openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /LowerCutScalarProjectionLemmas.lean
| import OAI.Computability.PerfectCompleteness.Decoding.CutProjectionAssembly | |
| import OAI.Computability.PerfectCompleteness.Decoding.LowerCutPairLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.CutNativeForms | |
| import OAI.Computability.PerfectCompleteness.Repetition.CleanPhysicalReplay | |
| import OAI.Computability.PerfectCompleteness.Sampling.SourceChildMarkedLaw | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.LowerCutScalarProjection | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open WholeArraySubtreeSplit WholeArrayInteriorExterior | |
| open scoped Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| theorem scalarEquiv_HPullback : | |
| {n m : Nat} → (p : Path branch n (m + 1)) → | |
| (slots projected : Slots branch n → Fin t → MixedSupport.Slot) → | |
| (q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) → | |
| (f : H (subtreeSlots p projected)) → | |
| LowerCutNodeCoordinates.scalarEquiv p slots | |
| (HPullback (fun s k => q (p.slotEmbedding s) k) f) = | |
| HPullback (ChildBlockProjection.nodeProjection q (upperNode p)) | |
| (LowerCutNodeCoordinates.scalarEquiv p projected f) | |
| | _, _, .refl _, _, _, _, _ => rfl | |
| | _, _, .step i p, slots, projected, q, f => | |
| scalarEquiv_HPullback p (childSlots slots i) (childSlots projected i) | |
| (fun s k => q (i, s) k) f | |
| theorem projectionCast_apply | |
| {a b c d : Slots branch m → Fin t → MixedSupport.Slot} | |
| (hab : a = b) (hcd : c = d) | |
| (q : ∀ s k, MixedSupport.Projection (a s k) (c s k)) | |
| (s : Slots branch m) (k : Fin t) : | |
| CutNativeForms.projectionCast hab hcd q s k = | |
| CutProjectionAssembly.castProjection | |
| (congrFun (congrFun hab s) k) (congrFun (congrFun hcd s) k) (q s k) := by | |
| cases hab | |
| cases hcd | |
| rfl | |
| theorem projectionCast_fill (p : Path branch n (m + 1)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (left right : Slots branch (m + 1) → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) : | |
| CutNativeForms.projectionCast | |
| (CleanPhysicalReplay.cutSlots_fill p outside left) | |
| (CleanPhysicalReplay.cutSlots_fill p outside right) | |
| (fun s k => CutProjectionAssembly.fillProjection p outside left right q | |
| (p.slotEmbedding s) k) = q := by | |
| funext s k | |
| rw [projectionCast_apply] | |
| exact CutProjectionAssembly.fillProjection_at_cut_cast p outside left right q s k | |
| theorem inverse_spaceCast_HPullback | |
| {a b c d : Slots branch m → Fin t → MixedSupport.Slot} | |
| (hab : a = b) (hcd : c = d) | |
| (q : ∀ s k, MixedSupport.Projection (a s k) (c s k)) (f : H d) : | |
| CutNativeForms.spaceCast hab.symm | |
| (HPullback (CutNativeForms.projectionCast hab hcd q) f) = | |
| HPullback q (CutNativeForms.spaceCast hcd.symm f) := by | |
| cases hab | |
| cases hcd | |
| rfl | |
| theorem fill_spaceCast_HPullback (p : Path branch n (m + 1)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (left right : Slots branch (m + 1) → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) (f : H right) : | |
| CutNativeForms.spaceCast (CleanPhysicalReplay.cutSlots_fill p outside left).symm | |
| (HPullback q f) = | |
| HPullback | |
| (fun s k => CutProjectionAssembly.fillProjection p outside left right q | |
| (p.slotEmbedding s) k) | |
| (CutNativeForms.spaceCast | |
| (CleanPhysicalReplay.cutSlots_fill p outside right).symm f) := by | |
| have h := inverse_spaceCast_HPullback | |
| (CleanPhysicalReplay.cutSlots_fill p outside left) | |
| (CleanPhysicalReplay.cutSlots_fill p outside right) | |
| (fun s k => CutProjectionAssembly.fillProjection p outside left right q | |
| (p.slotEmbedding s) k) f | |
| rw [projectionCast_fill] at h | |
| exact h | |
| theorem scalarEquiv_fill_HPullback (p : Path branch n (m + 1)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (left right : Slots branch (m + 1) → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) (f : H right) : | |
