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import OAI.Computability.PerfectCompleteness.Sampling.NumberedUniformCut
namespace OAI
section
namespace PerfectCompleteness.ExtraCutCall
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays
open OriginalWholeCutTape OriginalWholeCut OriginalWholeCutLaw
open UniqueGamesTheorem.Foundations.Games
open scoped BigOperators Classical
noncomputable section
def splitLast (calls : Nat) (A : Type*) :
(Fin (calls + 1) → A) ≃ (Fin calls → A) × A where
toFun x := (fun i => x i.castSucc, x (Fin.last calls))
invFun z := Fin.lastCases z.2 z.1
left_inv x := by
funext i
refine Fin.lastCases ?_ (fun j => ?_) i
· simp only [Fin.lastCases_last]
· simp only [Fin.lastCases_castSucc]
right_inv z := by
apply Prod.ext
· funext i
simp only [Fin.lastCases_castSucc]
· simp only [Fin.lastCases_last]
theorem splitLast_law {A : Type*} [Fintype A] (calls : Nat) (μ : FiniteDistribution A) :
(FiniteProduct.law (fun _ : Fin (calls + 1) => μ)).pushforward (splitLast calls A) =
(FiniteProduct.law (fun _ : Fin calls => μ)).product μ := by
rw [← FiniteDistribution.transport_eq_pushforward]
apply FiniteDistribution.eq_of_weight_eq
rintro ⟨values, last⟩
change (∏ i : Fin (calls + 1), μ.weight (Fin.lastCases last values i)) =
(∏ i : Fin calls, μ.weight (values i)) * μ.weight last
rw [Fin.prod_univ_castSucc]
simp only [Fin.lastCases_castSucc, Fin.lastCases_last]
private def middleSwap (A B C : Type*) : (A × B) × C ≃ (A × C) × B where
toFun z := ((z.1.1, z.2), z.1.2)
invFun z := ((z.1.1, z.2), z.1.2)
left_inv _ := rfl
right_inv _ := rfl
private theorem product_assoc_law {E A B : Type*}
[Fintype E] [Fintype A] [Fintype B]
(μ : FiniteDistribution E) (ν : FiniteDistribution A) (ξ : FiniteDistribution B) :
(μ.product (ν.product ξ)).pushforward (fun z => ((z.1, z.2.1), z.2.2)) =
(μ.product ν).product ξ := by
change (μ.product (ν.product ξ)).pushforward (Equiv.prodAssoc E A B).symm = _
rw [← FiniteDistribution.transport_eq_pushforward]
apply FiniteDistribution.eq_of_weight_eq
intro z
exact (mul_assoc _ _ _).symm
private theorem lift_split_law {E D A B : Type*}
[Fintype E] [Fintype D] [Fintype A] [Fintype B]
(μ : FiniteDistribution E) (ν : FiniteDistribution D)
(ρ : FiniteDistribution A) (ξ : FiniteDistribution B) (split : D → A × B)
(hsplit : ν.pushforward split = ρ.product ξ) :
(μ.product ν).pushforward (fun z => ((z.1, (split z.2).1), (split z.2).2)) =
(μ.product ρ).product ξ := by
have hmap : (μ.product ν).pushforward (fun z => (z.1, split z.2)) =
μ.product (ρ.product ξ) := by
have h := FiniteDistribution.product_pushforward μ ν id split
simpa only [id_eq, FiniteDistribution.pushforward_id, hsplit] using h
calc
_ = ((μ.product ν).pushforward (fun z => (z.1, split z.2))).pushforward
(fun z => ((z.1, z.2.1), z.2.2)) :=
(FiniteDistribution.pushforward_comp _ _ _).symm
_ = (μ.product (ρ.product ξ)).pushforward
(fun z => ((z.1, z.2.1), z.2.2)) :=
congrArg (fun law : FiniteDistribution (E × (A × B)) =>
law.pushforward (fun z => ((z.1, z.2.1), z.2.2))) hmap
_ = _ := product_assoc_law μ ρ ξ
variable {branch : Nat → Nat} {n m k t : Nat}
def splitAssembled (calls : Nat) (rows : Nat → Nat)
(slots : Slots branch (m + 1) → Fin t → MixedSupport.Slot) :
CutChildGrouping.Assembled (C := Fin (calls + 1)) slots rows ≃
CutChildGrouping.Assembled (C := Fin calls) slots rows × H slots :=
(OriginalWholeCutBridge.assembledEquiv (calls + 1) rows slots).symm.trans
((Equiv.prodCongr (splitLast calls (H slots)) (Equiv.refl (ChildArrays rows slots))).trans
((middleSwap (Fin calls → H slots) (H slots) (ChildArrays rows slots)).trans
(Equiv.prodCongr (OriginalWholeCutBridge.assembledEquiv calls rows slots)
(Equiv.refl (H slots)))))
def observationLaw (calls : Nat) (rows repeats : Nat → Nat) (chosen : Fin (branch m))
(q : Path branch m k) (slots : Slots branch (m + 1) → Fin t → MixedSupport.Slot) :
FiniteDistribution (CutChildGrouping.Assembled (C := Fin calls) slots rows) :=
