File size: 3,785 Bytes
cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | 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
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