| LowerCutNodeCoordinates.scalarEquiv p (CutSlotAssembly.fill p outside left) | |
| (CutNativeForms.spaceCast (CleanPhysicalReplay.cutSlots_fill p outside left).symm | |
| (HPullback q f)) = | |
| HPullback (ChildBlockProjection.nodeProjection | |
| (CutProjectionAssembly.fillProjection p outside left right q) (upperNode p)) | |
| (LowerCutNodeCoordinates.scalarEquiv p (CutSlotAssembly.fill p outside right) | |
| (CutNativeForms.spaceCast | |
| (CleanPhysicalReplay.cutSlots_fill p outside right).symm f)) := by | |
| rw [fill_spaceCast_HPullback] | |
| exact scalarEquiv_HPullback p _ _ _ _ | |
| def nodeEquiv (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| {a b : Nodes branch n} (hab : a = b) : | |
| H (nodeSlots slots a) ≃ₗ[F2] H (nodeSlots slots b) := by | |
| rw [hab] | |
| theorem nodeEquiv_HPullback | |
| (slots projected : Slots branch n → Fin t → MixedSupport.Slot) | |
| (q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| {a b : Nodes branch n} (hab : a = b) (f : H (nodeSlots projected a)) : | |
| nodeEquiv slots hab (HPullback (ChildBlockProjection.nodeProjection q a) f) = | |
| HPullback (ChildBlockProjection.nodeProjection q b) (nodeEquiv projected hab f) := by | |
| cases hab | |
| rfl | |
| theorem spaceEquiv_HPullback (p : Path branch n (m + 1)) | |
| (slots projected : Slots branch n → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch n) (level : Nat) | |
| (d : HierarchicalFrozenTables.LowerNodes upper level) | |
| (hnode : upperNode p = HierarchicalLeftDecoder.LowerNode upper level d) | |
| (q : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| (f : H (subtreeSlots p projected)) : | |
| LowerCutDecoderForm.spaceEquiv p slots upper level d hnode | |
| (HPullback (fun s k => q (p.slotEmbedding s) k) f) = | |
| HPullback (ChildBlockProjection.nodeProjection q | |
| (HierarchicalLeftDecoder.LowerNode upper level d)) | |
| (LowerCutDecoderForm.spaceEquiv p projected upper level d hnode f) := by | |
| change nodeEquiv slots hnode | |
| (LowerCutNodeCoordinates.scalarEquiv p slots | |
| (HPullback (fun s k => q (p.slotEmbedding s) k) f)) = | |
| HPullback (ChildBlockProjection.nodeProjection q | |
| (HierarchicalLeftDecoder.LowerNode upper level d)) | |
| (nodeEquiv projected hnode (LowerCutNodeCoordinates.scalarEquiv p projected f)) | |
| rw [scalarEquiv_HPullback, nodeEquiv_HPullback] | |
| theorem spaceEquiv_fill_HPullback (p : Path branch n (m + 1)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (left right : Slots branch (m + 1) → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch n) (level : Nat) | |
| (d : HierarchicalFrozenTables.LowerNodes upper level) | |
| (hnode : upperNode p = HierarchicalLeftDecoder.LowerNode upper level d) | |
| (q : ∀ s k, MixedSupport.Projection (left s k) (right s k)) (f : H right) : | |
| LowerCutDecoderForm.spaceEquiv p (CutSlotAssembly.fill p outside left) | |
| upper level d hnode | |
| (CutNativeForms.spaceCast (CleanPhysicalReplay.cutSlots_fill p outside left).symm | |
| (HPullback q f)) = | |
| HPullback (ChildBlockProjection.nodeProjection | |
| (CutProjectionAssembly.fillProjection p outside left right q) | |
| (HierarchicalLeftDecoder.LowerNode upper level d)) | |
| (LowerCutDecoderForm.spaceEquiv p (CutSlotAssembly.fill p outside right) | |
| upper level d hnode | |
| (CutNativeForms.spaceCast | |
| (CleanPhysicalReplay.cutSlots_fill p outside right).symm f)) := by | |
| rw [fill_spaceCast_HPullback] | |
| exact spaceEquiv_HPullback p _ _ upper level d hnode _ _ | |
| end | |
| end PerfectCompleteness.LowerCutScalarProjection | |
| end | |
| section | |
| namespace PerfectCompleteness.SourceQuestionLowerForms | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| section Casts | |
| variable {branch : Nat → Nat} {height t : Nat} {C : Type*} [Fintype C] | |
| (rows : Nat → Nat) | |
| {source target : Slots branch (height + 1) → Fin t → MixedSupport.Slot} | |
