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import OAI.Computability.PerfectCompleteness.Construction.OriginalCutBucketReplacement
import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerReplayTransportLemmas
import OAI.Computability.PerfectCompleteness.Sampling.OriginalWholeCutLaw
namespace OAI
section
namespace PerfectCompleteness.OriginalOwnBucketSplit
open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays TerminalCalls
open OriginalWholeCut OriginalWholeCutTape
open UniqueGamesTheorem.Foundations.Games
open scoped BigOperators Classical
noncomputable section
variable {branch : Nat → Nat} {n m k t : Nat}
@[instance_reducible] def valuesChildrenFintype (C : Type*) [Fintype C]
(slots : Slots branch (m + 1) → Fin t → MixedSupport.Slot) (rows : Nat → Nat) :
Fintype ((C → H slots) × ChildArrays rows slots) :=
@instFintypeProd (C → H slots) (ChildArrays rows slots) inferInstance inferInstance
attribute [local instance 2000] valuesChildrenFintype
def OtherIndex (rows repeats : Nat → Nat) : {n m : Nat} → Path branch n m → Type
| _, _, .refl _ => Empty
| n + 1, _, .step i p =>
(BucketSampler.Direction (rows (n + 1)) × TerminalIndex repeats (.step i p)) ⊕
OtherIndex rows repeats p
instance otherIndexFintype (rows repeats : Nat → Nat) (p : Path branch n m) :
Fintype (OtherIndex rows repeats p) := by
induction p with
| refl m => exact inferInstanceAs (Fintype Empty)
| @step n m i p ih =>
letI : Fintype (OtherIndex rows repeats p) := ih
exact inferInstanceAs (Fintype
((BucketSampler.Direction (rows (n + 1)) × TerminalIndex repeats (.step i p)) ⊕
OtherIndex rows repeats p))
private def sumEmpty (A : Type*) : A ≃ A ⊕ Empty where
toFun := Sum.inl
invFun := Sum.elim id Empty.elim
left_inv _ := rfl
right_inv x := by
cases x with
| inl a => rfl
| inr e => exact Empty.elim e
private def swapFirstSum (R A O : Type*) : R ⊕ (A ⊕ O) ≃ A ⊕ (R ⊕ O) where
toFun
| .inl r => .inr (.inl r)
| .inr (.inl a) => .inl a
| .inr (.inr o) => .inr (.inr o)
invFun
| .inl a => .inr (.inl a)
| .inr (.inl r) => .inl r
| .inr (.inr o) => .inr (.inr o)
left_inv x := by rcases x with r | (a | o) <;> rfl
right_inv x := by rcases x with a | (r | o) <;> rfl
def indexEquiv (rows repeats : Nat → Nat) : {n m : Nat} → (p : Path branch n m) →
OriginalCutCalls.Index rows repeats p ≃
BucketSampler.Direction (rows m) ⊕ OtherIndex rows repeats p
| _, _, .refl _ => sumEmpty _
| n + 1, _, .step i p =>
(Equiv.sumCongr
(Equiv.refl (BucketSampler.Direction (rows (n + 1)) ×
TerminalIndex repeats (.step i p)))
(indexEquiv rows repeats p)).trans (swapFirstSum _ _ _)
@[simp] theorem indexEquiv_ownBucket (rows repeats : Nat → Nat)
(p : Path branch n m) (a : BucketSampler.Direction (rows m)) :
indexEquiv rows repeats p (OriginalCutCalls.ownBucket rows repeats p a) = .inl a := by
revert a
induction p with
| refl m => intro a; rfl
| @step n m i p ih =>
intro a
change swapFirstSum _ _ _
(.inr (indexEquiv rows repeats p (OriginalCutCalls.ownBucket rows repeats p a))) = _
rw [ih a]
rfl
@[simp] theorem indexEquiv_symm_inl (rows repeats : Nat → Nat)
(p : Path branch n m) (a : BucketSampler.Direction (rows m)) :
(indexEquiv rows repeats p).symm (.inl a) =
OriginalCutCalls.ownBucket rows repeats p a := by
apply (indexEquiv rows repeats p).injective
rw [Equiv.apply_symm_apply, indexEquiv_ownBucket]
def otherCall (rows repeats : Nat → Nat) (p : Path branch n m)
(call : OtherIndex rows repeats p) : OriginalCutCalls.Index rows repeats p :=
(indexEquiv rows repeats p).symm (.inr call)
@[simp] theorem indexEquiv_otherCall (rows repeats : Nat → Nat)
(p : Path branch n m) (call : OtherIndex rows repeats p) :
indexEquiv rows repeats p (otherCall rows repeats p call) = .inr call :=
(indexEquiv rows repeats p).apply_symm_apply (.inr call)
theorem otherCall_injective (rows repeats : Nat → Nat) (p : Path branch n m) :
