openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Construction /OriginalChildBlocks.lean
| import OAI.Computability.PerfectCompleteness.Construction.CutChildGrouping | |
| import OAI.Computability.PerfectCompleteness.Construction.OriginalChildScalarAssemblyLemmas | |
| import OAI.Computability.PerfectCompleteness.Construction.OriginalChildScalars | |
| import OAI.Computability.PerfectCompleteness.Foundations.OriginalBelowCut | |
| import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerReplayTransportLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.OriginalChildBlocks | |
| open PointwiseSpaces RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open OriginalChildScalars | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| private theorem uniform_product {A B : Type*} [Fintype A] [Fintype B] | |
| [Nonempty A] [Nonempty B] : | |
| (FiniteDistribution.uniform A).product (FiniteDistribution.uniform B) = | |
| FiniteDistribution.uniform (A × B) := by | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro x | |
| simp only [FiniteDistribution.product, FiniteDistribution.uniform, | |
| Fintype.card_prod, Nat.cast_mul, one_div_mul_one_div] | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| def rawEquiv (calls : Nat) (rows : Nat → Nat) | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) (child : Fin (branch n)) : | |
| ((Fin calls → Square slots child) × Arrays (childSlots slots child) rows) ≃ | |
| ChildBlockCardinality.Raw calls rows (childSlots slots child) := Equiv.refl _ | |
| def arrayComponent (rows repeats : Nat → Nat) (chosen : Fin (branch n)) | |
| (q : Path branch n m) (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) : | |
| (child : Fin (branch n)) → FiniteDistribution (Arrays (childSlots slots child) rows) := | |
| ChildBlockMixture.component | |
| (fun child => WholeArraySampler.law rows repeats q (childSlots slots child)) chosen | |
| def specialLaw (calls : Nat) (rows repeats : Nat → Nat) (q : Path branch n m) | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) (child : Fin (branch n)) : | |
| FiniteDistribution (ChildBlockCardinality.Raw calls rows (childSlots slots child)) := | |
| @FiniteDistribution.transport | |
| ((Fin calls → Square slots child) × Arrays (childSlots slots child) rows) | |
| (ChildBlockCardinality.Raw calls rows (childSlots slots child)) | |
| (instFintypeProd _ _) (ChildBlockCardinality.rawFintype calls rows (childSlots slots child)) | |
| ((FiniteProduct.law (fun _ : Fin calls => specialSquareLaw repeats q slots child)).product | |
| (WholeArraySampler.law rows repeats q (childSlots slots child))) | |
| (rawEquiv calls rows slots child) | |
| def componentLaw (calls : Nat) (rows repeats : Nat → Nat) (chosen : Fin (branch n)) | |
| (q : Path branch n m) (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) | |
| (child : Fin (branch n)) : | |
| FiniteDistribution (ChildBlockCardinality.Raw calls rows (childSlots slots child)) := | |
| @FiniteDistribution.transport | |
| ((Fin calls → Square slots child) × Arrays (childSlots slots child) rows) | |
| (ChildBlockCardinality.Raw calls rows (childSlots slots child)) | |
| (instFintypeProd _ _) (ChildBlockCardinality.rawFintype calls rows (childSlots slots child)) | |
| ((callComponent calls repeats chosen q slots child).product | |
| (arrayComponent rows repeats chosen q slots child)) (rawEquiv calls rows slots child) | |
| theorem callComponent_uniform (calls : Nat) (repeats : Nat → Nat) | |
| (chosen : Fin (branch n)) (q : Path branch n m) | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) | |
| (child : Fin (branch n)) (hne : child ≠ chosen) : | |
| callComponent calls repeats chosen q slots child = | |
| FiniteDistribution.uniform (Fin calls → Square slots child) := by | |
| have h := squareComponent_other repeats q slots chosen ⟨child, hne⟩ | |
| unfold callComponent | |
| rw [h] | |
| exact UniformLinearImage.law_uniform | |
| theorem componentLaw_eq (calls : Nat) (rows repeats : Nat → Nat) | |