| def rawCast (h : source = target) : | |
| CutChildGrouping.Raw (C := C) source rows ≃ CutChildGrouping.Raw (C := C) target rows := by | |
| cases h | |
| exact Equiv.refl _ | |
| theorem rawCast_law (h : source = target) : | |
| (CutChildGrouping.rawLaw (C := C) source rows).pushforward (rawCast rows h) = | |
| CutChildGrouping.rawLaw (C := C) target rows := by | |
| cases h | |
| exact FiniteDistribution.pushforward_id _ | |
| theorem formCast_injective (h : source = target) : | |
| Function.Injective (CutNativeForms.formCast h) := by | |
| cases h | |
| exact Function.injective_id | |
| theorem fullCollision_formCast (h : source = target) | |
| (F G : Option (ChildBilinearCollisionTransfer.ParentForm source)) : | |
| BilinearCollisionTransfer.fullCollision (F.map (CutNativeForms.formCast h)) | |
| (G.map (CutNativeForms.formCast h)) = | |
| BilinearCollisionTransfer.fullCollision F G := | |
| MultiplicationFormInjective.definedEqual_map_of_injective | |
| (CutNativeForms.formCast h) (formCast_injective h) F G | |
| end Casts | |
| variable {branch rows repeats : Nat → Nat} {n height t v m : Nat} | |
| (path : Path branch n (height + 1)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (placeholder : Slots branch (height + 1) → Fin t → MixedSupport.Slot) | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| abbrev Questions := SourceChildMarkedLaw.Questions | |
| (branch := branch) (n := height) (t := t) (m := m) | |
| abbrev Calls := Option (WholeCutCalls.Index rows repeats path) | |
| def slots (q : Questions (branch := branch) (height := height) (t := t) (m := m)) : | |
| Slots branch n → Fin t → MixedSupport.Slot := | |
| CutSlotAssembly.fill path outside (SourceChildMarkedLaw.questionSlots clauses q) | |
| abbrev Raw (q : Questions (branch := branch) (height := height) (t := t) (m := m)) := | |
| CutChildGrouping.Raw (C := Calls (rows := rows) (repeats := repeats) path) | |
| (SourceChildMarkedLaw.questionSlots clauses q) rows | |
| abbrev Form (q : Questions (branch := branch) (height := height) (t := t) (m := m)) := | |
| ChildBilinearCollisionTransfer.ParentForm (SourceChildMarkedLaw.questionSlots clauses q) | |
| theorem cutSlots_eq (q : Questions (branch := branch) (height := height) (t := t) (m := m)) : | |
| WholeCutGrouping.cutSlots path (slots path outside clauses q) = | |
| SourceChildMarkedLaw.questionSlots clauses q := | |
| CleanPhysicalReplay.cutSlots_fill path outside _ | |
| def rawAtCut (q : Questions (branch := branch) (height := height) (t := t) (m := m)) | |
| (raw : Raw (rows := rows) (repeats := repeats) path clauses q) : | |
| LowerCutPair.Raw rows repeats path (slots path outside clauses q) := | |
| rawCast rows (cutSlots_eq path outside clauses q).symm raw | |
| theorem rawAtCut_law | |
| (q : Questions (branch := branch) (height := height) (t := t) (m := m)) : | |
| (CutChildGrouping.rawLaw (C := Calls (rows := rows) (repeats := repeats) path) | |
| (SourceChildMarkedLaw.questionSlots clauses q) rows).pushforward | |
| (rawAtCut (rows := rows) (repeats := repeats) path outside clauses q) = | |
| CutChildGrouping.rawLaw (C := Calls (rows := rows) (repeats := repeats) path) | |
| (WholeCutGrouping.cutSlots path (slots path outside clauses q)) rows := | |
| rawCast_law rows (cutSlots_eq path outside clauses q).symm | |
| variable (upper : Nodes branch n) (level : Nat) | |
| (d : HierarchicalFrozenTables.LowerNodes upper level) | |
| (hnode : WholeArrayInteriorExterior.upperNode path = | |
| HierarchicalLeftDecoder.LowerNode upper level d) | |
| (hbranch : ∀ k < n, 0 < branch k) | |
| {adviceRows : Nat} (A : ManyGoodRows.RowMap (Block rows upper) adviceRows) | |
| (exterior : CleanPhysicalReplay.Exterior rows repeats path outside placeholder) | |
| abbrev UpperTable (q : Questions (branch := branch) (height := height) (t := t) (m := m)) := | |
| HierarchicalAllDecoderTables.Input (rows := rows) | |
| (slots path outside clauses q) upper level adviceRows → | |