(OriginalChildBlocks.sourceLaw calls rows repeats chosen q slots).pushforward
(OriginalChildBlocks.observed calls rows repeats chosen q slots)
theorem observation_split_law (calls : Nat) (rows repeats : Nat → Nat)
(chosen : Fin (branch m)) (q : Path branch m k)
(slots : Slots branch (m + 1) → Fin t → MixedSupport.Slot) :
(observationLaw (calls + 1) rows repeats chosen q slots).pushforward
(splitAssembled calls rows slots) =
(observationLaw calls rows repeats chosen q slots).product
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain slots)) := by
simp only [observationLaw, OriginalWholeCutBridge.observed_product_law]
rw [← FiniteDistribution.transport_eq_pushforward]
apply FiniteDistribution.eq_of_weight_eq
intro z
change ((∏ i : Fin (calls + 1),
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain slots)).weight
(Fin.lastCases z.2 z.1.1 i)) *
(belowArraysLaw rows repeats chosen q slots).weight z.1.2) =
((∏ i : Fin calls,
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain slots)).weight (z.1.1 i)) *
(belowArraysLaw rows repeats chosen q slots).weight z.1.2) *
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain slots)).weight z.2
rw [Fin.prod_univ_castSucc]
simp only [Fin.lastCases_castSucc, Fin.lastCases_last]
ring
theorem uniform_split_law (calls : Nat) (rows : Nat → Nat)
(slots : Slots branch (m + 1) → Fin t → MixedSupport.Slot) :
(FiniteDistribution.uniform (CutChildGrouping.Assembled (C := Fin (calls + 1)) slots rows)).pushforward
(splitAssembled calls rows slots) =
(FiniteDistribution.uniform (CutChildGrouping.Assembled (C := Fin calls) slots rows)).product
(FiniteDistribution.uniform (H slots)) := by
rw [← FiniteDistribution.transport_eq_pushforward, UniformConditioning.uniform_transport]
exact WholeCutSampler.uniform_product.symm
abbrev EnlargedRecord (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) :=
Exterior rows repeats p slots × CutChildGrouping.Assembled
(C := Fin (OriginalCutCalls.count rows repeats n (m + 1) + 1)) (cutSlots p slots) rows
def splitRecord (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(record : EnlargedRecord rows repeats p slots) :
OriginalWholeCutBridge.NumberedRecord rows repeats p slots × H (cutSlots p slots) :=
let split := splitAssembled (OriginalCutCalls.count rows repeats n (m + 1)) rows
(cutSlots p slots) record.2
((record.1, split.1), split.2)
@[simp] theorem splitRecord_oldCall (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(record : EnlargedRecord rows repeats p slots)
(call : Fin (OriginalCutCalls.count rows repeats n (m + 1))) :
(splitRecord rows repeats p slots record).1.2.1 call = record.2.1 call.castSucc := rfl
@[simp] theorem splitRecord_fresh (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(record : EnlargedRecord rows repeats p slots) :
(splitRecord rows repeats p slots record).2 =
record.2.1 (Fin.last (OriginalCutCalls.count rows repeats n (m + 1))) := rfl
attribute [local instance 2000] OriginalWholeCutLaw.valuesBelowFintype
theorem record_fresh_law (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(chosen : Fin (branch m)) (q : Path branch m k)
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
((exteriorLaw rows repeats p slots).product
(observationLaw (OriginalCutCalls.count rows repeats n (m + 1) + 1)
rows repeats chosen q (cutSlots p slots))).pushforward (splitRecord rows repeats p slots) =
((recordLaw rows repeats p chosen q slots).pushforward
(OriginalWholeCutBridge.numberRecord rows repeats p slots)).product
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain (cutSlots p slots))) := by
rw [OriginalWholeCutBridge.numberRecord_law]
exact lift_split_law (exteriorLaw rows repeats p slots) _ _ _
(splitAssembled (OriginalCutCalls.count rows repeats n (m + 1)) rows (cutSlots p slots))
(observation_split_law (OriginalCutCalls.count rows repeats n (m + 1))
rows repeats chosen q (cutSlots p slots))
theorem uniform_record_fresh_law (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
