openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /SourceChildMarkedLaw.lean
| import OAI.Computability.PerfectCompleteness.Construction.SourceQuestionKernelJoint | |
| 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 | |