Function.Injective (otherCall rows repeats p) := by
intro a b h
have h' := congrArg (indexEquiv rows repeats p) h
simpa only [indexEquiv_otherCall, Sum.inr.injEq] using h'
theorem otherCall_ne_ownBucket (rows repeats : Nat → Nat) (p : Path branch n m)
(call : OtherIndex rows repeats p) (a : BucketSampler.Direction (rows m)) :
otherCall rows repeats p call ≠ OriginalCutCalls.ownBucket rows repeats p a := by
intro h
have h' := congrArg (indexEquiv rows repeats p) h
rw [indexEquiv_otherCall, indexEquiv_ownBucket] at h'
cases h'
def valuesEquiv (rows repeats : Nat → Nat) (p : Path branch n m) (A : Type*) :
(OriginalCutCalls.Index rows repeats p → A) ≃
(BucketSampler.Direction (rows m) → A) × (OtherIndex rows repeats p → A) :=
(Equiv.arrowCongr (indexEquiv rows repeats p) (Equiv.refl A)).trans
(Equiv.sumArrowEquivProdArrow _ _ _)
@[simp] theorem valuesEquiv_own (rows repeats : Nat → Nat) (p : Path branch n m)
(A : Type*) (values : OriginalCutCalls.Index rows repeats p → A)
(a : BucketSampler.Direction (rows m)) :
(valuesEquiv rows repeats p A values).1 a =
values (OriginalCutCalls.ownBucket rows repeats p a) := by
change values ((indexEquiv rows repeats p).symm (.inl a)) = _
rw [indexEquiv_symm_inl]
@[simp] theorem valuesEquiv_other (rows repeats : Nat → Nat) (p : Path branch n m)
(A : Type*) (values : OriginalCutCalls.Index rows repeats p → A)
(call : OtherIndex rows repeats p) :
(valuesEquiv rows repeats p A values).2 call =
values (otherCall rows repeats p call) := rfl
@[simp] theorem valuesEquiv_symm_own (rows repeats : Nat → Nat) (p : Path branch n m)
(A : Type*) (own : BucketSampler.Direction (rows m) → A)
(other : OtherIndex rows repeats p → A) (a : BucketSampler.Direction (rows m)) :
(valuesEquiv rows repeats p A).symm (own, other)
(OriginalCutCalls.ownBucket rows repeats p a) = own a := by
have h := congrArg (fun z => z.1 a)
((valuesEquiv rows repeats p A).apply_symm_apply (own, other))
simpa only [valuesEquiv_own] using h
@[simp] theorem valuesEquiv_symm_other (rows repeats : Nat → Nat) (p : Path branch n m)
(A : Type*) (own : BucketSampler.Direction (rows m) → A)
(other : OtherIndex rows repeats p → A) (call : OtherIndex rows repeats p) :
(valuesEquiv rows repeats p A).symm (own, other)
(otherCall rows repeats p call) = other call := by
have h := congrArg (fun z => z.2 call)
((valuesEquiv rows repeats p A).apply_symm_apply (own, other))
simpa only [valuesEquiv_other] using h
theorem valuesEquiv_law (rows repeats : Nat → Nat) (p : Path branch n m)
{A : Type*} [Fintype A] (μ : FiniteDistribution A) :
(FiniteProduct.law (fun _ : OriginalCutCalls.Index rows repeats p => μ)).pushforward
(valuesEquiv rows repeats p A) =
(FiniteProduct.law (fun _ : BucketSampler.Direction (rows m) => μ)).product
(FiniteProduct.law (fun _ : OtherIndex rows repeats p => μ)) := by
rw [← FiniteDistribution.transport_eq_pushforward]
apply FiniteDistribution.eq_of_weight_eq
intro z
change (∏ call : OriginalCutCalls.Index rows repeats p,
μ.weight (Sum.elim z.1 z.2 (indexEquiv rows repeats p call))) =
(∏ a : BucketSampler.Direction (rows m), μ.weight (z.1 a)) *
∏ other : OtherIndex rows repeats p, μ.weight (z.2 other)
rw [(indexEquiv rows repeats p).prod_comp
(fun tag => μ.weight (Sum.elim z.1 z.2 tag)), Fintype.prod_sum_type]
simp only [Sum.elim_inl, Sum.elim_inr]
abbrev BucketTape (rows : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) :=
BucketSampler.Tape (rows (m + 1)) (H (cutSlots p slots))
abbrev OtherValues (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) :=
OtherIndex rows repeats p → H (cutSlots p slots)
abbrev Complement (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) :=
Exterior rows repeats p slots × (OtherValues rows repeats p slots × BelowArrays rows p slots)
def recordEquiv (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
