openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Foundations /CutNativeForms.lean
| import OAI.Computability.PerfectCompleteness.Construction.SourceLocalOddLists | |
| import OAI.Computability.PerfectCompleteness.Foundations.CutSlotAssembly | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CutNativeForms | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays PointwiseSpaces | |
| variable {branch : Nat → Nat} {n k t : Nat} | |
| theorem nodeSlots_fill (lower : Nodes branch n) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (inside : Slots branch (Nodes.height lower) → Fin t → MixedSupport.Slot) : | |
| nodeSlots (CutSlotAssembly.fill (Nodes.path lower) outside inside) lower = inside := by | |
| funext s | |
| exact CutSlotAssembly.fill_at_cut (Nodes.path lower) outside inside s | |
| theorem nodeSlots_fill_of_agree (lower : Nodes branch n) | |
| (p : Path branch n (Nodes.height lower)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (inside : Slots branch (Nodes.height lower) → Fin t → MixedSupport.Slot) | |
| (hag : ∀ s, p.slotEmbedding s = (Nodes.path lower).slotEmbedding s) : | |
| nodeSlots (CutSlotAssembly.fill p outside inside) lower = inside := by | |
| funext s | |
| change CutSlotAssembly.fill p outside inside ((Nodes.path lower).slotEmbedding s) = inside s | |
| rw [← hag s] | |
| exact CutSlotAssembly.fill_at_cut p outside inside s | |
| section Equality | |
| variable {a b c d : Slots branch k → Fin t → MixedSupport.Slot} | |
| def spaceCast (hab : a = b) : H a ≃ₗ[F2] H b := by | |
| cases hab | |
| exact LinearEquiv.refl F2 (H a) | |
| def squareCast (hab : a = b) : squareSpace (H a) ≃ₗ[F2] squareSpace (H b) := by | |
| cases hab | |
| exact LinearEquiv.refl F2 (squareSpace (H a)) | |
| def dualCast (hab : a = b) (z : Module.Dual F2 (squareSpace (H a))) : | |
| Module.Dual F2 (squareSpace (H b)) := by | |
| cases hab | |
| exact z | |
| def formCast (hab : a = b) (F : SourceLocalOddLists.Form a) : SourceLocalOddLists.Form b := by | |
| cases hab | |
| exact F | |
| @[simp] theorem dualCast_apply (hab : a = b) | |
| (z : Module.Dual F2 (squareSpace (H a))) (f : squareSpace (H a)) : | |
| dualCast hab z (squareCast hab f) = z f := by | |
| cases hab | |
| rfl | |
| @[simp] theorem formCast_apply (hab : a = b) (F : SourceLocalOddLists.Form a) | |
| (f g : H a) : formCast hab F (spaceCast hab f) (spaceCast hab g) = F f g := by | |
| cases hab | |
| rfl | |
| theorem squareCast_product (hab : a = b) (f g : H a) : | |
| squareCast hab (OddListExtraction.productElement (H a) f g) = | |
| OddListExtraction.productElement (H b) (spaceCast hab f) (spaceCast hab g) := by | |
| cases hab | |
| rfl | |
| theorem formCast_multiplication (hab : a = b) | |
| (z : Module.Dual F2 (squareSpace (H a))) : | |
| formCast hab (OddListExtraction.multiplicationForm (H a) z) = | |
| OddListExtraction.multiplicationForm (H b) (dualCast hab z) := by | |
| cases hab | |
| rfl | |
| theorem formRank_formCast (hab : a = b) (F : SourceLocalOddLists.Form a) : | |
| OddListExtraction.formRank (H b) (formCast hab F) = | |
| OddListExtraction.formRank (H a) F := by | |
| cases hab | |
| rfl | |
| theorem squareCast_one (hab : a = b) | |
| (ha : (1 : Domain a → F2) ∈ squareSpace (H a)) | |
| (hb : (1 : Domain b → F2) ∈ squareSpace (H b)) : | |
| squareCast hab ⟨1, ha⟩ = ⟨1, hb⟩ := by | |
| cases hab | |
| rfl | |
| theorem dualCast_one (hab : a = b) (z : Module.Dual F2 (squareSpace (H a))) | |
| (ha : (1 : Domain a → F2) ∈ squareSpace (H a)) | |
| (hb : (1 : Domain b → F2) ∈ squareSpace (H b)) : | |
