| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.GlobalProfile |
| import OAI.Probability.Ballisticity.Estimates.StoppedSplice |
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| namespace OAI |
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| section |
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|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal Classical |
| namespace DirectionalTransience |
|
|
| lemma stoppedRowsEvent_mono {d : ℕ} (S : ℕ → Set (Lattice d)) (hS : Monotone S) |
| (τ σ : Environment d → ℕ) (hτσ : ∀ ω, τ ω≤σ ω) |
| (hσ : ∀ n, MeasurableSet[rowSigma (S n)] {ω | σ ω=n}) |
| (A : Set (Environment d)) (hA : StoppedRowsEvent S τ A) : StoppedRowsEvent S σ A := by |
| intro n |
| have he : A∩{ω | σ ω=n}= |
| (⋃ k : Fin (n+1), A∩{ω | τ ω=k.val})∩{ω | σ ω=n} := by |
| ext ω |
| constructor |
| · rintro ⟨ha,hs⟩ |
| refine ⟨Set.mem_iUnion.mpr ⟨⟨τ ω,by have hh := hτσ ω; change σ ω = n at hs; omega⟩,ha,rfl⟩,hs⟩ |
| · rintro ⟨hk,hs⟩ |
| obtain ⟨k,ha,_⟩ := Set.mem_iUnion.mp hk |
| exact ⟨ha,hs⟩ |
| rw [he] |
| have hAk (k : Fin (n+1)) : MeasurableSet[rowSigma (S n)] (A∩{ω | τ ω=k.val}) := by |
| have hk : k.val≤n := Nat.le_of_lt_succ k.isLt |
| exact (rowSigma_mono (hS hk)) _ (hA k.val) |
| exact (MeasurableSet.iUnion hAk).inter (hσ n) |
|
|
| lemma stoppedRowsEvent_measurable {d : ℕ} (S : ℕ → Set (Lattice d)) (τ : Environment d → ℕ) |
| (A : Set (Environment d)) (hA : StoppedRowsEvent S τ A) : MeasurableSet A := by |
| have he : A=⋃ n, A∩{ω | τ ω=n} := by ext ω; simp |
| rw [he] |
| exact MeasurableSet.iUnion fun n => (rowSigma_le _) _ (hA n) |
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| theorem stopped_fresh_joint_map {d : ℕ} {D F : Type*} [MeasurableSpace D] [MeasurableSpace F] |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (S : ℕ → Set (Lattice d)) (τ : Environment d → ℕ) |
| (hτ : ∀ n, MeasurableSet[rowSigma (S n)] {ω | τ ω=n}) |
| (U : Environment d → D) (hUm : Measurable U) |
| (hU : ∀ B, MeasurableSet B → StoppedRowsEvent S τ (U ⁻¹' B)) |
| (V : ℕ → Environment d → F) |
| (hV : ∀ n, @Measurable _ _ (rowSigma (S n)ᶜ) _ (V n)) |
| (R : Measure F) [IsProbabilityMeasure R] |
| (hR : ∀ n, (environmentLaw ν).map (V n)=R) : |
| (environmentLaw ν).map (fun ω => (U ω,V (τ ω) ω))=((environmentLaw ν).map U).prod R := by |
| have hVm (n : ℕ) : Measurable (V n) := (hV n).mono (rowSigma_le _) le_rfl |
| have hVm' : Measurable (fun p : Environment d×ℕ => V p.2 p.1) := |
| measurable_from_prod_countable_left hVm |
| have hmap : Measurable (fun ω => (U ω,V (τ ω) ω)) := |
| hUm.prodMk (hVm'.comp (measurable_id.prodMk (measurable_of_stop_events S τ hτ))) |
| apply Measure.ext_prod |
| intro A B hA hB |
| rw [Measure.map_apply hmap (hA.prod hB),Measure.prod_prod,Measure.map_apply hUm hA] |
| let E (n : ℕ) := (U ⁻¹' A)∩{ω | τ ω=n} |
| let K (n : ℕ) := E n∩(V n ⁻¹' B) |
| have hE (n : ℕ) : MeasurableSet[rowSigma (S n)] (E n) := hU A hA n |
| have hEm (n : ℕ) : MeasurableSet (E n) := (rowSigma_le _) _ (hE n) |
