File size: 7,154 Bytes
2f887db | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 | import OAI.Computability.PerfectCompleteness.Construction.FactoredFunctionsLemmas
import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerRefinement
namespace OAI
section
namespace PerfectCompleteness.CutSamplerLocality
open scoped BigOperators Classical
open RecursiveSpaces DescendantSpaces RecursiveSampler PointwiseSpaces
open CutSamplerRefinement
noncomputable section
universe u w
variable {branch : Nat → Nat} {n m : Nat}
def View : {n m : Nat} → (p : Path branch n (m + 1)) →
(A : Slots branch n → Type u) → (clean : Fin (branch m) → Prop) → Type u
| _, m, .refl _, A, clean =>
(i : {i : Fin (branch m) // ¬ clean i}) → Assignment (childFamily A i.val)
| _, _, .step i p, A, clean =>
((j : OffPath i) → Assignment (childFamily A j.val)) ×
View p (childFamily A i) clean
def view : {n m : Nat} → (p : Path branch n (m + 1)) →
(A : Slots branch n → Type u) → (clean : Fin (branch m) → Prop) →
Assignment A → View p A clean
| _, _, .refl _, A, _, x => fun i => childRestriction A i.val x
| _, _, .step i p, A, clean, x =>
(fun j => childRestriction A j.val x,
view p (childFamily A i) clean (childRestriction A i x))
def subTape (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat)
(i : Fin (branch n)) (p : Path branch n m)
(A : Slots branch (n + 1) → Type u)
(tape : CutSamplerRefinement.Tape 𝕜 repeats (.step i p) A)
(h : Fin (repeats (n + 1))) (b : Bool) :
CutSamplerRefinement.Tape 𝕜 repeats p (childFamily A i) :=
fun j => tape (Sum.inr ((h, b), j))
def ZeroAtClean (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat) :
{n m : Nat} → (p : Path branch n (m + 1)) → (A : Slots branch n → Type u) →
(clean : Fin (branch m) → Prop) → CutSamplerRefinement.Tape 𝕜 repeats p A → Prop
| _, _, .refl _, _, clean, tape => ∀ i, clean i → tape () i = 0
| _ + 1, _, .step i p, A, clean, tape =>
∀ h b, ZeroAtClean 𝕜 repeats p (childFamily A i) clean
(subTape 𝕜 repeats i p A tape h b)
theorem evaluate_refl (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat)
(A : Slots branch (m + 1) → Type u)
(tape : CutSamplerRefinement.Tape 𝕜 repeats (.refl (m + 1)) A) :
CutSamplerRefinement.evaluate 𝕜 repeats (.refl (m + 1)) A tape =
UniformChildSum.recursiveSum A (tape ()) := rfl
theorem evaluate_step (𝕜 : Type w) [Field 𝕜] (repeats : Nat → Nat)
(i : Fin (branch n)) (p : Path branch n m)
(A : Slots branch (n + 1) → Type u)
(tape : CutSamplerRefinement.Tape 𝕜 repeats (.step i p) A) :
CutSamplerRefinement.evaluate 𝕜 repeats (.step i p) A tape =
RecursiveSampler.combine 𝕜 A i (repeats (n + 1))
(fun j => tape (Sum.inl j))
(fun h => CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i)
(subTape 𝕜 repeats i p A tape h false))
(fun h => CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i)
(subTape 𝕜 repeats i p A tape h true)) := rfl
theorem evaluate_eq_of_view_eq (𝕜 : Type w) [Field 𝕜]
(repeats : Nat → Nat) (p : Path branch n (m + 1)) :
∀ (A : Slots branch n → Type u) (clean : Fin (branch m) → Prop)
(tape : CutSamplerRefinement.Tape 𝕜 repeats p A),
ZeroAtClean 𝕜 repeats p A clean tape →
∀ x y : Assignment A, view p A clean x = view p A clean y →
(CutSamplerRefinement.evaluate 𝕜 repeats p A tape).val x =
(CutSamplerRefinement.evaluate 𝕜 repeats p A tape).val y := by
induction n generalizing m with
| zero =>
have h := p.height_le
omega
| succ n ih =>
cases p with
| refl =>
intro A clean tape hzero x y hview
have heval (z : Assignment A) :
(UniformChildSum.recursiveSum (𝕜 := 𝕜) A (tape ())).val z =
∑ i : Fin (branch n), (tape () i).val (childRestriction A i z) := by
have hsum := congrFun
(UniformChildSum.childSum_apply (𝕜 := 𝕜) (childRestriction A)
(fun i => space 𝕜 branch n (childFamily A i)) (tape ())) z
simp only [Finset.sum_apply, PointwiseSpaces.pullback_apply] at hsum
refine hsum.trans ?_
refine Finset.sum_congr (ι := Fin (branch n)) ?_ ?_
· ext i
simp only [Finset.mem_univ]
· intro i _
rfl
rw [evaluate_refl, heval x, heval y]
apply Finset.sum_congr rfl
intro i _
by_cases hi : clean i
· rw [hzero i hi]
rfl
· exact congrArg (tape () i).val (congrFun hview ⟨i, hi⟩)
| step i p =>
intro A clean tape hzero x y hview
have hout : ∀ j : OffPath i, childRestriction A j.val x = childRestriction A j.val y :=
congrFun (congrArg Prod.fst hview)
have hin : view p (childFamily A i) clean (childRestriction A i x) =
view p (childFamily A i) clean (childRestriction A i y) :=
congrArg Prod.snd hview
rw [evaluate_step]
simp only [RecursiveSampler.combine, RecursiveSampler.combineFunction,
Finset.sum_apply, Pi.add_apply, Pi.mul_apply, PointwiseSpaces.pullback_apply]
change
(∑ j : OffPath i, (tape (Sum.inl j)).val (childRestriction A j.val x)) +
(∑ h, (CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i)
(subTape 𝕜 repeats i p A tape h false)).val (childRestriction A i x) *
(CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i)
(subTape 𝕜 repeats i p A tape h true)).val (childRestriction A i x)) =
(∑ j : OffPath i, (tape (Sum.inl j)).val (childRestriction A j.val y)) +
(∑ h, (CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i)
(subTape 𝕜 repeats i p A tape h false)).val (childRestriction A i y) *
(CutSamplerRefinement.evaluate 𝕜 repeats p (childFamily A i)
(subTape 𝕜 repeats i p A tape h true)).val (childRestriction A i y))
congr 1
· exact Finset.sum_congr rfl (fun j _ => congrArg (tape (Sum.inl j)).val (hout j))
· apply Finset.sum_congr rfl
intro h _
rw [ih p (childFamily A i) clean (subTape 𝕜 repeats i p A tape h false)
(hzero h false) _ _ hin,
ih p (childFamily A i) clean (subTape 𝕜 repeats i p A tape h true)
(hzero h true) _ _ hin]
theorem evaluate_factors (𝕜 : Type w) [Field 𝕜]
(repeats : Nat → Nat) (p : Path branch n (m + 1))
(A : Slots branch n → Type u) (clean : Fin (branch m) → Prop)
(tape : CutSamplerRefinement.Tape 𝕜 repeats p A)
(hzero : ZeroAtClean 𝕜 repeats p A clean tape) :
(CutSamplerRefinement.evaluate 𝕜 repeats p A tape).val ∈
FactoredFunctions.factoringSpace (view p A clean) :=
evaluate_eq_of_view_eq 𝕜 repeats p A clean tape hzero
end
end PerfectCompleteness.CutSamplerLocality
end
end OAI
|