((exteriorLaw rows repeats p slots).product
(FiniteDistribution.uniform (CutChildGrouping.Assembled
(C := Fin (OriginalCutCalls.count rows repeats n (m + 1) + 1)) (cutSlots p slots) rows))).pushforward
(splitRecord rows repeats p slots) =
((OriginalUniformCut.referenceLaw rows repeats p slots).pushforward
(OriginalWholeCutBridge.numberRecord rows repeats p slots)).product
(FiniteDistribution.uniform (H (cutSlots p slots))) := by
rw [NumberedUniformCut.reference_numberRecord_law]
exact lift_split_law (exteriorLaw rows repeats p slots) _ _ _
(splitAssembled (OriginalCutCalls.count rows repeats n (m + 1)) rows (cutSlots p slots))
(uniform_split_law (OriginalCutCalls.count rows repeats n (m + 1)) rows (cutSlots p slots))
def arraysAndFresh (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(record : EnlargedRecord rows repeats p slots) : Arrays slots rows × H (cutSlots p slots) :=
(OriginalWholeCutBridge.reconstruct rows repeats p slots
(splitRecord rows repeats p slots record).1, (splitRecord rows repeats p slots record).2)
theorem arrays_fresh_law (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(chosen : Fin (branch m)) (q : Path branch m k)
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
((exteriorLaw rows repeats p slots).product
(observationLaw (OriginalCutCalls.count rows repeats n (m + 1) + 1)
rows repeats chosen q (cutSlots p slots))).pushforward (arraysAndFresh rows repeats p slots) =
(WholeArraySampler.law rows repeats (p.append (.step chosen q)) slots).product
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain (cutSlots p slots))) := by
calc
_ = (((exteriorLaw rows repeats p slots).product
(observationLaw (OriginalCutCalls.count rows repeats n (m + 1) + 1)
rows repeats chosen q (cutSlots p slots))).pushforward
(splitRecord rows repeats p slots)).pushforward
(fun z => (OriginalWholeCutBridge.reconstruct rows repeats p slots z.1, z.2)) :=
(FiniteDistribution.pushforward_comp _ _ _).symm
_ = _ := by
rw [record_fresh_law, OriginalWholeCutBridge.numberRecord_law]
have h := FiniteDistribution.product_pushforward
((exteriorLaw rows repeats p slots).product
(observationLaw (OriginalCutCalls.count rows repeats n (m + 1))
rows repeats chosen q (cutSlots p slots)))
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain (cutSlots p slots)))
(OriginalWholeCutBridge.reconstruct rows repeats p slots) id
simpa only [id_eq, FiniteDistribution.pushforward_id,
observationLaw, OriginalWholeCutBridge.reconstruct_child_law] using h
theorem uniform_arrays_fresh_law (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
((exteriorLaw rows repeats p slots).product
(FiniteDistribution.uniform (CutChildGrouping.Assembled
(C := Fin (OriginalCutCalls.count rows repeats n (m + 1) + 1)) (cutSlots p slots) rows))).pushforward
(arraysAndFresh rows repeats p slots) =
(WholeArraySampler.law rows repeats p slots).product
(FiniteDistribution.uniform (H (cutSlots p slots))) := by
calc
_ = (((exteriorLaw rows repeats p slots).product
(FiniteDistribution.uniform (CutChildGrouping.Assembled
(C := Fin (OriginalCutCalls.count rows repeats n (m + 1) + 1)) (cutSlots p slots) rows))).pushforward
(splitRecord rows repeats p slots)).pushforward
(fun z => (OriginalWholeCutBridge.reconstruct rows repeats p slots z.1, z.2)) :=
(FiniteDistribution.pushforward_comp _ _ _).symm
_ = _ := by
rw [uniform_record_fresh_law, NumberedUniformCut.reference_numberRecord_law]
have h := FiniteDistribution.product_pushforward
((exteriorLaw rows repeats p slots).product
(FiniteDistribution.uniform (CutChildGrouping.Assembled
(C := Fin (OriginalCutCalls.count rows repeats n (m + 1))) (cutSlots p slots) rows)))
(FiniteDistribution.uniform (H (cutSlots p slots)))
(OriginalWholeCutBridge.reconstruct rows repeats p slots) id
simpa only [id_eq, FiniteDistribution.pushforward_id,
NumberedUniformCut.reconstruct_product_uniform] using h
end
end PerfectCompleteness.ExtraCutCall
end
end OAI
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