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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import OAI.Computability.PerfectCompleteness.Construction.SourceChildKernelLemmas
import OAI.Computability.PerfectCompleteness.Construction.SourceQuestionPositionSplit
namespace OAI
section
namespace PerfectCompleteness.SourceQuestionKernelJoint
open scoped Classical
open RecursiveSpaces SourceQuestionPositionSplit
open UniqueGamesTheorem.Foundations.Games
noncomputable section
private theorem kernelJoint_pushforward_first {S T R : Type*}
[Fintype S] [Fintype T] [Fintype R]
(μ : FiniteDistribution S) (kernel : S → FiniteDistribution R) (e : S ≃ T) :
(CleanConditioning.kernelJoint μ kernel).pushforward
(Equiv.prodCongr e (Equiv.refl R)) =
CleanConditioning.kernelJoint (μ.pushforward e) (fun s => kernel (e.symm s)) := by
simp only [FiniteDistribution.pushforward_equiv]
apply FiniteDistribution.eq_of_weight_eq
intro point
rfl
variable {branch : Nat → Nat} {n t v m : Nat}
abbrev ChoiceTuple :=
Fin (branch n) → ChildPositions (branch := branch) (n := n) (t := t)
def sourceEquiv (designated : Fin (branch n) → Slots branch n) :
SourceChildKernel.Sources (m := m) (t := t) designated ≃
Questions (branch := branch) (n := n) (t := t) (m := m) ×
ChoiceTuple (branch := branch) (n := n) (t := t) :=
(SourceQuestionPositionSplit.split designated).trans
(Equiv.prodCongr (Equiv.refl _) SourceQuestionPositionSplit.byChild)
def sources (designated : Fin (branch n) → Slots branch n)
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(choices : ChoiceTuple (branch := branch) (n := n) (t := t)) :
SourceChildKernel.Sources (m := m) (t := t) designated :=
(sourceEquiv designated).symm (q, choices)
@[simp] theorem sources_eq_join (designated : Fin (branch n) → Slots branch n)
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(choices : ChoiceTuple (branch := branch) (n := n) (t := t)) :
sources designated q choices =
SourceQuestionPositionSplit.join designated q
(SourceQuestionPositionSplit.byChild.symm choices) := rfl
theorem sourceLaw_sourceEquiv [NeZero m]
(designated : Fin (branch n) → Slots branch n) :
(SourceChildQuestionLaw.sourceLaw m t designated).pushforward (sourceEquiv designated) =
(PreliminarySampler.questionsLaw (branch := branch) (n := n + 1) (t := t) (m := m)).product
(FiniteProduct.law (fun _ : Fin (branch n) =>
FiniteDistribution.uniform (ChildPositions (branch := branch) (n := n) (t := t)))) := by
calc
_ = ((SourceChildQuestionLaw.sourceLaw m t designated).pushforward
(SourceQuestionPositionSplit.split designated)).pushforward
(Equiv.prodCongr (Equiv.refl _) SourceQuestionPositionSplit.byChild) := by
rw [FiniteDistribution.pushforward_comp]
rfl
_ = _ := by
rw [SourceQuestionPositionSplit.sourceLaw_split]
change ((PreliminarySampler.questionsLaw
(branch := branch) (n := n + 1) (t := t) (m := m)).product
(FiniteDistribution.uniform (Positions (branch := branch) (n := n) (t := t)))).pushforward
(fun x => (id x.1, SourceQuestionPositionSplit.byChild x.2)) = _
rw [FiniteDistribution.product_pushforward, FiniteDistribution.pushforward_id,
SourceQuestionPositionSplit.positions_byChild]
variable {C : Type*} [Fintype C]
(rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch n) → Slots branch n)
def groupEquiv :
SourceChildKernel.Sample (C := C) (t := t) rows clauses designated ≃
(Questions (branch := branch) (n := n) (t := t) (m := m) ×
ChoiceTuple (branch := branch) (n := n) (t := t)) ×
SourceChildKernelJoint.RawTuple (C := C) (t := t) rows clauses designated :=
(SourceChildKernelJoint.groupEquiv rows clauses designated).trans
(Equiv.prodCongr (sourceEquiv designated) (Equiv.refl _))
@[simp] theorem groupEquiv_raw
(sample : SourceChildKernel.Sample (C := C) (t := t) rows clauses designated)
(i : Fin (branch n)) :
(groupEquiv rows clauses designated sample).2 i = (sample i).2.2 := rfl
theorem originalLaw_group [NeZero m]
(flag : Fin (branch n) → FiniteDistribution Bool) :
(SourceChildKernel.originalLaw (C := C) (t := t) rows clauses designated flag).pushforward
(groupEquiv rows clauses designated) =
CleanConditioning.kernelJoint
((PreliminarySampler.questionsLaw
(branch := branch) (n := n + 1) (t := t) (m := m)).product
(FiniteProduct.law (fun _ : Fin (branch n) =>
FiniteDistribution.uniform
(ChildPositions (branch := branch) (n := n) (t := t)))))
(fun qp => FiniteProduct.law (fun i =>
SourceChildKernel.kernel (C := C) (t := t) rows clauses designated flag i
(sources designated qp.1 qp.2 i))) := by
calc
_ = ((SourceChildKernel.originalLaw (C := C) (t := t) rows clauses designated flag).pushforward
(SourceChildKernelJoint.groupEquiv rows clauses designated)).pushforward
(Equiv.prodCongr (sourceEquiv designated) (Equiv.refl _)) := by
rw [FiniteDistribution.pushforward_comp]
rfl
_ = (CleanConditioning.kernelJoint (SourceChildQuestionLaw.sourceLaw m t designated)
(fun source => FiniteProduct.law (fun i =>
SourceChildKernel.kernel (C := C) (t := t) rows clauses designated flag i
(source i)))).pushforward
(Equiv.prodCongr (sourceEquiv designated) (Equiv.refl _)) := by
rw [SourceChildKernelJoint.originalLaw_group]
rfl
_ = _ := by
rw [kernelJoint_pushforward_first, sourceLaw_sourceEquiv]
rfl
end
end PerfectCompleteness.SourceQuestionKernelJoint
end
end OAI
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