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import OAI.Computability.PerfectCompleteness.Foundations.MultiplicationFormInjectiveLemmas
import OAI.Computability.PerfectCompleteness.Foundations.WholeCutCallsLemmas
import OAI.Computability.PerfectCompleteness.Sampling.ChildCallNumberingLaws
import OAI.Computability.PerfectCompleteness.Sampling.SourceChildKernelMarginalLemmas
namespace OAI
section
namespace PerfectCompleteness.SourceChildMarkedLaw
open UniqueGamesTheorem.Foundations.Games
open RecursiveSpaces TreeSourceSpaces HierarchicalArrays SourceChildKernel
open scoped Classical
noncomputable section
private theorem uniform_equiv {A B : Type*}
[Fintype A] [Fintype B] [Nonempty A] [Nonempty B] (e : A ≃ B) :
(FiniteDistribution.uniform A).pushforward e = FiniteDistribution.uniform B := by
rw [FiniteDistribution.pushforward_equiv]
apply FiniteDistribution.eq_of_weight_eq
intro x
change 1 / (Fintype.card A : ℝ) = 1 / (Fintype.card B : ℝ)
rw [Fintype.card_congr e]
private theorem kernelJoint_map {S A B : Type*}
[Fintype S] [Fintype A] [Fintype B]
(μ : FiniteDistribution S) (P : S → FiniteDistribution A) (f : A → B) :
(CleanConditioning.kernelJoint μ P).pushforward (fun x => (x.1, f x.2)) =
CleanConditioning.kernelJoint μ (fun s => (P s).pushforward f) := by
apply SigmaObservation.eq_of_probability_eq
intro event
rw [FiniteDistribution.probability_pushforward, CleanConditioning.probability_kernelJoint,
CleanConditioning.probability_kernelJoint]
simp only [FiniteDistribution.probability_pushforward]
private theorem kernelJoint_observe {S A B : Type*}
[Fintype S] [Fintype A] [Fintype B]
(μ : FiniteDistribution S) (P : S → FiniteDistribution A) (f : S × A → B) :
(CleanConditioning.kernelJoint μ P).pushforward f =
μ.mixture (fun s => (P s).pushforward (fun a => f (s, a))) := by
apply SigmaObservation.eq_of_probability_eq
intro event
rw [FiniteDistribution.probability_pushforward, CleanConditioning.probability_kernelJoint,
FiniteDistribution.probability_mixture]
simp only [FiniteDistribution.probability_pushforward]
private theorem joint_product_observe {S Z A : Type*} {Y : S → Type*}
[Fintype S] [Fintype Z] [Fintype A] [∀ s, Fintype (Y s)]
(μ : FiniteDistribution S) (ν : FiniteDistribution Z)
(P : S → Z → FiniteDistribution A) (f : (s : S) → Z × A → Y s) :
(CleanConditioning.kernelJoint (μ.product ν) (fun x => P x.1 x.2)).pushforward
(fun x => (⟨x.1.1, f x.1.1 (x.1.2, x.2)⟩ : Σ s, Y s)) =
CompletionSoundness.sigmaLaw μ
(fun s => (CleanConditioning.kernelJoint ν (P s)).pushforward (f s)) := by
apply SigmaObservation.eq_of_probability_eq
intro event
rw [FiniteDistribution.probability_pushforward, CleanConditioning.probability_kernelJoint,
CompletionSoundness.sigmaLaw_probability]
simp only [FiniteDistribution.product, FiniteDistribution.expectation, Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro s _
rw [FiniteDistribution.probability_pushforward, CleanConditioning.probability_kernelJoint,
Finset.mul_sum]
apply Finset.sum_congr rfl
intro z _
exact mul_assoc _ _ _
variable {branch : Nat → Nat} {n t v m : Nat} {C : Type*} [Fintype C]
(rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch n) → Slots branch n)
def projectedLaw (i : Fin (branch n)) (s : Source (m := m) (t := t) designated i) :
FiniteDistribution (Native (C := C) (t := t) rows clauses designated i s) :=
(FiniteDistribution.uniform (Projected (C := C) (t := t) rows clauses designated i s)).pushforward
(SourceChildNativeLaw.pullback rows clauses designated i s true)
def markedChild (i : Fin (branch n)) (s : Source (m := m) (t := t) designated i)
(raw : Raw (C := C) (t := t) rows clauses designated i) :
Bool × Native (C := C) (t := t) rows clauses designated i s :=
(rawProjected rows clauses designated i raw,
SourceChildNativeLaw.decodeLeft rows clauses designated i s raw)
theorem kernel_markedChild (flag : Fin (branch n) → FiniteDistribution Bool)
(i : Fin (branch n)) (s : Source (m := m) (t := t) designated i) :
(kernel (C := C) (t := t) rows clauses designated flag i s).pushforward
(markedChild rows clauses designated i s) =
CleanConditioning.kernelJoint (flag i) (fun b => if b then