Record rows repeats p slots ≃ BucketTape rows p slots × Complement rows repeats p slots where
toFun record :=
((valuesEquiv rows repeats p (H (cutSlots p slots)) record.2.1).1,
(record.1,
((valuesEquiv rows repeats p (H (cutSlots p slots)) record.2.1).2, record.2.2)))
invFun z :=
(z.2.1, ((valuesEquiv rows repeats p (H (cutSlots p slots))).symm (z.1, z.2.2.1),
z.2.2.2))
left_inv record := by
rcases record with ⟨exterior, values, below⟩
change (exterior, ((valuesEquiv rows repeats p (H (cutSlots p slots))).symm
(valuesEquiv rows repeats p (H (cutSlots p slots)) values), below)) = _
rw [Equiv.symm_apply_apply]
right_inv z := by
rcases z with ⟨own, exterior, other, below⟩
change
(((valuesEquiv rows repeats p (H (cutSlots p slots)))
((valuesEquiv rows repeats p (H (cutSlots p slots))).symm (own, other))).1,
(exterior,
(((valuesEquiv rows repeats p (H (cutSlots p slots)))
((valuesEquiv rows repeats p (H (cutSlots p slots))).symm (own, other))).2,
below))) = _
rw [Equiv.apply_symm_apply]
def assembleRecord (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(own : BucketTape rows p slots) (complement : Complement rows repeats p slots) :
Record rows repeats p slots :=
(recordEquiv rows repeats p slots).symm (own, complement)
@[simp] theorem recordEquiv_own (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(record : Record rows repeats p slots) (a : BucketSampler.Direction (rows (m + 1))) :
(recordEquiv rows repeats p slots record).1 a =
record.2.1 (OriginalCutCalls.ownBucket rows repeats p a) :=
valuesEquiv_own rows repeats p (H (cutSlots p slots)) record.2.1 a
@[simp] theorem split_assembleRecord (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(own : BucketTape rows p slots) (complement : Complement rows repeats p slots) :
recordEquiv rows repeats p slots (assembleRecord rows repeats p slots own complement) =
(own, complement) :=
(recordEquiv rows repeats p slots).apply_symm_apply (own, complement)
@[simp] theorem assembleRecord_split (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(record : Record rows repeats p slots) :
assembleRecord rows repeats p slots (recordEquiv rows repeats p slots record).1
(recordEquiv rows repeats p slots record).2 = record :=
(recordEquiv rows repeats p slots).symm_apply_apply record
@[simp] theorem assembleRecord_exterior (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(own : BucketTape rows p slots) (complement : Complement rows repeats p slots) :
(assembleRecord rows repeats p slots own complement).1 = complement.1 := rfl
@[simp] theorem assembleRecord_below (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(own : BucketTape rows p slots) (complement : Complement rows repeats p slots) :
(assembleRecord rows repeats p slots own complement).2.2 = complement.2.2 := rfl
@[simp] theorem assembleRecord_own (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(own : BucketTape rows p slots) (complement : Complement rows repeats p slots)
(a : BucketSampler.Direction (rows (m + 1))) :
(assembleRecord rows repeats p slots own complement).2.1
(OriginalCutCalls.ownBucket rows repeats p a) = own a :=
valuesEquiv_symm_own rows repeats p (H (cutSlots p slots)) own complement.2.1 a
@[simp] theorem assembleRecord_other (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(own : BucketTape rows p slots) (complement : Complement rows repeats p slots)
(call : OtherIndex rows repeats p) :
(assembleRecord rows repeats p slots own complement).2.1 (otherCall rows repeats p call) =
complement.2.1 call :=
valuesEquiv_symm_other rows repeats p (H (cutSlots p slots)) own complement.2.1 call
theorem assembleRecord_nonown_eq (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(own₁ own₂ : BucketTape rows p slots) (complement : Complement rows repeats p slots)