| (chosen : Fin (branch n)) (q : Path branch n m) | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) (child : Fin (branch n)) : | |
| componentLaw calls rows repeats chosen q slots child = | |
| ChildBlockMixture.component (specialLaw calls rows repeats q slots) chosen child := by | |
| by_cases h : child = chosen | |
| · subst child | |
| simp only [componentLaw, callComponent, squareComponent_self, arrayComponent, | |
| ChildBlockMixture.component, dite_eq_left rfl, dite_true, specialLaw] | |
| · have harray : arrayComponent rows repeats chosen q slots child = | |
| FiniteDistribution.uniform (Arrays (childSlots slots child) rows) := | |
| dite_eq_right (Ne.symm h) | |
| rw [componentLaw, callComponent_uniform calls repeats chosen q slots child h, harray] | |
| rw [uniform_product] | |
| have huniform := @UniformConditioning.uniform_transport | |
| ((Fin calls → Square slots child) × Arrays (childSlots slots child) rows) | |
| (ChildBlockCardinality.Raw calls rows (childSlots slots child)) | |
| (instFintypeProd _ _) (ChildBlockCardinality.rawFintype calls rows (childSlots slots child)) | |
| inferInstance inferInstance (rawEquiv calls rows slots child) | |
| have hcomponent : | |
| ChildBlockMixture.component (specialLaw calls rows repeats q slots) chosen child = | |
| FiniteDistribution.uniform | |
| (ChildBlockCardinality.Raw calls rows (childSlots slots child)) := by | |
| simp only [ChildBlockMixture.component, dite_eq_right (Ne.symm h)] | |
| exact huniform.trans hcomponent.symm | |
| abbrev SourceTape (calls : Nat) (rows repeats : Nat → Nat) (chosen : Fin (branch n)) | |
| (q : Path branch n m) (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) := | |
| (Fin calls → ScalarTape repeats chosen q slots) × | |
| WholeArraySampler.Tape rows repeats (.step chosen q) slots | |
| def sourceLaw (calls : Nat) (rows repeats : Nat → Nat) (chosen : Fin (branch n)) | |
| (q : Path branch n m) (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) : | |
| FiniteDistribution (SourceTape calls rows repeats chosen q slots) := | |
| (callsLaw calls repeats chosen q slots).product | |
| (WholeArraySampler.tapeLaw rows repeats (.step chosen q) slots) | |
| def rawMap (calls : Nat) (rows repeats : Nat → Nat) (chosen : Fin (branch n)) | |
| (q : Path branch n m) (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) | |
| (ω : SourceTape calls rows repeats chosen q slots) : | |
| (child : Fin (branch n)) → ChildBlockCardinality.Raw calls rows (childSlots slots child) := | |
| fun child => rawEquiv calls rows slots child | |
| (callContributions calls repeats chosen q slots ω.1 child, | |
| OriginalBelowCut.below rows repeats chosen q slots ω.2 child) | |
| @[simp] theorem rawMap_call (calls : Nat) (rows repeats : Nat → Nat) | |
| (chosen : Fin (branch n)) (q : Path branch n m) | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) | |
| (ω : SourceTape calls rows repeats chosen q slots) (child : Fin (branch n)) (call : Fin calls) : | |
| (rawMap calls rows repeats chosen q slots ω child).1 call = | |
| stepSquareMap repeats chosen q slots (ω.1 call) child := rfl | |
| @[simp] theorem rawMap_array (calls : Nat) (rows repeats : Nat → Nat) | |
| (chosen : Fin (branch n)) (q : Path branch n m) | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) | |
| (ω : SourceTape calls rows repeats chosen q slots) (child : Fin (branch n)) | |
| (node : Nodes branch n) : | |
| (rawMap calls rows repeats chosen q slots ω child).2 node = | |
| WholeArraySampler.evaluate rows repeats (.step chosen q) slots ω.2 (.inr (child, node)) := rfl | |
| theorem rawMap_product_law (calls : Nat) (rows repeats : Nat → Nat) | |
| (chosen : Fin (branch n)) (q : Path branch n m) | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) : | |
| (sourceLaw calls rows repeats chosen q slots).pushforward | |
| (rawMap calls rows repeats chosen q slots) = | |
| FiniteProduct.law (componentLaw calls rows repeats chosen q slots) := by | |
| calc | |