projectedLaw (C := C) (t := t) rows clauses designated i s
else FiniteDistribution.uniform (Native (C := C) (t := t) rows clauses designated i s)) := by
calc
_ = ((kernel (C := C) (t := t) rows clauses designated flag i s).pushforward
(SourceChildKernelLaw.physical rows clauses designated i s)).pushforward
(fun x => (x.1, SourceChildNativeLaw.physicalLeft rows clauses designated i s x)) :=
(FiniteDistribution.pushforward_comp _ _ _).symm
_ = CleanConditioning.kernelJoint (flag i) (fun b =>
(FiniteDistribution.uniform (Block C rows (mixedSlots clauses designated i s b))).pushforward
(SourceChildNativeLaw.pullback rows clauses designated i s b)) := by
rw [SourceChildKernelLaw.kernel_physical]
exact MarkedKernelProduct.physical_marked (flag i)
(fun b => FiniteDistribution.uniform (Block C rows (mixedSlots clauses designated i s b)))
(SourceChildNativeLaw.pullback rows clauses designated i s)
_ = _ := by
apply congrArg (CleanConditioning.kernelJoint (flag i))
funext b
cases b with
| false =>
have hf : SourceChildNativeLaw.pullback (C := C) rows clauses designated i s false =
(id : Native (C := C) (t := t) rows clauses designated i s → _) :=
funext (SourceChildNativeLaw.pullback_false rows clauses designated i s)
change (FiniteDistribution.uniform
(Native (C := C) (t := t) rows clauses designated i s)).pushforward
(SourceChildNativeLaw.pullback rows clauses designated i s false) = _
rw [hf, FiniteDistribution.pushforward_id]
rfl
| true => rfl
abbrev Questions := SourceQuestionPositionSplit.Questions
(branch := branch) (n := n) (t := t) (m := m)
abbrev Choice := SourceQuestionPositionSplit.ChildPositions (branch := branch) (n := n) (t := t)
def childSource (q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t)) :
Source (m := m) (t := t) designated i :=
SourceQuestionPositionSplit.join designated q (fun leaf k => z leaf.2 k) i
def questionSlots (q : Questions (branch := branch) (n := n) (t := t) (m := m)) :
Slots branch (n + 1) → Fin t → MixedSupport.Slot :=
sourceSlots clauses (PreliminarySampler.endpoints q)
theorem nativeSlots_childSource
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t)) :
nativeSlots clauses designated i (childSource designated q i z) =
childSlots (questionSlots clauses q) i :=
congrArg (fun slots => childSlots slots i)
(SourceQuestionPositionSplit.parentLeftSlots_join clauses designated q (fun leaf k => z leaf.2 k))
theorem childSource_sources
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(zs : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t))
(i : Fin (branch n)) :
childSource designated q i (zs i) = SourceQuestionKernelJoint.sources designated q zs i := rfl
def nativeEquiv (q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t)) :
Native (C := C) (t := t) rows clauses designated i (childSource designated q i z) ≃
CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i :=
Equiv.cast (congrArg (Block C rows) (nativeSlots_childSource clauses designated q i z))
def projectedSlots (q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t)) :
Slots branch n → Fin t → MixedSupport.Slot :=
mixedSlots clauses designated i (childSource designated q i z) true
def questionProjection
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t)) :
∀ leaf k, MixedSupport.Projection (childSlots (questionSlots clauses q) i leaf k)
(projectedSlots clauses designated q i z leaf k) :=
cast (congrArg
(fun slots : Slots branch n → Fin t → MixedSupport.Slot =>
∀ leaf k, MixedSupport.Projection (slots leaf k)
(projectedSlots clauses designated q i z leaf k))
(nativeSlots_childSource clauses designated q i z))
(projection clauses designated i (childSource designated q i z) true)
omit [Fintype C] in
theorem nativeEquiv_pullback
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t))
(block : Projected (C := C) (t := t) rows clauses designated i (childSource designated q i z)) :
nativeEquiv rows clauses designated q i z
(SourceChildNativeLaw.pullback rows clauses designated i (childSource designated q i z) true block) =
ChildCallNumbering.pullback rows (questionProjection clauses designated q i z) block :=
ChildCallNumbering.pullback_cast rows (nativeSlots_childSource clauses designated q i z)
