| import Mathlib |
| import OAI.Probability.Ballisticity.Stationary.ArrayMap |
| import OAI.Probability.Ballisticity.Walk.UpperKernel |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal NNReal Classical |
| namespace DirectionalTransience |
|
|
| lemma hitKernel_translation {d : ℕ} (ω : Environment d) (z x : Lattice d) |
| (S T U : Set (Lattice d)) : |
| hitKernel ((fun y => z+y) ⁻¹' S) ((fun y => z+y) ⁻¹' T) ((fun y => ω (z+y)),x) U = |
| hitKernel S T (ω,z+x) ((fun y => y-z) ⁻¹' U) := by |
| simp only [hitKernel_apply] |
| apply tsum_congr |
| intro n |
| rw [←quenched_translation ω z x, |
| Measure.map_apply (by fun_prop) (measurableSet_hitAt _ _ _)] |
| congr 1 |
| ext X |
| simp only [Set.mem_preimage,HitAt,Set.mem_ofPred_eq,Set.mem_inter_iff] |
| simp only [add_sub_cancel] |
|
|
| lemma strip_translation {d : ℕ} (ℓ : Vector d) (z x : Lattice d) (H : ℝ) : |
| (fun y => z+y) ⁻¹' Strip ℓ (z+x) H=Strip ℓ x H := by |
| ext y |
| simp only [Set.mem_preimage,Strip,Set.mem_ofPred_eq,dot_realPosition_add] |
| constructor <;> rintro ⟨h₁,h₂⟩ <;> constructor <;> linarith |
|
|
| lemma upper_translation {d : ℕ} (ℓ : Vector d) (z x : Lattice d) (H : ℝ) : |
| (fun y => z+y) ⁻¹' Upper ℓ (z+x) H=Upper ℓ x H := by |
| ext y |
| simp only [Set.mem_preimage,Upper,Set.mem_ofPred_eq,dot_realPosition_add] |
| constructor <;> intro h <;> linarith |
|
|
| lemma variableHitKernel_translation {d : ℕ} (ℓ : Vector d) (H : ℕ) |
| (ω : Environment d) (z x : Lattice d) (U : Set (Lattice d)) : |
| variableHitKernel ℓ H ((fun y => ω (z+y)),x) U = |
| variableHitKernel ℓ H (ω,z+x) ((fun y => y-z) ⁻¹' U) := by |
| simpa only [variableHitKernel,Kernel.coe_mk,strip_translation,upper_translation] using |
| hitKernel_translation ω z x (Strip ℓ (z+x) H) (Upper ℓ (z+x) H) U |
|
|
| lemma horizontalProjection_add {d : ℕ} (e : Direction d) (x y : Lattice d) : |
| horizontalProjection e (x+y)=horizontalProjection e x+horizontalProjection e y := by |
| apply Subtype.ext |
| funext j |
| by_cases hj : j=e.1 |
| · subst j |
| simp [horizontalProjection] |
| · simp [horizontalProjection,Function.update_of_ne hj] |
|
|
| lemma horizontalProjection_sub {d : ℕ} (e : Direction d) (x y : Lattice d) : |
| horizontalProjection e (x-y)=horizontalProjection e x-horizontalProjection e y := by |
| apply Subtype.ext |
| funext j |
| by_cases hj : j=e.1 |
| · subst j |
| simp [horizontalProjection] |
| · simp [horizontalProjection,Function.update_of_ne hj] |
|
|
| lemma horizontalLift_add {d : ℕ} (e : Direction d) (a b : ℤ) (x y : HorizontalSpace e) : |
| horizontalLift e a x+horizontalLift e b y=horizontalLift e (a+b) (x+y) := by |
| simp only [horizontalLift,AddSubgroup.coe_add,add_zsmul] |
| abel |
|
|
| lemma episodeSpliced_kernel {d : ℕ} (e : Direction d) |
| (t : Environment d → ℤ → ℕ) (X : EpisodeInput e) (i : ℤ) (m a : ℕ) |
| (z : HorizontalSpace e) (A : Set (HorizontalSpace e)) : |
| upperHorizontalKernel e (t X.1.1 (i+m)-t X.1.1 i) |
| (fun p => episodeSplicedRow e t X i m a p.1 p.2) z A = |
| variableHitKernel (realPosition (step e)) (t X.1.1 (i+m)-t X.1.1 i) |
| (X.1.1,horizontalLift e (t X.1.1 i) (X.1.2 (i,a)+z)) |
| {y | horizontalProjection e y-X.1.2 (i,a)∈A} := by |
| let b := horizontalLift e (t X.1.1 i) (X.1.2 (i,a)) |
| let H := t X.1.1 (i+m)-t X.1.1 i |
| have he : hitKernel (Strip (realPosition (step e)) (horizontalLift e 0 z) H) |
| (Upper (realPosition (step e)) (horizontalLift e 0 z) H) |
| (upperEnvironment e (fun p => episodeSplicedRow e t X i m a p.1 p.2),horizontalLift e 0 z)= |
| hitKernel (Strip (realPosition (step e)) (horizontalLift e 0 z) H) |
| (Upper (realPosition (step e)) (horizontalLift e 0 z) H) |
| ((fun y => X.1.1 (b+y)),horizontalLift e 0 z) := by |
| apply hitKernel_congr |
| intro y hy |
| have hy0 : 0 ≤ signedHeight e y := by |
| have := hy.1 |
| simp only [signedHeight_projection,signedHeight_horizontalLift,Int.cast_zero] at this |
| exact_mod_cast this |
| have hyH : (signedHeight e y).toNat<H := by |
| have := hy.2 |
| simp only [signedHeight_projection,signedHeight_horizontalLift,Int.cast_zero,zero_add] at this |
| have hylt : signedHeight e y<(H:ℤ) := by exact_mod_cast this |
| omega |
| change episodeSplicedRow e t X i m a (signedHeight e y).toNat (horizontalProjection e y)=_ |
| rw [episodeSplicedRow,ite_eq_left hyH] |
| simp only [Int.toNat_of_nonneg hy0] |
| apply congrArg X.1.1 |
| calc |
| _ = horizontalLift e (t X.1.1 i) (X.1.2 (i,a))+ |
| horizontalLift e (signedHeight e y) (horizontalProjection e y) := |
| (horizontalLift_add e _ _ _ _).symm |
| _ = b+y := by rw [horizontalLift_projection] |
| rw [upperHorizontalKernel_apply,he] |
| change variableHitKernel (realPosition (step e)) H ((fun y => X.1.1 (b+y)),horizontalLift e 0 z) |
| (horizontalProjection e ⁻¹' A)=_ |
| rw [variableHitKernel_translation] |
| dsimp only [b] |
| rw [horizontalLift_add,add_zero] |
| congr 1 |
| ext y |
| simp only [Set.mem_preimage,Set.mem_ofPred_eq,horizontalProjection_sub,horizontalProjection_lift] |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end OAI |
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|