| _ = ((sourceLaw calls rows repeats chosen q slots).pushforward | |
| (fun ω => (callContributions calls repeats chosen q slots ω.1, | |
| OriginalBelowCut.below rows repeats chosen q slots ω.2))).pushforward | |
| (fun z child => rawEquiv calls rows slots child (z.1 child, z.2 child)) := | |
| (FiniteDistribution.pushforward_comp _ _ _).symm | |
| _ = ((FiniteProduct.law (callComponent calls repeats chosen q slots)).product | |
| (FiniteProduct.law (arrayComponent rows repeats chosen q slots))).pushforward | |
| (fun z child => rawEquiv calls rows slots child (z.1 child, z.2 child)) := by | |
| rw [sourceLaw, FiniteDistribution.product_pushforward, | |
| callContributions_law, OriginalBelowCut.below_pushforward, arrayComponent] | |
| _ = _ := by | |
| have hpair := OriginalChildBlockProducts.pair_transport_law | |
| (Ω := fun child : Fin (branch n) => Fin calls → ↥(Square slots child)) | |
| (Γ := fun child => Arrays (childSlots slots child) rows) | |
| (Δ := fun child => ChildBlockCardinality.Raw calls rows (childSlots slots child)) | |
| (rawEquiv calls rows slots) (callComponent calls repeats chosen q slots) | |
| (arrayComponent rows repeats chosen q slots) | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro x | |
| have hw := congrFun (congrArg FiniteDistribution.weight hpair) x | |
| simpa only [componentLaw, FiniteProduct.law, FiniteDistribution.transport] using hw | |
| theorem rawMap_law (calls : Nat) (rows repeats : Nat → Nat) | |
| (chosen : Fin (branch n)) (q : Path branch n m) | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) : | |
| (sourceLaw calls rows repeats chosen q slots).pushforward | |
| (rawMap calls rows repeats chosen q slots) = | |
| ChildBlockMixture.special (specialLaw calls rows repeats q slots) chosen := by | |
| rw [rawMap_product_law] | |
| apply congrArg (fun laws : (child : Fin (branch n)) → | |
| FiniteDistribution (ChildBlockCardinality.Raw calls rows (childSlots slots child)) => | |
| FiniteProduct.law laws) | |
| funext child | |
| exact componentLaw_eq calls rows repeats chosen q slots child | |
| def observed (calls : Nat) (rows repeats : Nat → Nat) (chosen : Fin (branch n)) | |
| (q : Path branch n m) (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) | |
| (ω : SourceTape calls rows repeats chosen q slots) : | |
| CutChildGrouping.Assembled (C := Fin calls) slots rows := | |
| (fun call => RecursiveSampler.evaluate F2 repeats (.step chosen q) (LeafDomain slots) (ω.1 call), | |
| OriginalBelowCut.below rows repeats chosen q slots ω.2) | |
| theorem assemble_rawMap (calls : Nat) (rows repeats : Nat → Nat) | |
| (chosen : Fin (branch n)) (q : Path branch n m) | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) | |
| (ω : SourceTape calls rows repeats chosen q slots) : | |
| CutChildGrouping.assemble slots rows (rawMap calls rows repeats chosen q slots ω) = | |
| observed calls rows repeats chosen q slots ω := by | |
| apply Prod.ext | |
| · funext call | |
| exact OriginalChildScalarAssembly.recursiveSum_callContributions | |
| calls repeats chosen q slots ω.1 call | |
| · rfl | |
| theorem observed_law (calls : Nat) (rows repeats : Nat → Nat) | |
| (chosen : Fin (branch n)) (q : Path branch n m) | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) : | |
| (sourceLaw calls rows repeats chosen q slots).pushforward | |
| (observed calls rows repeats chosen q slots) = | |
| (ChildBlockMixture.special (specialLaw calls rows repeats q slots) chosen).pushforward | |
| (CutChildGrouping.assemble slots rows) := by | |
| calc | |
| _ = ((sourceLaw calls rows repeats chosen q slots).pushforward | |
| (rawMap calls rows repeats chosen q slots)).pushforward | |
| (CutChildGrouping.assemble slots rows) := by | |
| rw [FiniteDistribution.pushforward_comp] | |
| congr 1 | |
| funext ω | |
| exact (assemble_rawMap calls rows repeats chosen q slots ω).symm | |
| _ = _ := by rw [rawMap_law] | |
| end | |
| end PerfectCompleteness.OriginalChildBlocks | |
| end | |
| end OAI | |