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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | 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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