openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /LowerCutFormCollision.lean
| import OAI.Computability.PerfectCompleteness.Decoding.LowerCutScalarProjectionLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.LowerCutFormCollision | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| variable {branch : Nat → Nat} {n height t : Nat} | |
| (path : Path branch n (height + 1)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (inside : Slots branch (height + 1) → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch n) (level : Nat) | |
| (d : HierarchicalFrozenTables.LowerNodes upper level) | |
| (hnode : WholeArrayInteriorExterior.upperNode path = | |
| HierarchicalLeftDecoder.LowerNode upper level d) | |
| def nativeScalar (f : H inside) : | |
| HierarchicalDecoderTables.LowerH (CutSlotAssembly.fill path outside inside) upper level d := | |
| LowerCutDecoderForm.spaceEquiv path (CutSlotAssembly.fill path outside inside) | |
| upper level d hnode | |
| (CutNativeForms.spaceCast (CleanPhysicalReplay.cutSlots_fill path outside inside).symm f) | |
| def nativeForm | |
| (answer : HierarchicalDecoderTables.Answer | |
| (CutSlotAssembly.fill path outside inside) upper level d) : | |
| ChildBilinearCollisionTransfer.ParentForm inside := | |
| CutNativeForms.formCast (CleanPhysicalReplay.cutSlots_fill path outside inside) | |
| (LowerCutDecoderForm.form path (CutSlotAssembly.fill path outside inside) | |
| upper level d hnode answer) | |
| private theorem formCast_apply_inverse | |
| {source target : Slots branch (height + 1) → Fin t → MixedSupport.Slot} | |
| (h : source = target) (F : ChildBilinearCollisionTransfer.ParentForm source) | |
| (f g : H target) : | |
| CutNativeForms.formCast h F f g = | |
| F (CutNativeForms.spaceCast h.symm f) (CutNativeForms.spaceCast h.symm g) := by | |
| cases h | |
| rfl | |
| theorem nativeForm_apply | |
| (answer : HierarchicalDecoderTables.Answer | |
| (CutSlotAssembly.fill path outside inside) upper level d) (f g : H inside) : | |
| nativeForm path outside inside upper level d hnode answer f g = | |
| OddListExtraction.multiplicationForm | |
| (HierarchicalDecoderTables.LowerH | |
| (CutSlotAssembly.fill path outside inside) upper level d) answer | |
| (nativeScalar path outside inside upper level d hnode f) | |
| (nativeScalar path outside inside upper level d hnode g) := by | |
| unfold nativeForm | |
| rw [formCast_apply_inverse] | |
| rfl | |
| variable (projected : Slots branch (height + 1) → Fin t → MixedSupport.Slot) | |
| (projection : ∀ s k, MixedSupport.Projection (inside s k) (projected s k)) | |
| theorem nativeScalar_HPullback (f : H projected) : | |
| nativeScalar path outside inside upper level d hnode (HPullback projection f) = | |
| HPullback (ChildBlockProjection.nodeProjection | |
| (CutProjectionAssembly.fillProjection path outside inside projected projection) | |
| (HierarchicalLeftDecoder.LowerNode upper level d)) | |
| (nativeScalar path outside projected upper level d hnode f) := | |
| LowerCutScalarProjection.spaceEquiv_fill_HPullback path outside inside projected | |
| upper level d hnode projection f | |
| theorem restricted_on_native | |
| (answer₀ answer₁ : Option (HierarchicalDecoderTables.Answer | |
| (CutSlotAssembly.fill path outside inside) upper level d)) | |
| (U : Submodule F2 (H inside)) (hU : U ≤ LinearMap.range (HPullback projection)) | |
| (hcollision : BilinearCollisionTransfer.restrictedCollision | |
| (answer₀.map (OddListExtraction.multiplicationForm | |
| (HierarchicalDecoderTables.LowerH | |
| (CutSlotAssembly.fill path outside inside) upper level d))) | |
| (answer₁.map (OddListExtraction.multiplicationForm | |
| (HierarchicalDecoderTables.LowerH | |
| (CutSlotAssembly.fill path outside inside) upper level d))) | |
| (LinearMap.range (HPullback (ChildBlockProjection.nodeProjection | |
| (CutProjectionAssembly.fillProjection path outside inside projected projection) | |
| (HierarchicalLeftDecoder.LowerNode upper level d)))) = true) : | |
| BilinearCollisionTransfer.restrictedCollision | |
| (answer₀.map (nativeForm path outside inside upper level d hnode)) | |
| (answer₁.map (nativeForm path outside inside upper level d hnode)) U = true := by | |
| obtain ⟨F, G, hF, hG, heq⟩ := of_decide_eq_true hcollision | |
| cases answer₀ with | |
| | none => cases hF | |
| | some a₀ => | |
| cases answer₁ with | |
| | none => cases hG | |
| | some a₁ => | |
| have hF' := Option.some.inj hF | |
| have hG' := Option.some.inj hG | |
| rw [← hF', ← hG'] at heq | |
| apply decide_eq_true | |
| refine ⟨nativeForm path outside inside upper level d hnode a₀, | |
| nativeForm path outside inside upper level d hnode a₁, rfl, rfl, ?_⟩ | |
| intro f hf g hg | |
| rw [nativeForm_apply, nativeForm_apply] | |
| obtain ⟨f', hf'⟩ := hU hf | |
| obtain ⟨g', hg'⟩ := hU hg | |
| have hfmem : nativeScalar path outside inside upper level d hnode f ∈ | |
| LinearMap.range (HPullback (ChildBlockProjection.nodeProjection | |
| (CutProjectionAssembly.fillProjection path outside inside projected projection) | |
| (HierarchicalLeftDecoder.LowerNode upper level d))) := by | |
| refine ⟨nativeScalar path outside projected upper level d hnode f', ?_⟩ | |
| rw [← nativeScalar_HPullback, hf'] | |
| have hgmem : nativeScalar path outside inside upper level d hnode g ∈ | |
| LinearMap.range (HPullback (ChildBlockProjection.nodeProjection | |
| (CutProjectionAssembly.fillProjection path outside inside projected projection) | |
| (HierarchicalLeftDecoder.LowerNode upper level d))) := by | |
| refine ⟨nativeScalar path outside projected upper level d hnode g', ?_⟩ | |
| rw [← nativeScalar_HPullback, hg'] | |
| exact heq _ hfmem _ hgmem | |
| end | |
| end PerfectCompleteness.LowerCutFormCollision | |
| end | |
| end OAI | |