| have hK (n : ℕ) : MeasurableSet (K n) := (hEm n).inter (hB.preimage (hVm n)) |
| have hEd : Pairwise (Function.onFun Disjoint E) := by |
| intro n m hnm |
| exact Set.disjoint_left.mpr fun ω hn hm => hnm (hn.2.symm.trans hm.2) |
| have hKd : Pairwise (Function.onFun Disjoint K) := by |
| intro n m hnm |
| exact (hEd hnm).mono Set.inter_subset_left Set.inter_subset_left |
| have he : (fun ω => (U ω,V (τ ω) ω)) ⁻¹' (A ×ˢ B)=⋃ n, K n := by |
| ext ω |
| constructor |
| · rintro ⟨ha,hb⟩ |
| exact Set.mem_iUnion.mpr ⟨τ ω,⟨ha,rfl⟩,hb⟩ |
| · intro h |
| obtain ⟨n,⟨ha,hn⟩,hb⟩ := Set.mem_iUnion.mp h |
| refine ⟨ha,?_⟩ |
| change τ ω=n at hn |
| change V n ω∈B at hb |
| simpa only [hn] using hb |
| rw [he,measure_iUnion hKd hK] |
| have hfactor (n : ℕ) : environmentLaw ν (K n)=environmentLaw ν (E n)*R B := by |
| have hind := (Indep_iff _ _ _).mp (environment_indep_rows ν (S:=S n) |
| (T:=(S n)ᶜ) disjoint_compl_right) (E n) (V n ⁻¹' B) (hE n) ((hV n) hB) |
| rw [←Measure.map_apply (hVm n) hB,hR n] at hind |
| exact hind |
| simp_rw [hfactor] |
| rw [ENNReal.tsum_mul_right,←measure_iUnion hEd hEm] |
| have he' : (⋃ n, E n)=U ⁻¹' A := by ext ω; simp [E] |
| rw [he'] |
|
|
| noncomputable def upperField {d : ℕ} (e : Direction d) (n : ℕ) (ω : Environment d) : |
| (ℕ×HorizontalSpace e) → Row d := fun p => ω (horizontalLift e (n+p.1) p.2) |
|
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| lemma upperField_index_injective {d : ℕ} (e : Direction d) (n : ℕ) : |
| Function.Injective (fun p : ℕ×HorizontalSpace e => horizontalLift e (n+p.1) p.2) := by |
| intro p q h |
| have hh := congrArg (signedHeight e) h |
| have hz := congrArg (horizontalProjection e) h |
| rw [signedHeight_horizontalLift,signedHeight_horizontalLift] at hh |
| rw [horizontalProjection_lift,horizontalProjection_lift] at hz |
| exact Prod.ext (by exact_mod_cast (add_left_cancel hh)) hz |
|
|
| lemma upperField_measurable_rows {d : ℕ} (e : Direction d) (n : ℕ) : |
| @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) n)ᶜ) _ (upperField e n) := by |
| apply @Measurable.of_eval (Environment d) (ℕ×HorizontalSpace e) (fun _ => Row d) |
| (rowSigma (BelowHeight (realPosition (step e)) n)ᶜ) (fun _ => inferInstance) |
| intro p |
| apply measurable_row_on |
| change ¬ dot (realPosition (horizontalLift e (n+p.1) p.2)) (realPosition (step e))<n |
| rw [signedHeight_projection,signedHeight_horizontalLift] |
| push_cast |
| linarith [Nat.cast_nonneg (α:=ℝ) p.1] |
|
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| lemma upperField_map {d : ℕ} (e : Direction d) (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (n : ℕ) : (environmentLaw ν).map (upperField e n)= |
| Measure.infinitePi (fun _ : ℕ×HorizontalSpace e => ν) := |
| Measure.map_infinitePi_infinitePi_of_inj (upperField_index_injective e n) |
|
|
| end DirectionalTransience |
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| end |
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| end OAI |
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