openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /CutProjectionAssembly.lean
| import OAI.Computability.PerfectCompleteness.Foundations.CutSlotAssembly | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CutProjectionAssembly | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces MixedSupport | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| def castProjection {a b c d : Slot} (hab : a = b) (hcd : c = d) | |
| (q : Projection a c) : Projection b d := by | |
| cases hab | |
| cases hcd | |
| exact q | |
| @[simp] theorem castProjection_heq {a b c d : Slot} (hab : a = b) (hcd : c = d) | |
| (q : Projection a c) : HEq (castProjection hab hcd q) q := by | |
| cases hab | |
| cases hcd | |
| rfl | |
| def fillProjection : {n m : Nat} → (p : Path branch n m) → | |
| (outside : Slots branch n → Fin t → Slot) → | |
| (left right : Slots branch m → Fin t → Slot) → | |
| (∀ s k, Projection (left s k) (right s k)) → | |
| ∀ s k, Projection (CutSlotAssembly.fill p outside left s k) | |
| (CutSlotAssembly.fill p outside right s k) | |
| | _, _, .refl _, _, _, _, q => q | |
| | _, _, .step i p, outside, left, right, q => fun s k => | |
| if hs : s.1 = i then | |
| castProjection | |
| (by simp only [CutSlotAssembly.fill, ite_eq_left hs]) | |
| (by simp only [CutSlotAssembly.fill, ite_eq_left hs]) | |
| (fillProjection p (fun j => outside (i, j)) left right q s.2 k) | |
| else | |
| castProjection | |
| (by simp only [CutSlotAssembly.fill, ite_eq_right hs]) | |
| (by simp only [CutSlotAssembly.fill, ite_eq_right hs]) | |
| (Projection.keep (outside s k)) | |
| private theorem dite_heq_of_pos {α β : Type} {P : Prop} [Decidable P] | |
| (f : P → α) (g : ¬ P → α) (b : β) (hP : P) | |
| (h : HEq (f hP) b) : HEq (dite P f g) b := by | |
| rw [dite_eq_left hP] | |
| exact h | |
| theorem fillProjection_at_cut (p : Path branch n m) | |
| (outside : Slots branch n → Fin t → Slot) | |
| (left right : Slots branch m → Fin t → Slot) | |
| (q : ∀ s k, Projection (left s k) (right s k)) | |
| (s : Slots branch m) (k : Fin t) : | |
| HEq (fillProjection p outside left right q (p.slotEmbedding s) k) (q s k) := by | |
| revert outside left right q s k | |
| induction p with | |
| | refl _ => intros; rfl | |
| | step i p ih => | |
| intro outside left right q s k | |
| refine @dite_heq_of_pos _ _ (i = i) _ _ _ (q s k) rfl ?_ | |
| exact (castProjection_heq _ _ _).trans | |
| (ih (fun j => outside (i, j)) left right q s k) | |
| theorem fillProjection_at_cut_cast (p : Path branch n m) | |
| (outside : Slots branch n → Fin t → Slot) | |
| (left right : Slots branch m → Fin t → Slot) | |
| (q : ∀ s k, Projection (left s k) (right s k)) | |
| (s : Slots branch m) (k : Fin t) : | |
| castProjection | |
| (congrFun (CutSlotAssembly.fill_at_cut p outside left s) k) | |
| (congrFun (CutSlotAssembly.fill_at_cut p outside right s) k) | |
| (fillProjection p outside left right q (p.slotEmbedding s) k) = q s k := by | |
| apply eq_of_heq | |
| exact (castProjection_heq _ _ _).trans (fillProjection_at_cut p outside left right q s k) | |
| end | |
| end PerfectCompleteness.CutProjectionAssembly | |
| end | |
| end OAI | |