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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 | import OAI.Computability.PerfectCompleteness.Foundations.CutSlotOutside
import OAI.Computability.PerfectCompleteness.Foundations.WholeCutExteriorTransportLemmas
import OAI.Computability.PerfectCompleteness.Foundations.WholeCutGrouping
namespace OAI
section
namespace PerfectCompleteness.FilledUniformSampler
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open UniqueGamesTheorem.Foundations.Games
open scoped Classical
noncomputable section
variable {branch : Nat → Nat} {n m t : Nat}
(rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(outside : Slots branch n → Fin t → MixedSupport.Slot)
(placeholder : Slots branch (m + 1) → Fin t → MixedSupport.Slot)
abbrev Exterior := WholeCutGrouping.Exterior rows repeats p
(CutSlotAssembly.fill p outside placeholder)
theorem cutSlots_fill (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) :
WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside) = inside := by
funext s
exact CutSlotAssembly.fill_at_cut p outside inside s
def exteriorEquiv (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) :
Exterior rows repeats p outside placeholder ≃
WholeCutGrouping.Exterior rows repeats p (CutSlotAssembly.fill p outside inside) :=
WholeCutExteriorTransport.equiv rows repeats p
(CutSlotAssembly.fill p outside placeholder) (CutSlotAssembly.fill p outside inside)
(CutSlotAssembly.fill_outside_eq p outside placeholder inside)
def childrenEquiv (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) :
CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows ≃
CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p)
(WholeCutGrouping.cutSlots p (CutSlotAssembly.fill p outside inside)) rows :=
Equiv.piCongrRight (fun i => Equiv.cast (congrArg
(fun slots => CutChildGrouping.Child (C := WholeCutCalls.Index rows repeats p) slots rows i)
(cutSlots_fill p outside inside).symm))
@[simp] theorem childrenEquiv_apply
(inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot)
(children : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows)
(i : Fin (branch m)) :
childrenEquiv rows repeats p outside inside children i =
cast (congrArg
(fun slots => CutChildGrouping.Child (C := WholeCutCalls.Index rows repeats p) slots rows i)
(cutSlots_fill p outside inside).symm) (children i) := rfl
theorem childrenEquiv_heq
(inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot)
(children : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows)
(i : Fin (branch m)) :
HEq (childrenEquiv rows repeats p outside inside children i) (children i) :=
cast_heq _ _
def filledSplitEquiv (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) :
(Exterior rows repeats p outside placeholder ×
CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) ≃
WholeCutSampler.Tape rows repeats p (CutSlotAssembly.fill p outside inside) :=
(Equiv.prodCongr (exteriorEquiv rows repeats p outside placeholder inside)
(childrenEquiv rows repeats p outside inside)).trans
(WholeCutGrouping.splitTape rows repeats p (CutSlotAssembly.fill p outside inside)).symm
theorem filledSplitEquiv_apply
(inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot)
(external : Exterior rows repeats p outside placeholder)
(children : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) :
filledSplitEquiv rows repeats p outside placeholder inside (external, children) =
(WholeCutGrouping.splitTape rows repeats p (CutSlotAssembly.fill p outside inside)).symm
(exteriorEquiv rows repeats p outside placeholder inside external,
childrenEquiv rows repeats p outside inside children) := rfl
theorem filledSplitEquiv_tapeLaw
(inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) :
((FiniteDistribution.uniform (Exterior rows repeats p outside placeholder)).product
(CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows)).pushforward
(filledSplitEquiv rows repeats p outside placeholder inside) =
WholeCutSampler.tapeLaw rows repeats p (CutSlotAssembly.fill p outside inside) := by
unfold CutChildGrouping.rawLaw
rw [UniformLinearImage.law_uniform, WholeCutSampler.uniform_product,
WholeCutSampler.tapeLaw_eq_uniform, FiniteDistribution.pushforward_equiv]
apply FiniteDistribution.eq_of_weight_eq
intro x
change 1 / (Fintype.card (Exterior rows repeats p outside placeholder ×
CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) : ℝ) =
1 / (Fintype.card
(WholeCutSampler.Tape rows repeats p (CutSlotAssembly.fill p outside inside)) : ℝ)
rw [Fintype.card_congr (filledSplitEquiv rows repeats p outside placeholder inside)]
def evaluate (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot)
(x : Exterior rows repeats p outside placeholder ×
CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) :
Arrays (CutSlotAssembly.fill p outside inside) rows :=
WholeCutSampler.evaluate rows repeats p (CutSlotAssembly.fill p outside inside)
(filledSplitEquiv rows repeats p outside placeholder inside x)
theorem evaluate_law (inside : Slots branch (m + 1) → Fin t → MixedSupport.Slot) :
((FiniteDistribution.uniform (Exterior rows repeats p outside placeholder)).product
(CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows)).pushforward
(evaluate rows repeats p outside placeholder inside) =
WholeArraySampler.law rows repeats p (CutSlotAssembly.fill p outside inside) := by
unfold evaluate
rw [← FiniteDistribution.pushforward_comp
((FiniteDistribution.uniform (Exterior rows repeats p outside placeholder)).product
(CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows))
(filledSplitEquiv rows repeats p outside placeholder inside)
(WholeCutSampler.evaluate rows repeats p (CutSlotAssembly.fill p outside inside)),
filledSplitEquiv_tapeLaw]
exact WholeCutSampler.evaluate_law rows repeats p (CutSlotAssembly.fill p outside inside)
end
end PerfectCompleteness.FilledUniformSampler
end
end OAI
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