| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.BaseConditionalReaders |
|
|
| namespace OAI |
|
|
| /-! Dual matrix multiplication exponents and finite rectangular constructions. -/ |
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|
| noncomputable section |
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|
| namespace MatrixMultiplication.DualWitness |
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|
| attribute [local instance 10000] Classical.propDecidable Classical.decEq |
| attribute [local instance 11000] instDecidableEqFin |
|
|
| open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels |
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|
| abbrev Word := Script.Coord (Fin 2) Script.dual |
| def source : FlaggedTensor Word Word Word := Script.tensor BaseTwo.flagged Script.dual |
| abbrev Support := RetainedLeaf source .B .C |
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|
| def readable : ReadableTensor source := |
| ReadableTensor.script BaseTwo.flagged BaseConditionalReaders.baseTwoReadable Script.dual |
|
|
| def program := terminationProgram source .B .C (by intro h; cases h) |
| readable.program readable.context_eq readable.view_eq |
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|
| theorem program_complete : program.Complete := |
| terminationProgram_complete source .B .C (by intro h; cases h) |
| readable.program readable.context_eq readable.view_eq readable.complete |
|
|
| theorem record_injective : Function.Injective |
| (labelRecordOf program.labels (1 + readable.depth)) := |
| terminationRecord_injective source .B .C (by intro h; cases h) |
| readable.program readable.context_eq readable.view_eq readable.complete |
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|
| def x (a : Support) : Word := a.val.val.1 |
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|
| theorem coord_card (U : Type*) [Fintype U] (s : Script) : |
| Fintype.card (Script.Coord U s) = Fintype.card U ^ s.factors := by |
| induction s with |
| | base => |
| change Fintype.card U = Fintype.card U ^ 1 |
| exact (pow_one _).symm |
| | step prior center m ih => |
| change Fintype.card (Fin m → Script.Coord U prior) = _ |
| simp only [Fintype.card_fun, Fintype.card_fin, ih, Script.factors, pow_mul] |
|
|
| theorem word_card : Fintype.card Word = 2 ^ 288 := by |
| simpa only [Fintype.card_fin, Script.dual_factors] using coord_card (Fin 2) Script.dual |
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| end MatrixMultiplication.DualWitness |
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| end |
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| end OAI |
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|
