openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /ChildCallNumberingLaws.lean
| import OAI.Computability.PerfectCompleteness.Construction.ChildCallNumbering | |
| import OAI.Computability.PerfectCompleteness.Foundations.DependentPredictionDifferenceLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.MultiplicationFormInjectiveLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.RationalFiniteLawLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.ChildCallNumberingLaws | |
| noncomputable section | |
| open scoped BigOperators Classical | |
| open RecursiveSpaces TreeSourceSpaces | |
| open UniqueGamesTheorem.Foundations.Games | |
| variable {branch : Nat → Nat} {n t : Nat} {C : Type*} [Fintype C] | |
| def liftedLaw (rows : Nat → Nat) | |
| {slots projected : Slots branch n → Fin t → MixedSupport.Slot} | |
| (p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) : | |
| FiniteDistribution (ChildCallNumbering.Raw (C := C) rows slots) := | |
| (FiniteDistribution.uniform (ChildCallNumbering.Raw (C := C) rows projected)).pushforward | |
| (ChildCallNumbering.pullback rows p) | |
| theorem liftedLaw_numbered (rows : Nat → Nat) | |
| {slots projected : Slots branch n → Fin t → MixedSupport.Slot} | |
| (p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) : | |
| (liftedLaw (C := C) rows p).pushforward (ChildCallNumbering.equiv rows slots) = | |
| ChildBlockProjection.liftedLaw (Fintype.card C) rows p := by | |
| unfold liftedLaw ChildBlockProjection.liftedLaw | |
| rw [← ChildCallNumbering.uniform_equiv (C := C) rows projected] | |
| rw [FiniteDistribution.pushforward_comp, FiniteDistribution.pushforward_comp] | |
| congr 1 | |
| def mixtureLaw {Z : Type*} [Fintype Z] (rows : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (projected : Z → Slots branch n → Fin t → MixedSupport.Slot) | |
| (p : ∀ z s k, MixedSupport.Projection (slots s k) (projected z s k)) | |
| (choiceLaw : FiniteDistribution Z) : | |
| FiniteDistribution (ChildCallNumbering.Raw (C := C) rows slots) := | |
| choiceLaw.mixture (fun z => liftedLaw rows (p z)) | |
| theorem mixtureLaw_numbered {Z : Type*} [Fintype Z] (rows : Nat → Nat) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (projected : Z → Slots branch n → Fin t → MixedSupport.Slot) | |
| (p : ∀ z s k, MixedSupport.Projection (slots s k) (projected z s k)) | |
| (choiceLaw : FiniteDistribution Z) : | |
| (mixtureLaw (C := C) rows slots projected p choiceLaw).pushforward | |
| (ChildCallNumbering.equiv rows slots) = | |
| ChildBlockProjection.mixtureLaw (Fintype.card C) rows slots projected p choiceLaw := by | |
| simp only [mixtureLaw, FiniteDistribution.pushforward_mixture, | |
| liftedLaw_numbered, ChildBlockProjection.mixtureLaw] | |
| variable {I : Type*} [Fintype I] [DecidableEq I] | |
| {Z : I → Type*} [∀ i, Fintype (Z i)] | |
| def replacementLaws (rows : Nat → Nat) | |
| (slots : I → Slots branch n → Fin t → MixedSupport.Slot) | |
| (projected : (i : I) → Z i → Slots branch n → Fin t → MixedSupport.Slot) | |
| (p : ∀ i z s k, MixedSupport.Projection (slots i s k) (projected i z s k)) | |
| (choiceLaw : (i : I) → FiniteDistribution (Z i)) : | |
| (i : I) → FiniteDistribution (ChildCallNumbering.Raw (C := C) rows (slots i)) := | |
| fun i => mixtureLaw rows (slots i) (projected i) (p i) (choiceLaw i) | |
| def number (rows : Nat → Nat) | |
| (slots : I → Slots branch n → Fin t → MixedSupport.Slot) | |
| (x : (i : I) → ChildCallNumbering.Raw (C := C) rows (slots i)) : | |
| (i : I) → ChildBlockCardinality.Raw (Fintype.card C) rows (slots i) := | |
| fun i => ChildCallNumbering.equiv rows (slots i) (x i) | |
| def unnumber (rows : Nat → Nat) | |
| (slots : I → Slots branch n → Fin t → MixedSupport.Slot) | |
| (x : (i : I) → ChildBlockCardinality.Raw (Fintype.card C) rows (slots i)) : | |
| (i : I) → ChildCallNumbering.Raw (C := C) rows (slots i) := | |
| fun i => (ChildCallNumbering.equiv rows (slots i)).symm (x i) | |
| omit [Fintype I] [DecidableEq I] in | |
| @[simp] theorem unnumber_number (rows : Nat → Nat) | |
| (slots : I → Slots branch n → Fin t → MixedSupport.Slot) | |
