openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Completion /HierarchyPrograms.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.HierarchyWords | |
| import OAI.LinearAlgebra.MatrixMultiplication.Completion.Labels | |
| namespace OAI | |
| /-! Readable tensor completion and its finite arithmetic realization. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CompletionHierarchyWords | |
| open MatrixMultiplication.Foundation | |
| universe u v p q | |
| variable {A : Type u} | |
| namespace ReadableHierarchy | |
| open RecursiveCompletion CompletionLabels.TopologicalFlatten | |
| def ofProgram {Coord : Color → Type v} {Context : Type q} {Code : Type p} | |
| [Fintype Code] {depth : ℕ} (s : Program A Coord Context Code depth) | |
| (ctx : Context) (hctx : ∀ a, s.context a = ctx) | |
| (joint : Function.Injective (fun a => (s.view .B a, s.view .C a, s.view .A a))) : | |
| ReadableHierarchy A (Coord .B) (Coord .C) (Coord .A) where | |
| Label := fun _ => Code | |
| fintypeLabel := fun _ => inferInstance | |
| labels := s.labels | |
| depth := depth | |
| pair := s.pair | |
| x := s.view .B | |
| y := s.view .C | |
| z := s.view .A | |
| coordinates_injective := joint | |
| readXYX n _ prior own := s.read n .B ctx (Program.recordArray n prior) own | |
| readXYY n _ prior own := s.read n .C ctx (Program.recordArray n prior) own | |
| readXZX n _ prior own := s.read n .B ctx (Program.recordArray n prior) own | |
| readXZZ n _ prior own := s.read n .A ctx (Program.recordArray n prior) own | |
| readYZY n _ prior own := s.read n .C ctx (Program.recordArray n prior) own | |
| readYZZ n _ prior own := s.read n .A ctx (Program.recordArray n prior) own | |
| readXYX_readable n hn hp a := by | |
| change s.read n .B ctx (Program.recordArray n (labelRecordOf s.labels n a)) | |
| (s.view .B a) = s.labels n a | |
| rw [Program.recordArray_labelRecordOf, ← hctx a] | |
| exact s.readable n hn a .B (Or.inl (by simp only [hp, firstSide])) | |
| readXYY_readable n hn hp a := by | |
| change s.read n .C ctx (Program.recordArray n (labelRecordOf s.labels n a)) | |
| (s.view .C a) = s.labels n a | |
| rw [Program.recordArray_labelRecordOf, ← hctx a] | |
| exact s.readable n hn a .C (Or.inr (by simp only [hp, secondSide])) | |
| readXZX_readable n hn hp a := by | |
| change s.read n .B ctx (Program.recordArray n (labelRecordOf s.labels n a)) | |
| (s.view .B a) = s.labels n a | |
| rw [Program.recordArray_labelRecordOf, ← hctx a] | |
| exact s.readable n hn a .B (Or.inl (by simp only [hp, firstSide])) | |
| readXZZ_readable n hn hp a := by | |
| change s.read n .A ctx (Program.recordArray n (labelRecordOf s.labels n a)) | |
| (s.view .A a) = s.labels n a | |
| rw [Program.recordArray_labelRecordOf, ← hctx a] | |
| exact s.readable n hn a .A (Or.inr (by simp only [hp, secondSide])) | |
| readYZY_readable n hn hp a := by | |
| change s.read n .C ctx (Program.recordArray n (labelRecordOf s.labels n a)) | |
| (s.view .C a) = s.labels n a | |
| rw [Program.recordArray_labelRecordOf, ← hctx a] | |
| exact s.readable n hn a .C (Or.inl (by simp only [hp, firstSide])) | |
| readYZZ_readable n hn hp a := by | |
| change s.read n .A ctx (Program.recordArray n (labelRecordOf s.labels n a)) | |
| (s.view .A a) = s.labels n a | |
| rw [Program.recordArray_labelRecordOf, ← hctx a] | |
| exact s.readable n hn a .A (Or.inr (by simp only [hp, secondSide])) | |
| theorem ofProgram_complete {Coord : Color → Type v} {Context : Type q} {Code : Type p} | |
| [Fintype Code] {depth : ℕ} (s : Program A Coord Context Code depth) | |
| (ctx : Context) (hctx : ∀ a, s.context a = ctx) | |
| (joint : Function.Injective (fun a => (s.view .B a, s.view .C a, s.view .A a))) | |
| (complete : s.Complete) : | |
| Function.Injective (labelRecordOf (ofProgram s ctx hctx joint).labels depth) := by | |
| intro a b h | |
| apply complete a b ((hctx a).trans (hctx b).symm) | |
| intro n hn | |
| exact label_eq_of_record_eq s.labels h hn | |
| end ReadableHierarchy | |
| end MatrixMultiplication.CompletionHierarchyWords | |
| end | |
| end OAI | |