, partialTraj_succ_self, ← map_comp_right, hp,
map_comp_right, ← snd_eq, snd_prod, ← map_comp_right]
all_goals measurability
rw [hf, map_comp_right _ (by fun_prop) (by fun_prop), traj_map_frestrictLe, partialTraj_succ_self,
← map_comp_right _ (by fun_prop) (by fun_prop), hp,
map_comp_right _ (by fun_prop) (by fun_prop), ← snd_eq, snd_prod,
← map_comp_right _ (by fun_prop) (by fun_prop)]
simp

-- Probability/Kernel/IonescuTulcea/Traj.lean
lemma partialTraj_compProd_kernel_eq_traj_map {a : ℕ} {x₀ : Π n : Iic 0, X n} :
(partialTraj κ 0 a x₀) ⊗ₘ (κ a) = (traj κ 0 x₀).map (fun x ↦ (frestrictLe a x, x (a + 1))) := by
set f := fun x ↦ (frestrictLe a x, x (a + 1))
set g := fun x ↦ (frestrictLe a x, x)
have hf : f = (Prod.map id (fun x ↦ x (a + 1))) ∘ g := rfl
rw [hf, ← Measure.map_map, ← partialTraj_compProd_traj, ← MeasureTheory.Measure.compProd_map,
traj_map_eq_kernel]
all_goals measurability
rw [hf, ← Measure.map_map (by fun_prop) (by fun_prop), ← partialTraj_compProd_traj zero_le',
← MeasureTheory.Measure.compProd_map (by fun_prop), traj_map_eq_kernel]

-- (Extract kernel lemmas from rewrites?) Probability/Kernel/IonescuTulcea/Traj.lean
lemma trajMeasure_map_frestrictLe_compProd_kernel_eq_trajMeasure_map {a : ℕ} :
(trajMeasure μ₀ κ).map (frestrictLe a) ⊗ₘ κ a =
(trajMeasure μ₀ κ).map (fun x ↦ (frestrictLe a x, x (a + 1))) := by
rw [Measure.compProd_eq_comp_prod, trajMeasure, Measure.map_comp, traj_map_frestrictLe,
Measure.comp_assoc, Measure.map_comp]
any_goals fun_prop
congr
ext1 x₀
rw [comp_apply, ← Measure.compProd_eq_comp_prod, map_apply,
rw [Measure.compProd_eq_comp_prod, trajMeasure, Measure.map_comp _ _ (by fun_prop),
traj_map_frestrictLe, Measure.comp_assoc, Measure.map_comp _ _ (by fun_prop)]
congr with x₀
rw [comp_apply, ← Measure.compProd_eq_comp_prod, map_apply _ (by fun_prop),
partialTraj_compProd_kernel_eq_traj_map]
fun_prop

-- Probability/Kernel/IonescuTulcea/Traj.lean
lemma condDistrib_trajMeasure_ae_eq_kernel {a : ℕ}
Expand Down