(projection clauses designated i (childSource designated q i z) true) block
omit [Fintype C] in
theorem nativeEquiv_pullback_call
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t))
(block : Projected (C := C) (t := t) rows clauses designated i (childSource designated q i z))
(call : C) (x : Domain (childSlots (questionSlots clauses q) i)) :
((nativeEquiv rows clauses designated q i z
(SourceChildNativeLaw.pullback rows clauses designated i (childSource designated q i z) true block)).1 call).val x =
(block.1 call).val (sourceProjection (questionProjection clauses designated q i z) x) := by
rw [nativeEquiv_pullback]; rfl
omit [Fintype C] in
theorem nativeEquiv_pullback_arrays
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t))
(block : Projected (C := C) (t := t) rows clauses designated i (childSource designated q i z))
(x : Domain (childSlots (questionSlots clauses q) i)) :
fullJoint (nativeEquiv rows clauses designated q i z
(SourceChildNativeLaw.pullback rows clauses designated i (childSource designated q i z) true block)).2 x =
fullJoint block.2 (sourceProjection (questionProjection clauses designated q i z) x) := by
rw [nativeEquiv_pullback]; rfl
def replacementAt (q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t)) :
FiniteDistribution (CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i) :=
(projectedLaw (C := C) (t := t) rows clauses designated i (childSource designated q i z)).pushforward
(nativeEquiv rows clauses designated q i z)
def replacementLaws (q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) :
FiniteDistribution (CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i) :=
(FiniteDistribution.uniform (Choice (branch := branch) (n := n) (t := t))).mixture
(replacementAt rows clauses designated q i)
theorem replacementAt_eq_liftedLaw
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t)) :
replacementAt (C := C) rows clauses designated q i z =
ChildCallNumberingLaws.liftedLaw (C := C) rows (questionProjection clauses designated q i z) := by
let μ : FiniteDistribution
(Projected (C := C) (t := t) rows clauses designated i (childSource designated q i z)) :=
FiniteDistribution.uniform _
let f : Projected (C := C) (t := t) rows clauses designated i (childSource designated q i z) →
Native (C := C) (t := t) rows clauses designated i (childSource designated q i z) :=
fun block => SourceChildNativeLaw.pullback rows clauses designated i
(childSource designated q i z) true block
let g : Native (C := C) (t := t) rows clauses designated i (childSource designated q i z) →
CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i :=
nativeEquiv rows clauses designated q i z
exact (FiniteDistribution.pushforward_comp μ f g).trans
(congrArg (fun k : Projected (C := C) (t := t) rows clauses designated i
(childSource designated q i z) →
CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i => μ.pushforward k)
(funext fun block => nativeEquiv_pullback rows clauses designated q i z block))
theorem replacementLaws_eq
(q : Questions (branch := branch) (n := n) (t := t) (m := m)) :
replacementLaws (C := C) rows clauses designated q =
ChildCallNumberingLaws.replacementLaws (C := C) rows (childSlots (questionSlots clauses q))
(projectedSlots clauses designated q) (questionProjection clauses designated q)
(fun _ => FiniteDistribution.uniform (Choice (branch := branch) (n := n) (t := t))) := by
funext i
unfold replacementLaws ChildCallNumberingLaws.replacementLaws ChildCallNumberingLaws.mixtureLaw
apply congrArg (FiniteDistribution.mixture
(FiniteDistribution.uniform (Choice (branch := branch) (n := n) (t := t))))
funext z
exact replacementAt_eq_liftedLaw rows clauses designated q i z
def questionMarkedChild
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t))
(raw : Raw (C := C) (t := t) rows clauses designated i) :
Bool × CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i :=
(rawProjected rows clauses designated i raw,
nativeEquiv rows clauses designated q i z
(SourceChildNativeLaw.decodeLeft rows clauses designated i (childSource designated q i z) raw))
theorem kernel_questionMarkedChild
(flag : Fin (branch n) → FiniteDistribution Bool)
