openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /LowerCutPairLemmas.lean
| import OAI.Computability.PerfectCompleteness.Algebra.ChildBilinearCollisionTransfer | |
| import OAI.Computability.PerfectCompleteness.Decoding.ChildAssemblyProjection | |
| import OAI.Computability.PerfectCompleteness.Decoding.LowerCutNodeCoordinates | |
| import OAI.Computability.PerfectCompleteness.Decoding.LowerCutRowsLemmas | |
| import OAI.Computability.PerfectCompleteness.Decoding.LowerDirectionFiber | |
| import OAI.Computability.PerfectCompleteness.Foundations.CutCallExtension | |
| import OAI.Computability.PerfectCompleteness.Foundations.MultiplicationFormInjectiveLemmas | |
| import OAI.Computability.PerfectCompleteness.Machines.LowerCutDecoderInput | |
| import OAI.Computability.PerfectCompleteness.Machines.OwnInputReferenceLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.LowerCutPair | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open WholeCutGrouping | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped Classical | |
| abbrev F2 := ZMod 2 | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| (rows repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| abbrev Calls := WholeCutCalls.Index rows repeats p | |
| abbrev Raw := CutCallExtension.Extended (C := Calls rows repeats p) (cutSlots p slots) rows | |
| abbrev NativeRows := Fin (rows (m + 1)) → H (cutSlots p slots) | |
| abbrev NativeMatrix := Module.Dual F2 (H (cutSlots p slots)) →ₗ[F2] (Fin (rows (m + 1)) → F2) | |
| def first (exterior : Exterior rows repeats p slots) (raw : Raw rows repeats p slots) : | |
| Arrays slots rows := | |
| LowerCutRows.reconstruct rows repeats p slots exterior | |
| (CutCallExtension.original (cutSlots p slots) rows raw) | |
| def fresh (raw : Raw rows repeats p slots) : H (cutSlots p slots) := | |
| CutCallExtension.fresh (cutSlots p slots) rows raw | |
| def second (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) : | |
| Arrays slots rows := | |
| SelectedArrayReplacement.replace rows p slots (first rows repeats p slots exterior raw) | |
| (LowerDirectionFiber.replace a | |
| (SelectedArrayReplacement.selectedRows rows p slots (first rows repeats p slots exterior raw)) | |
| (fresh rows repeats p slots raw)) | |
| def arraysPair (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) : | |
| Arrays slots rows × Arrays slots rows := | |
| (first rows repeats p slots exterior raw, second rows repeats p slots exterior a raw) | |
| @[simp] theorem selected_second (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) : | |
| SelectedArrayReplacement.selectedRows rows p slots (second rows repeats p slots exterior a raw) = | |
| LowerDirectionFiber.replace a | |
| (SelectedArrayReplacement.selectedRows rows p slots (first rows repeats p slots exterior raw)) | |
| (fresh rows repeats p slots raw) := | |
| SelectedArrayReplacement.selectedRows_replace rows p slots _ _ | |
| theorem second_outside (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) | |
| (node : Nodes branch n) (hne : node ≠ WholeArrayInteriorExterior.upperNode p) : | |
| second rows repeats p slots exterior a raw node = | |
| first rows repeats p slots exterior raw node := | |
| SelectedArrayReplacement.replace_outside rows p slots _ _ node hne | |
| theorem selected_quotient (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) : | |
| LowerDirectionFiber.quotientQuery (H (cutSlots p slots)) a | |
| (SelectedArrayReplacement.selectedRows rows p slots | |
| (second rows repeats p slots exterior a raw)) = | |