| dualCast hab z ⟨1, hb⟩ = z ⟨1, ha⟩ := by | |
| cases hab | |
| rfl | |
| def oneElement (slots : Slots branch k → Fin t → MixedSupport.Slot) | |
| (hbranch : ∀ j < k, 0 < branch j) : squareSpace (H slots) := | |
| ⟨(1 : Domain slots → F2), TreeSourceSpaces.one_mem_squareSpace slots hbranch⟩ | |
| theorem dualCast_normalized (hab : a = b) (hbranch : ∀ j < k, 0 < branch j) | |
| (z : Module.Dual F2 (squareSpace (H a))) | |
| (hnorm : z (oneElement a hbranch) = 1) : | |
| dualCast hab z (oneElement b hbranch) = 1 := by | |
| cases hab | |
| exact hnorm | |
| def projectionCast (hab : a = b) (hcd : c = d) | |
| (p : ∀ s j, MixedSupport.Projection (a s j) (c s j)) : | |
| ∀ s j, MixedSupport.Projection (b s j) (d s j) := by | |
| cases hab | |
| cases hcd | |
| exact p | |
| theorem spaceCast_HPullback (hab : a = b) (hcd : c = d) | |
| (p : ∀ s j, MixedSupport.Projection (a s j) (c s j)) (f : H c) : | |
| spaceCast hab (HPullback p f) = | |
| HPullback (projectionCast hab hcd p) (spaceCast hcd f) := by | |
| cases hab | |
| cases hcd | |
| rfl | |
| theorem formCast_projection_agreement (hab : a = b) (hcd : c = d) | |
| (p : ∀ s j, MixedSupport.Projection (a s j) (c s j)) | |
| (F : SourceLocalOddLists.Form a) (G : SourceLocalOddLists.Form c) | |
| (hagree : ∀ f g, G f g = F (HPullback p f) (HPullback p g)) : | |
| ∀ f g, formCast hcd G f g = formCast hab F | |
| (HPullback (projectionCast hab hcd p) f) | |
| (HPullback (projectionCast hab hcd p) g) := by | |
| cases hab | |
| cases hcd | |
| exact hagree | |
| theorem dualCast_projection_agreement (hab : a = b) (hcd : c = d) | |
| (p : ∀ s j, MixedSupport.Projection (a s j) (c s j)) | |
| (z : Module.Dual F2 (squareSpace (H a))) (G : SourceLocalOddLists.Form c) | |
| (hagree : ∀ f g, G f g = OddListExtraction.multiplicationForm (H a) z | |
| (HPullback p f) (HPullback p g)) : | |
| ∀ f g, formCast hcd G f g = | |
| OddListExtraction.multiplicationForm (H b) (dualCast hab z) | |
| (HPullback (projectionCast hab hcd p) f) | |
| (HPullback (projectionCast hab hcd p) g) := by | |
| cases hab | |
| cases hcd | |
| exact hagree | |
| end Equality | |
| section Native | |
| variable (lower : Nodes branch n) (p : Path branch n (Nodes.height lower)) | |
| (outside : Slots branch n → Fin t → MixedSupport.Slot) | |
| (inside : Slots branch (Nodes.height lower) → Fin t → MixedSupport.Slot) | |
| (hag : ∀ s, p.slotEmbedding s = (Nodes.path lower).slotEmbedding s) | |
| def nativeDual | |
| (z : Module.Dual F2 (squareSpace (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)))) : | |
| Module.Dual F2 (squareSpace (H inside)) := | |
| dualCast (nodeSlots_fill_of_agree lower p outside inside hag) z | |
| def nativeForm | |
| (F : SourceLocalOddLists.Form (nodeSlots (CutSlotAssembly.fill p outside inside) lower)) : | |
| SourceLocalOddLists.Form inside := | |
| formCast (nodeSlots_fill_of_agree lower p outside inside hag) F | |
| theorem nativeForm_multiplication | |
| (z : Module.Dual F2 (squareSpace (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)))) : | |
| nativeForm lower p outside inside hag | |
| (OddListExtraction.multiplicationForm | |
| (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)) z) = | |
| OddListExtraction.multiplicationForm (H inside) (nativeDual lower p outside inside hag z) := | |
| formCast_multiplication (nodeSlots_fill_of_agree lower p outside inside hag) z | |
| theorem nativeForm_rank | |
| (F : SourceLocalOddLists.Form (nodeSlots (CutSlotAssembly.fill p outside inside) lower)) : | |