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(i : Fin (branch n)) (z : Choice (branch := branch) (n := n) (t := t)) :
(kernel (C := C) (t := t) rows clauses designated flag i (childSource designated q i z)).pushforward
(questionMarkedChild rows clauses designated q i z) =
CleanConditioning.kernelJoint (flag i) (fun b => if b then
replacementAt (C := C) rows clauses designated q i z
else FiniteDistribution.uniform (CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i)) := by
calc
_ = ((kernel (C := C) (t := t) rows clauses designated flag i (childSource designated q i z)).pushforward
(markedChild rows clauses designated i (childSource designated q i z))).pushforward
(fun x => (x.1, nativeEquiv rows clauses designated q i z x.2)) :=
(FiniteDistribution.pushforward_comp _ _ _).symm
_ = _ := by
rw [kernel_markedChild, kernelJoint_map]
apply congrArg (CleanConditioning.kernelJoint (flag i))
funext b
cases b with
| false => exact uniform_equiv (nativeEquiv rows clauses designated q i z)
| true => rfl
abbrev Marked (q : Questions (branch := branch) (n := n) (t := t) (m := m)) :=
(Fin (branch n) → Bool) × CutChildGrouping.Raw (C := C) (questionSlots clauses q) rows
def markedBlocks (q : Questions (branch := branch) (n := n) (t := t) (m := m))
(zs : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t))
(raw : SourceChildKernelJoint.RawTuple (C := C) (t := t) rows clauses designated) :
Marked (C := C) rows clauses q :=
MarkedKernelProduct.markEquiv (fun i => questionMarkedChild rows clauses designated q i (zs i) (raw i))
theorem kernels_markedBlocks
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(zs : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t))
(β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) :
(FiniteProduct.law (fun i => kernel (C := C) (t := t) rows clauses designated
(fun _ => ProjectionPosterior.bernoulli β hβ hβ') i
(SourceQuestionKernelJoint.sources designated q zs i))).pushforward
(markedBlocks rows clauses designated q zs) =
SparseReplacement.markedLaw
(fun i => FiniteDistribution.uniform (CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i))
(fun i => replacementAt rows clauses designated q i (zs i)) β hβ hβ' := by
let P : (i : Fin (branch n)) → FiniteDistribution (Raw (C := C) (t := t) rows clauses designated i) :=
fun i => kernel rows clauses designated
(fun _ => ProjectionPosterior.bernoulli β hβ hβ') i
(SourceQuestionKernelJoint.sources designated q zs i)
let f : (i : Fin (branch n)) → Raw (C := C) (t := t) rows clauses designated i →
Bool × CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i :=
fun i => questionMarkedChild rows clauses designated q i (zs i)
let e : ((i : Fin (branch n)) →
Bool × CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i) ≃
Marked (C := C) rows clauses q := MarkedKernelProduct.markEquiv
calc
_ = ((FiniteProduct.law P).pushforward (fun raw i => f i (raw i))).pushforward e :=
(FiniteDistribution.pushforward_comp (FiniteProduct.law P)
(fun raw i => f i (raw i)) e).symm
_ = (FiniteProduct.law (fun i => (P i).pushforward (f i))).pushforward e :=
congrArg (fun law : FiniteDistribution ((i : Fin (branch n)) →
Bool × CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i) =>
law.pushforward e)
(FiniteProduct.pushforward_map P f)
_ = (FiniteProduct.law (fun i => CleanConditioning.kernelJoint
(ProjectionPosterior.bernoulli β hβ hβ') (fun b => if b then
replacementAt (C := C) rows clauses designated q i (zs i)
else FiniteDistribution.uniform
(CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i)))).pushforward e :=
congrArg (fun laws : (i : Fin (branch n)) → FiniteDistribution
(Bool × CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i) =>
(FiniteProduct.law laws).pushforward e)
(funext fun i => kernel_questionMarkedChild rows clauses designated
(fun _ => ProjectionPosterior.bernoulli β hβ hβ') q i (zs i))
_ = _ := MarkedKernelProduct.marked_product _ _ β hβ hβ'
theorem conditional_markedLaw
(q : Questions (branch := branch) (n := n) (t := t) (m := m))
(β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) :
(CleanConditioning.kernelJoint
(FiniteProduct.law (fun _ : Fin (branch n) =>