(call : OriginalCutCalls.Index rows repeats p)
(hcall : ∀ a, call ≠ OriginalCutCalls.ownBucket rows repeats p a) :
(assembleRecord rows repeats p slots own₁ complement).2.1 call =
(assembleRecord rows repeats p slots own₂ complement).2.1 call := by
cases he : indexEquiv rows repeats p call with
| inl a =>
exfalso
apply hcall a
apply (indexEquiv rows repeats p).injective
rw [indexEquiv_ownBucket]
exact he
| inr other =>
have hc : call = otherCall rows repeats p other := by
apply (indexEquiv rows repeats p).injective
rw [indexEquiv_otherCall]
exact he
rw [hc, assembleRecord_other, assembleRecord_other]
def complementLaw (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(scalar : FiniteDistribution (H (cutSlots p slots)))
(exterior : FiniteDistribution (Exterior rows repeats p slots))
(below : FiniteDistribution (BelowArrays rows p slots)) :
FiniteDistribution (Complement rows repeats p slots) :=
exterior.product ((FiniteProduct.law
(fun _ : OtherIndex rows repeats p => scalar)).product below)
theorem independent_record_split (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(scalar : FiniteDistribution (H (cutSlots p slots)))
(exterior : FiniteDistribution (Exterior rows repeats p slots))
(below : FiniteDistribution (BelowArrays rows p slots)) :
(exterior.product ((FiniteProduct.law
(fun _ : OriginalCutCalls.Index rows repeats p => scalar)).product below)).pushforward
(recordEquiv rows repeats p slots) =
(BucketSampler.tapeLaw (rows (m + 1)) scalar).product
(complementLaw rows repeats p slots scalar exterior below) := by
rw [← FiniteDistribution.transport_eq_pushforward]
apply FiniteDistribution.eq_of_weight_eq
intro z
change exterior.weight z.2.1 *
((∏ call : OriginalCutCalls.Index rows repeats p,
scalar.weight (Sum.elim z.1 z.2.2.1 (indexEquiv rows repeats p call))) *
below.weight z.2.2.2) =
(∏ a : BucketSampler.Direction (rows (m + 1)), scalar.weight (z.1 a)) *
(exterior.weight z.2.1 *
((∏ other : OtherIndex rows repeats p, scalar.weight (z.2.2.1 other)) *
below.weight z.2.2.2))
rw [(indexEquiv rows repeats p).prod_comp
(fun tag => scalar.weight (Sum.elim z.1 z.2.2.1 tag)), Fintype.prod_sum_type]
simp only [Sum.elim_inl, Sum.elim_inr]
ring
def actualComplementLaw (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) :
FiniteDistribution (Complement rows repeats p slots) :=
complementLaw rows repeats p slots
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain (cutSlots p slots)))
(exteriorLaw rows repeats p slots)
(OriginalWholeCutLaw.belowArraysLaw rows repeats chosen q (cutSlots p slots))
theorem recordLaw_split (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) :
(OriginalWholeCutLaw.recordLaw rows repeats p chosen q slots).pushforward
(recordEquiv rows repeats p slots) =
(BucketSampler.tapeLaw (rows (m + 1))
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain (cutSlots p slots)))).product
(actualComplementLaw rows repeats p chosen q slots) :=
independent_record_split rows repeats p slots
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain (cutSlots p slots)))
(exteriorLaw rows repeats p slots)
(OriginalWholeCutLaw.belowArraysLaw rows repeats chosen q (cutSlots p slots))
theorem uniform_record_split (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot) :
(FiniteDistribution.uniform (Record rows repeats p slots)).pushforward
(recordEquiv rows repeats p slots) =
(FiniteDistribution.uniform (BucketTape rows p slots)).product
(FiniteDistribution.uniform (Complement rows repeats p slots)) := by
rw [← FiniteDistribution.transport_eq_pushforward,
UniformConditioning.uniform_transport, WholeCutSampler.uniform_product]
theorem uniform_record_split_bucketLaw (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot) :
(FiniteDistribution.uniform (Record rows repeats p slots)).pushforward
(recordEquiv rows repeats p slots) =