| OddListExtraction.formRank (H inside) (nativeForm lower p outside inside hag F) = | |
| OddListExtraction.formRank (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)) F := | |
| formRank_formCast (nodeSlots_fill_of_agree lower p outside inside hag) F | |
| theorem nativeDual_normalized (hbranch : ∀ j < Nodes.height lower, 0 < branch j) | |
| (z : Module.Dual F2 (squareSpace (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)))) | |
| (hnorm : z (oneElement | |
| (nodeSlots (CutSlotAssembly.fill p outside inside) lower) hbranch) = 1) : | |
| nativeDual lower p outside inside hag z (oneElement inside hbranch) = 1 := | |
| dualCast_normalized (nodeSlots_fill_of_agree lower p outside inside hag) hbranch z hnorm | |
| variable (outside' : Slots branch n → Fin t → MixedSupport.Slot) | |
| (inside' : Slots branch (Nodes.height lower) → Fin t → MixedSupport.Slot) | |
| def nativeProjection | |
| (projection : ∀ s j, MixedSupport.Projection | |
| (nodeSlots (CutSlotAssembly.fill p outside inside) lower s j) | |
| (nodeSlots (CutSlotAssembly.fill p outside' inside') lower s j)) : | |
| ∀ s j, MixedSupport.Projection (inside s j) (inside' s j) := | |
| projectionCast (nodeSlots_fill_of_agree lower p outside inside hag) | |
| (nodeSlots_fill_of_agree lower p outside' inside' hag) projection | |
| theorem nativeForm_projection_agreement | |
| (projection : ∀ s j, MixedSupport.Projection | |
| (nodeSlots (CutSlotAssembly.fill p outside inside) lower s j) | |
| (nodeSlots (CutSlotAssembly.fill p outside' inside') lower s j)) | |
| (F : SourceLocalOddLists.Form (nodeSlots (CutSlotAssembly.fill p outside inside) lower)) | |
| (G : SourceLocalOddLists.Form (nodeSlots (CutSlotAssembly.fill p outside' inside') lower)) | |
| (hagree : ∀ f g, G f g = F (HPullback projection f) (HPullback projection g)) : | |
| ∀ f g, nativeForm lower p outside' inside' hag G f g = | |
| nativeForm lower p outside inside hag F | |
| (HPullback (nativeProjection lower p outside inside hag outside' inside' projection) f) | |
| (HPullback (nativeProjection lower p outside inside hag outside' inside' projection) g) := | |
| formCast_projection_agreement (nodeSlots_fill_of_agree lower p outside inside hag) | |
| (nodeSlots_fill_of_agree lower p outside' inside' hag) projection F G hagree | |
| theorem nativeDual_projection_agreement | |
| (projection : ∀ s j, MixedSupport.Projection | |
| (nodeSlots (CutSlotAssembly.fill p outside inside) lower s j) | |
| (nodeSlots (CutSlotAssembly.fill p outside' inside') lower s j)) | |
| (z : Module.Dual F2 (squareSpace (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)))) | |
| (G : SourceLocalOddLists.Form (nodeSlots (CutSlotAssembly.fill p outside' inside') lower)) | |
| (hagree : ∀ f g, G f g = OddListExtraction.multiplicationForm | |
| (H (nodeSlots (CutSlotAssembly.fill p outside inside) lower)) z | |
| (HPullback projection f) (HPullback projection g)) : | |
| ∀ f g, nativeForm lower p outside' inside' hag G f g = | |
| OddListExtraction.multiplicationForm (H inside) (nativeDual lower p outside inside hag z) | |
| (HPullback (nativeProjection lower p outside inside hag outside' inside' projection) f) | |
| (HPullback (nativeProjection lower p outside inside hag outside' inside' projection) g) := | |
| dualCast_projection_agreement (nodeSlots_fill_of_agree lower p outside inside hag) | |
| (nodeSlots_fill_of_agree lower p outside' inside' hag) projection z G hagree | |
| end Native | |
| end | |
| end PerfectCompleteness.CutNativeForms | |
| end | |
| end OAI | |