FiniteDistribution.uniform (Choice (branch := branch) (n := n) (t := t))))
(fun zs => FiniteProduct.law (fun i => kernel (C := C) (t := t) rows clauses designated
(fun _ => ProjectionPosterior.bernoulli β hβ hβ') i
(SourceQuestionKernelJoint.sources designated q zs i)))).pushforward
(fun x => markedBlocks rows clauses designated q x.1 x.2) =
SparseReplacement.markedLaw
(fun i => FiniteDistribution.uniform (CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i))
(replacementLaws rows clauses designated q) β hβ hβ' := by
rw [kernelJoint_observe]
simp only [kernels_markedBlocks]
exact MarkedKernelProduct.mixture_markedLaw _ _ _ β hβ hβ'
abbrev Observation :=
Σ q : Questions (branch := branch) (n := n) (t := t) (m := m),
Marked (C := C) rows clauses q
def observe (sample : Sample (C := C) (t := t) rows clauses designated) :
Observation (branch := branch) (n := n) (C := C) (t := t) rows clauses :=
let grouped := SourceQuestionKernelJoint.groupEquiv rows clauses designated sample
⟨grouped.1.1, markedBlocks rows clauses designated grouped.1.1 grouped.1.2 grouped.2⟩
@[simp] theorem observe_mask
(sample : Sample (C := C) (t := t) rows clauses designated) (i : Fin (branch n)) :
(observe rows clauses designated sample).2.1 i =
rawProjected rows clauses designated i (sample i).2.2 := rfl
theorem original_marked_law [NeZero m]
(β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) :
(originalLaw (C := C) (t := t) rows clauses designated
(fun _ => ProjectionPosterior.bernoulli β hβ hβ')).pushforward
(observe rows clauses designated) =
CompletionSoundness.sigmaLaw
(PreliminarySampler.questionsLaw (branch := branch) (n := n + 1) (t := t) (m := m))
(fun q => SparseReplacement.markedLaw
(fun i => FiniteDistribution.uniform
(CutChildGrouping.Child (C := C) (questionSlots clauses q) rows i))
(replacementLaws rows clauses designated q) β hβ hβ') := by
unfold observe
rw [← FiniteDistribution.pushforward_comp
(originalLaw (C := C) (t := t) rows clauses designated
(fun _ => ProjectionPosterior.bernoulli β hβ hβ'))
(SourceQuestionKernelJoint.groupEquiv rows clauses designated)
(fun x => (⟨x.1.1, markedBlocks rows clauses designated x.1.1 x.1.2 x.2⟩ :
Observation (branch := branch) (n := n) (C := C) (t := t) rows clauses)),
SourceQuestionKernelJoint.originalLaw_group]
exact (joint_product_observe
(S := Questions (branch := branch) (n := n) (t := t) (m := m))
(Z := SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t))
(A := SourceChildKernelJoint.RawTuple (C := C) (t := t) rows clauses designated)
(Y := fun q => Marked (C := C) rows clauses q)
(PreliminarySampler.questionsLaw (branch := branch) (n := n + 1) (t := t) (m := m))
(FiniteProduct.law (fun _ : Fin (branch n) =>
FiniteDistribution.uniform (Choice (branch := branch) (n := n) (t := t))))
(fun q zs => FiniteProduct.law (fun i => kernel (C := C) (t := t) rows clauses designated
(fun _ => ProjectionPosterior.bernoulli β hβ hβ') i
(SourceQuestionKernelJoint.sources designated q zs i)))
(fun q x => markedBlocks rows clauses designated q x.1 x.2)).trans
(congrArg (CompletionSoundness.sigmaLaw
(PreliminarySampler.questionsLaw (branch := branch) (n := n + 1) (t := t) (m := m)))
(funext fun q => conditional_markedLaw rows clauses designated q β hβ hβ'))
theorem original_extended_marked_law [NeZero m] {root : Nat}
(repeats : Nat → Nat) (p : DescendantSpaces.Path branch root (n + 1))
(β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) :
(originalLaw (C := Option (WholeCutCalls.Index rows repeats p)) (t := t)
rows clauses designated (fun _ => ProjectionPosterior.bernoulli β hβ hβ')).pushforward
(observe rows clauses designated) =
CompletionSoundness.sigmaLaw
(PreliminarySampler.questionsLaw (branch := branch) (n := n + 1) (t := t) (m := m))
(fun q => SparseReplacement.markedLaw
(fun i => FiniteDistribution.uniform
(CutChildGrouping.Child (C := Option (WholeCutCalls.Index rows repeats p))
(questionSlots clauses q) rows i))
(replacementLaws rows clauses designated q) β hβ hβ') :=
original_marked_law rows clauses designated β hβ hβ'
end
end PerfectCompleteness.SourceChildMarkedLaw
end
end OAI
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