| LowerDirectionFiber.quotientQuery (H (cutSlots p slots)) a | |
| (SelectedArrayReplacement.selectedRows rows p slots | |
| (first rows repeats p slots exterior raw)) := by | |
| rw [selected_second] | |
| exact LowerDirectionFiber.quotientQuery_replace _ _ _ _ | |
| def readRowsFresh (exterior : Exterior rows repeats p slots) (raw : Raw rows repeats p slots) : | |
| NativeRows rows p slots × H (cutSlots p slots) := | |
| (SelectedArrayReplacement.selectedRows rows p slots (first rows repeats p slots exterior raw), | |
| fresh rows repeats p slots raw) | |
| def selectPair : | |
| (CutChildGrouping.Assembled (C := Calls rows repeats p) (cutSlots p slots) rows × | |
| H (cutSlots p slots)) →ₗ[F2] (NativeRows rows p slots × H (cutSlots p slots)) := | |
| (LowerCutRows.selectRows rows repeats p slots).prodMap LinearMap.id | |
| theorem selectPair_surjective : Function.Surjective (selectPair rows repeats p slots) := by | |
| rintro ⟨S, h⟩ | |
| obtain ⟨record, hrecord⟩ := LowerCutRows.selectRows_surjective rows repeats p slots S | |
| exact ⟨(record, h), Prod.ext hrecord rfl⟩ | |
| @[simp] theorem readRowsFresh_eq (exterior : Exterior rows repeats p slots) | |
| (raw : Raw rows repeats p slots) : | |
| readRowsFresh rows repeats p slots exterior raw = | |
| selectPair rows repeats p slots | |
| (CutCallExtension.observe (cutSlots p slots) rows raw) := by | |
| apply Prod.ext | |
| · exact LowerCutRows.selectedRows_reconstruct_eq rows repeats p slots exterior _ | |
| · rfl | |
| theorem readRowsFresh_law (exterior : Exterior rows repeats p slots) : | |
| (CutChildGrouping.rawLaw (C := Option (Calls rows repeats p)) (cutSlots p slots) rows).pushforward | |
| (readRowsFresh rows repeats p slots exterior) = | |
| (FiniteDistribution.uniform (NativeRows rows p slots)).product | |
| (FiniteDistribution.uniform (H (cutSlots p slots))) := by | |
| change (CutChildGrouping.rawLaw (C := Option (Calls rows repeats p)) (cutSlots p slots) rows).pushforward | |
| (fun raw => readRowsFresh rows repeats p slots exterior raw) = _ | |
| simp only [readRowsFresh_eq] | |
| rw [← FiniteDistribution.pushforward_comp, CutCallExtension.observe_law, | |
| WholeCutSampler.uniform_product, | |
| UniformLinearImage.uniform_pushforward_linearMap | |
| (selectPair rows repeats p slots) (selectPair_surjective rows repeats p slots)] | |
| exact WholeCutSampler.uniform_product.symm | |
| def selectedPair (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) : | |
| NativeRows rows p slots × NativeRows rows p slots := | |
| (SelectedArrayReplacement.selectedRows rows p slots (first rows repeats p slots exterior raw), | |
| SelectedArrayReplacement.selectedRows rows p slots (second rows repeats p slots exterior a raw)) | |
| theorem selectedPair_law (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) : | |
| (CutChildGrouping.rawLaw (C := Option (Calls rows repeats p)) (cutSlots p slots) rows).pushforward | |
| (selectedPair rows repeats p slots exterior a) = | |
| LowerDirectionFiber.fiberPairLaw (H := H (cutSlots p slots)) a := by | |
| have h := congrArg | |
| (fun μ : FiniteDistribution (NativeRows rows p slots × H (cutSlots p slots)) => | |
| μ.pushforward (fun z => (z.1, LowerDirectionFiber.replace a z.1 z.2))) | |
| (readRowsFresh_law rows repeats p slots exterior) | |
| rw [FiniteDistribution.pushforward_comp, LowerDirectionFiber.two_copy_law] at h | |
| change (CutChildGrouping.rawLaw (C := Option (Calls rows repeats p)) (cutSlots p slots) rows).pushforward | |
| (fun raw => | |
| (SelectedArrayReplacement.selectedRows rows p slots (first rows repeats p slots exterior raw), | |