| HierarchicalAllDecoderTables.UpperAnswer (slots path outside clauses q) upper | |
| variable (tables : (q : Questions (branch := branch) (height := height) (t := t) (m := m)) → | |
| UpperTable (rows := rows) path outside clauses upper level (adviceRows := adviceRows) q) | |
| (cutoff : Nat) | |
| def first (q : Questions (branch := branch) (height := height) (t := t) (m := m)) | |
| (raw : Raw (rows := rows) (repeats := repeats) path clauses q) : Option (Form clauses q) := | |
| (LowerCutDecoderForm.first path (slots path outside clauses q) upper level d hnode hbranch A | |
| (tables q) cutoff | |
| (CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder | |
| (SourceChildMarkedLaw.questionSlots clauses q) exterior) | |
| (rawAtCut path outside clauses q raw)).map | |
| (CutNativeForms.formCast (cutSlots_eq path outside clauses q)) | |
| def second (direction : BucketSampler.Direction (rows (height + 1))) | |
| (q : Questions (branch := branch) (height := height) (t := t) (m := m)) | |
| (raw : Raw (rows := rows) (repeats := repeats) path clauses q) : Option (Form clauses q) := | |
| (LowerCutDecoderForm.second path (slots path outside clauses q) upper level d hnode hbranch A | |
| (tables q) cutoff | |
| (CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder | |
| (SourceChildMarkedLaw.questionSlots clauses q) exterior) | |
| direction (rawAtCut path outside clauses q raw)).map | |
| (CutNativeForms.formCast (cutSlots_eq path outside clauses q)) | |
| theorem fullCollision_probability | |
| (direction : BucketSampler.Direction (rows (height + 1))) | |
| (q : Questions (branch := branch) (height := height) (t := t) (m := m)) : | |
| (CutChildGrouping.rawLaw (C := Calls (rows := rows) (repeats := repeats) path) | |
| (SourceChildMarkedLaw.questionSlots clauses q) rows).probability | |
| (fun raw => BilinearCollisionTransfer.fullCollision | |
| (first path outside placeholder clauses upper level d hnode hbranch A exterior tables cutoff q raw) | |
| (second path outside placeholder clauses upper level d hnode hbranch A exterior tables cutoff | |
| direction q raw)) = | |
| (CutChildGrouping.rawLaw (C := Calls (rows := rows) (repeats := repeats) path) | |
| (WholeCutGrouping.cutSlots path (slots path outside clauses q)) rows).probability | |
| (fun raw => BilinearCollisionTransfer.fullCollision | |
| (LowerCutDecoderForm.first path (slots path outside clauses q) upper level d hnode hbranch A | |
| (tables q) cutoff (CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder | |
| (SourceChildMarkedLaw.questionSlots clauses q) exterior) raw) | |
| (LowerCutDecoderForm.second path (slots path outside clauses q) upper level d hnode hbranch A | |
| (tables q) cutoff (CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder | |
| (SourceChildMarkedLaw.questionSlots clauses q) exterior) direction raw)) := by | |
| have h := congrArg (fun μ : FiniteDistribution | |
| (CutChildGrouping.Raw (C := Calls (rows := rows) (repeats := repeats) path) | |
| (WholeCutGrouping.cutSlots path (slots path outside clauses q)) rows) => μ.probability | |
| (fun raw => BilinearCollisionTransfer.fullCollision | |
| (LowerCutDecoderForm.first path (slots path outside clauses q) upper level d hnode hbranch A | |
| (tables q) cutoff (CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder | |
| (SourceChildMarkedLaw.questionSlots clauses q) exterior) raw) | |
| (LowerCutDecoderForm.second path (slots path outside clauses q) upper level d hnode hbranch A | |
| (tables q) cutoff (CleanPhysicalReplay.exteriorAt rows repeats path outside placeholder | |
| (SourceChildMarkedLaw.questionSlots clauses q) exterior) direction raw))) | |
| (rawAtCut_law (rows := rows) (repeats := repeats) path outside clauses q) | |
| rw [FiniteDistribution.probability_pushforward] at h | |
| simpa only [first, second, fullCollision_formCast] using h | |
| end | |
| end PerfectCompleteness.SourceQuestionLowerForms | |
| end | |
| end OAI | |