(BucketSampler.tapeLaw (rows (m + 1))
(FiniteDistribution.uniform (H (cutSlots p slots)))).product
(FiniteDistribution.uniform (Complement rows repeats p slots)) := by
rw [BucketSampler.tapeLaw, UniformLinearImage.law_uniform]
exact uniform_record_split rows repeats p slots
theorem uniform_assembleRecord (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot) :
((FiniteDistribution.uniform (BucketTape rows p slots)).product
(FiniteDistribution.uniform (Complement rows repeats p slots))).pushforward
(fun z => assembleRecord rows repeats p slots z.1 z.2) =
FiniteDistribution.uniform (Record rows repeats p slots) := by
rw [WholeCutSampler.uniform_product]
change (FiniteDistribution.uniform
(BucketTape rows p slots × Complement rows repeats p slots)).pushforward
(recordEquiv rows repeats p slots).symm = _
rw [← FiniteDistribution.transport_eq_pushforward]
exact UniformConditioning.uniform_transport (recordEquiv rows repeats p slots).symm
def zeroOwnReconstruction (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(complement : Complement rows repeats p slots) : Arrays slots rows :=
reconstructRecord rows repeats p slots (assembleRecord rows repeats p slots 0 complement)
theorem reconstruct_assembleRecord_outside (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(own₁ own₂ : BucketTape rows p slots) (complement : Complement rows repeats p slots)
(node : Nodes branch n) (hnode : node ≠ WholeArrayInteriorExterior.upperNode p) :
reconstructRecord rows repeats p slots (assembleRecord rows repeats p slots own₁ complement)
node =
reconstructRecord rows repeats p slots (assembleRecord rows repeats p slots own₂ complement)
node := by
change reconstruct rows repeats p slots complement.1
(assembleRecord rows repeats p slots own₁ complement).2.1 complement.2.2 node =
reconstruct rows repeats p slots complement.1
(assembleRecord rows repeats p slots own₂ complement).2.1 complement.2.2 node
exact OriginalCutBucketReplacement.reconstruct_outside_eq_of_nonown_eq rows repeats p slots
complement.1 _ _ complement.2.2
(assembleRecord_nonown_eq rows repeats p slots own₁ own₂ complement) node hnode
def background (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot)
(complement : Complement rows repeats p slots) :
HierarchicalMatrixTable.Background (rows := rows) slots (WholeArrayInteriorExterior.upperNode p) :=
HierarchicalMatrixTable.backgroundOf slots (WholeArrayInteriorExterior.upperNode p)
(zeroOwnReconstruction rows repeats p slots complement)
theorem background_assembleRecord (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(own : BucketTape rows p slots) (complement : Complement rows repeats p slots) :
HierarchicalMatrixTable.backgroundOf slots (WholeArrayInteriorExterior.upperNode p)
(reconstructRecord rows repeats p slots (assembleRecord rows repeats p slots own complement)) =
background rows repeats p slots complement := by
funext node
exact reconstruct_assembleRecord_outside rows repeats p slots own 0 complement node.val node.property
theorem background_record (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot)
(record : Record rows repeats p slots) :
HierarchicalMatrixTable.backgroundOf slots (WholeArrayInteriorExterior.upperNode p)
(reconstructRecord rows repeats p slots record) =
background rows repeats p slots (recordEquiv rows repeats p slots record).2 := by
have h := background_assembleRecord rows repeats p slots
(recordEquiv rows repeats p slots record).1 (recordEquiv rows repeats p slots record).2
rw [assembleRecord_split] at h
exact h
local instance backgroundFintype (rows : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
Fintype (HierarchicalMatrixTable.Background (rows := rows) slots
(WholeArrayInteriorExterior.upperNode p)) := Fintype.ofFinite _
def backgroundLaw (rows repeats : Nat → Nat) (p : Path branch n (m + 1))