| (x : (i : I) → ChildCallNumbering.Raw (C := C) rows (slots i)) : | |
| unnumber rows slots (number rows slots x) = x := by | |
| funext i | |
| exact (ChildCallNumbering.equiv rows (slots i)).symm_apply_apply (x i) | |
| omit [Fintype I] [DecidableEq I] in | |
| @[simp] theorem number_unnumber (rows : Nat → Nat) | |
| (slots : I → Slots branch n → Fin t → MixedSupport.Slot) | |
| (x : (i : I) → ChildBlockCardinality.Raw (Fintype.card C) rows (slots i)) : | |
| number rows slots (unnumber (C := C) rows slots x) = x := by | |
| funext i | |
| exact (ChildCallNumbering.equiv rows (slots i)).apply_symm_apply (x i) | |
| theorem uniform_product_numbered (rows : Nat → Nat) | |
| (slots : I → Slots branch n → Fin t → MixedSupport.Slot) : | |
| (FiniteProduct.law (fun i => FiniteDistribution.uniform | |
| (ChildCallNumbering.Raw (C := C) rows (slots i)))).pushforward (number rows slots) = | |
| FiniteProduct.law (fun i => FiniteDistribution.uniform | |
| (ChildBlockCardinality.Raw (Fintype.card C) rows (slots i))) := by | |
| unfold number | |
| rw [FiniteProduct.pushforward_map] | |
| congr 1 | |
| funext i | |
| exact ChildCallNumbering.uniform_equiv rows (slots i) | |
| theorem conditional_product_numbered (rows : Nat → Nat) | |
| (slots : I → Slots branch n → Fin t → MixedSupport.Slot) | |
| (projected : (i : I) → Z i → Slots branch n → Fin t → MixedSupport.Slot) | |
| (p : ∀ i z s k, MixedSupport.Projection (slots i s k) (projected i z s k)) | |
| (choiceLaw : (i : I) → FiniteDistribution (Z i)) (mask : I → Bool) : | |
| (FiniteProduct.law (fun i => if mask i then | |
| replacementLaws (C := C) rows slots projected p choiceLaw i else | |
| FiniteDistribution.uniform (ChildCallNumbering.Raw (C := C) rows (slots i)))).pushforward | |
| (number rows slots) = | |
| FiniteProduct.law (fun i => if mask i then | |
| ChildBlockProjection.replacementLaws (Fintype.card C) rows slots projected p choiceLaw i else | |
| FiniteDistribution.uniform (ChildBlockCardinality.Raw (Fintype.card C) rows (slots i))) := by | |
| unfold number | |
| rw [FiniteProduct.pushforward_map] | |
| congr 1 | |
| funext i | |
| cases mask i with | |
| | false => | |
| simp only [Bool.false_eq_true, ite_false] | |
| exact ChildCallNumbering.uniform_equiv rows (slots i) | |
| | true => | |
| simp only [ite_true] | |
| exact mixtureLaw_numbered rows (slots i) (projected i) (p i) (choiceLaw i) | |
| theorem markedLaw_numbered (rows : Nat → Nat) | |
| (slots : I → Slots branch n → Fin t → MixedSupport.Slot) | |
| (projected : (i : I) → Z i → Slots branch n → Fin t → MixedSupport.Slot) | |
| (p : ∀ i z s k, MixedSupport.Projection (slots i s k) (projected i z s k)) | |
| (choiceLaw : (i : I) → FiniteDistribution (Z i)) | |
| (β : ℝ) (hβ : 0 ≤ β) (hβ' : β ≤ 1) : | |
| (SparseReplacement.markedLaw | |
| (fun i => FiniteDistribution.uniform (ChildCallNumbering.Raw (C := C) rows (slots i))) | |
| (replacementLaws rows slots projected p choiceLaw) β hβ hβ').pushforward | |
| (fun z => (z.1, number rows slots z.2)) = | |
| SparseReplacement.markedLaw | |
| (fun i => FiniteDistribution.uniform | |
| (ChildBlockCardinality.Raw (Fintype.card C) rows (slots i))) | |
| (ChildBlockProjection.replacementLaws (Fintype.card C) rows slots projected p choiceLaw) | |
| β hβ hβ' := by | |
| apply SigmaObservation.eq_of_probability_eq | |
| intro event | |
| rw [FiniteDistribution.probability_pushforward] | |
| unfold SparseReplacement.markedLaw | |
| rw [CleanConditioning.probability_kernelJoint, CleanConditioning.probability_kernelJoint] | |
| apply Finset.sum_congr rfl | |
| intro mask _ | |
| apply congrArg (fun x : ℝ => | |
| (FiniteProduct.law (fun _ : I => ProjectionPosterior.bernoulli β hβ hβ')).weight mask * x) | |
| have h := congrArg (fun μ => μ.probability (fun x => event (mask, x))) | |
| (conditional_product_numbered (C := C) rows slots projected p choiceLaw mask) | |
| simpa only [FiniteDistribution.probability_pushforward] using h | |
| end | |
| end PerfectCompleteness.ChildCallNumberingLaws | |
| end | |
| end OAI | |