| SelectedArrayReplacement.selectedRows rows p slots (second rows repeats p slots exterior a raw))) = _ | |
| simpa only [readRowsFresh, selected_second] using h | |
| def matrixPair (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) (raw : Raw rows repeats p slots) : | |
| NativeMatrix rows p slots × NativeMatrix rows p slots := | |
| let z := selectedPair rows repeats p slots exterior a raw | |
| (EvaluationMatrix.ofRows (H (cutSlots p slots)) z.1, | |
| EvaluationMatrix.ofRows (H (cutSlots p slots)) z.2) | |
| local instance nativeMatrixFintype : Fintype (NativeMatrix rows p slots) := | |
| Fintype.ofEquiv (NativeRows rows p slots) (EvaluationMatrix.rowsEquiv (H (cutSlots p slots))).symm | |
| theorem matrixPair_law (exterior : Exterior rows repeats p slots) | |
| (a : BucketSampler.Direction (rows (m + 1))) : | |
| (CutChildGrouping.rawLaw (C := Option (Calls rows repeats p)) (cutSlots p slots) rows).pushforward | |
| (matrixPair rows repeats p slots exterior a) = | |
| ((FiniteDistribution.uniform (NativeMatrix rows p slots)).product | |
| (FiniteDistribution.uniform (H (cutSlots p slots)))).pushforward | |
| (fun z => (z.1, EvaluationMatrix.shift (H (cutSlots p slots)) z.1 a.val z.2)) := by | |
| have h := congrArg | |
| (fun μ : FiniteDistribution (NativeRows rows p slots × NativeRows rows p slots) => | |
| μ.pushforward (fun z => (EvaluationMatrix.ofRows (H (cutSlots p slots)) z.1, | |
| EvaluationMatrix.ofRows (H (cutSlots p slots)) z.2))) | |
| (selectedPair_law rows repeats p slots exterior a) | |
| rw [FiniteDistribution.pushforward_comp, LowerDirectionFiber.matrix_pair_law] at h | |
| exact h | |
| end | |
| end PerfectCompleteness.LowerCutPair | |
| end | |
| section | |
| namespace PerfectCompleteness.LowerCutDecoderForm | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open WholeCutGrouping | |
| variable {branch rows repeats : Nat → Nat} {n height t : Nat} | |
| (path : Path branch n (height + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch n) (lowerLevel : Nat) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (hnode : WholeArrayInteriorExterior.upperNode path = | |
| HierarchicalLeftDecoder.LowerNode upper lowerLevel d) | |
| def spaceEquiv : H (cutSlots path slots) ≃ₗ[ZMod 2] | |
| HierarchicalDecoderTables.LowerH slots upper lowerLevel d := | |
| (LowerCutNodeCoordinates.scalarEquiv path slots).trans (by | |
| rw [hnode]) | |
| def form (answer : HierarchicalDecoderTables.Answer slots upper lowerLevel d) : | |
| ChildBilinearCollisionTransfer.ParentForm (cutSlots path slots) where | |
| toFun f := (OddListExtraction.multiplicationForm | |
| (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) answer | |
| (spaceEquiv path slots upper lowerLevel d hnode f)).comp | |
| (spaceEquiv path slots upper lowerLevel d hnode).toLinearMap | |
| map_add' f g := by | |
| ext h | |
| simp only [LinearMap.comp_apply, map_add, LinearMap.add_apply] | |
| map_smul' c f := by | |
| ext h | |
| simp only [LinearMap.comp_apply, map_smul, | |
| LinearMap.smul_apply, RingHom.id_apply] | |
| @[simp] theorem form_apply | |
| (answer : HierarchicalDecoderTables.Answer slots upper lowerLevel d) | |
| (f g : H (cutSlots path slots)) : | |
| form path slots upper lowerLevel d hnode answer f g = | |
| OddListExtraction.multiplicationForm | |
| (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) answer | |
| (spaceEquiv path slots upper lowerLevel d hnode f) | |
| (spaceEquiv path slots upper lowerLevel d hnode g) := rfl | |
| theorem form_injective : | |