(slots : Slots branch n → Fin t → MixedSupport.Slot) :
FiniteDistribution (HierarchicalMatrixTable.Background (rows := rows) slots
(WholeArrayInteriorExterior.upperNode p)) :=
(FiniteDistribution.uniform (Complement rows repeats p slots)).pushforward
(background rows repeats p slots)
def actualBackgroundLaw (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) :
FiniteDistribution (HierarchicalMatrixTable.Background (rows := rows) slots
(WholeArrayInteriorExterior.upperNode p)) :=
(actualComplementLaw rows repeats p chosen q slots).pushforward
(background rows repeats p slots)
theorem recordLaw_own_background (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) :
(OriginalWholeCutLaw.recordLaw rows repeats p chosen q slots).pushforward
(fun record =>
((recordEquiv rows repeats p slots record).1,
HierarchicalMatrixTable.backgroundOf slots (WholeArrayInteriorExterior.upperNode p)
(reconstructRecord rows repeats p slots record))) =
(BucketSampler.tapeLaw (rows (m + 1))
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain (cutSlots p slots)))).product
(actualBackgroundLaw rows repeats p chosen q slots) := by
calc
_ = ((OriginalWholeCutLaw.recordLaw rows repeats p chosen q slots).pushforward
(recordEquiv rows repeats p slots)).pushforward
(fun z => (z.1, background rows repeats p slots z.2)) := by
rw [FiniteDistribution.pushforward_comp]
apply congrArg (OriginalWholeCutLaw.recordLaw rows repeats p chosen q slots).pushforward
funext record
exact Prod.ext rfl (background_record rows repeats p slots record)
_ = ((BucketSampler.tapeLaw (rows (m + 1))
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain (cutSlots p slots)))).product
(actualComplementLaw rows repeats p chosen q slots)).pushforward
(fun z => (z.1, background rows repeats p slots z.2)) := by
rw [recordLaw_split]
_ = _ := by
simpa only [id_eq, FiniteDistribution.pushforward_id, actualBackgroundLaw] using
(FiniteDistribution.product_pushforward
(BucketSampler.tapeLaw (rows (m + 1))
(RecursiveSampler.law F2 repeats (.step chosen q) (LeafDomain (cutSlots p slots))))
(actualComplementLaw rows repeats p chosen q slots)
(id : BucketTape rows p slots → BucketTape rows p slots)
(background rows repeats p slots))
theorem uniform_own_background (rows repeats : Nat → Nat)
(p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot) :
(FiniteDistribution.uniform (Record rows repeats p slots)).pushforward
(fun record =>
((recordEquiv rows repeats p slots record).1,
HierarchicalMatrixTable.backgroundOf slots (WholeArrayInteriorExterior.upperNode p)
(reconstructRecord rows repeats p slots record))) =
(FiniteDistribution.uniform (BucketTape rows p slots)).product
(backgroundLaw rows repeats p slots) := by
calc
_ = ((FiniteDistribution.uniform (Record rows repeats p slots)).pushforward
(recordEquiv rows repeats p slots)).pushforward
(fun z => (z.1, background rows repeats p slots z.2)) := by
rw [FiniteDistribution.pushforward_comp]
apply congrArg ((FiniteDistribution.uniform (Record rows repeats p slots)).pushforward)
funext record
exact Prod.ext rfl (background_record rows repeats p slots record)
_ = ((FiniteDistribution.uniform (BucketTape rows p slots)).product
(FiniteDistribution.uniform (Complement rows repeats p slots))).pushforward
(fun z => (z.1, background rows repeats p slots z.2)) := by
rw [uniform_record_split]
_ = _ := by
simpa only [id_eq, FiniteDistribution.pushforward_id, backgroundLaw] using
(FiniteDistribution.product_pushforward
(FiniteDistribution.uniform (BucketTape rows p slots))
(FiniteDistribution.uniform (Complement rows repeats p slots))
(id : BucketTape rows p slots → BucketTape rows p slots)
(background rows repeats p slots))
end
end PerfectCompleteness.OriginalOwnBucketSplit
end
end OAI
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