| Function.Injective (form path slots upper lowerLevel d hnode) := by | |
| intro answer answer' hform | |
| apply MultiplicationFormInjective.multiplicationForm_injective | |
| (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) | |
| ext f g | |
| obtain ⟨f', rfl⟩ := (spaceEquiv path slots upper lowerLevel d hnode).surjective f | |
| obtain ⟨g', rfl⟩ := (spaceEquiv path slots upper lowerLevel d hnode).surjective g | |
| exact congrArg (fun F : ChildBilinearCollisionTransfer.ParentForm | |
| (cutSlots path slots) => F f' g') hform | |
| variable (hbranch : ∀ k < n, 0 < branch k) {adviceRows : Nat} | |
| (A : ManyGoodRows.RowMap (Block rows upper) adviceRows) | |
| (table : HierarchicalAllDecoderTables.Input (rows := rows) | |
| slots upper lowerLevel adviceRows → HierarchicalAllDecoderTables.UpperAnswer slots upper) | |
| (cutoff : Nat) | |
| def answer (arrays : Arrays slots rows) : | |
| Option (HierarchicalDecoderTables.Answer slots upper lowerLevel d) := | |
| HierarchicalDecoderTables.decode slots upper lowerLevel hbranch d cutoff | |
| (table (LowerCutDecoderInput.actualInput slots upper lowerLevel arrays A)) | |
| def readForm (arrays : Arrays slots rows) : | |
| Option (ChildBilinearCollisionTransfer.ParentForm (cutSlots path slots)) := | |
| (answer slots upper lowerLevel d hbranch A table cutoff arrays).map | |
| (form path slots upper lowerLevel d hnode) | |
| def first (exterior : Exterior rows repeats path slots) | |
| (raw : LowerCutPair.Raw rows repeats path slots) : | |
| Option (ChildBilinearCollisionTransfer.ParentForm (cutSlots path slots)) := | |
| readForm path slots upper lowerLevel d hnode hbranch A table cutoff | |
| (LowerCutPair.first rows repeats path slots exterior raw) | |
| def second (exterior : Exterior rows repeats path slots) | |
| (direction : BucketSampler.Direction (rows (height + 1))) | |
| (raw : LowerCutPair.Raw rows repeats path slots) : | |
| Option (ChildBilinearCollisionTransfer.ParentForm (cutSlots path slots)) := | |
| readForm path slots upper lowerLevel d hnode hbranch A table cutoff | |
| (LowerCutPair.second rows repeats path slots exterior direction raw) | |
| theorem fullCollision_readForm (arrays arrays' : Arrays slots rows) : | |
| BilinearCollisionTransfer.fullCollision | |
| (readForm path slots upper lowerLevel d hnode hbranch A table cutoff arrays) | |
| (readForm path slots upper lowerLevel d hnode hbranch A table cutoff arrays') = | |
| DecoderTableCoupling.definedEqual | |
| (answer slots upper lowerLevel d hbranch A table cutoff arrays) | |
| (answer slots upper lowerLevel d hbranch A table cutoff arrays') := | |
| MultiplicationFormInjective.definedEqual_map_of_injective | |
| (form path slots upper lowerLevel d hnode) | |
| (form_injective path slots upper lowerLevel d hnode) _ _ | |
| theorem fullCollision_pair (exterior : Exterior rows repeats path slots) | |
| (direction : BucketSampler.Direction (rows (height + 1))) | |
| (raw : LowerCutPair.Raw rows repeats path slots) : | |
| BilinearCollisionTransfer.fullCollision | |
| (first path slots upper lowerLevel d hnode hbranch A table cutoff exterior raw) | |
| (second path slots upper lowerLevel d hnode hbranch A table cutoff exterior direction raw) = | |
| TwoResponseCollision.collision | |
| (answer slots upper lowerLevel d hbranch A table cutoff) | |
| (LowerCutPair.arraysPair rows repeats path slots exterior direction raw) := | |
| fullCollision_readForm path slots upper lowerLevel d hnode hbranch A table cutoff _ _ | |
| end | |
| end PerfectCompleteness.LowerCutDecoderForm | |
| end | |
| end OAI | |