openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /CanonicalCW.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.Canonicalization | |
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWCoarseIndices | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointCanonicalCW | |
| open MatrixMultiplication.Foundation JointPopulation JointCanonicalization | |
| open CWStrands HistorySymmetry InheritedMasks | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| variable {H F : Type*} [Fintype H] [DecidableEq H] [CommRing F] | |
| (counts : H → Shape → ℕ) (leftLength rightLength : H → ℕ) | |
| (parentShape : H → Fin 3 → ℕ) | |
| (hleft : ∀ h, leftLength h ≤ 8) | |
| abbrev Left (h : H) := Fin (leftLength h) → Fin 7 | |
| abbrev Right (h : H) := Fin (rightLength h) → Fin 7 | |
| def shapeNat (u : Shape) : Fin 3 → ℕ := fun s => (shapeSide s u).val | |
| def parentTensor (h : H) : Tensor F | |
| (Left leftLength h × Right rightLength h) | |
| (Left leftLength h × Right rightLength h) | |
| (Left leftLength h × Right rightLength h) := | |
| fun x y z => shapeTensor (Fin (leftLength h) ⊕ Fin (rightLength h)) (parentShape h) | |
| (Sum.elim x.1 x.2) (Sum.elim y.1 y.2) (Sum.elim z.1 z.2) | |
| def coarse (_s : Fin 3) (h : H) (w : Left leftLength h × Right rightLength h) : Fin 17 := | |
| CWCoarseIndices.coarseIndex (by simpa using hleft h) w.1 | |
| def leftTensor (c : ClassKey (H := H)) : | |
| Tensor F (Left leftLength c.1) (Left leftLength c.1) (Left leftLength c.1) := | |
| shapeTensor (Fin (leftLength c.1)) (shapeNat c.2) | |
| def rightTensor (c : ClassKey (H := H)) : | |
| Tensor F (Right rightLength c.1) (Right rightLength c.1) (Right rightLength c.1) := | |
| shapeTensor (Fin (rightLength c.1)) (parentShape c.1 - shapeNat c.2) | |
| omit [Fintype H] [DecidableEq H] in | |
| theorem block_eq_product (h : H) (u : Shape) | |
| (hu : ∀ s, shapeNat u s ≤ parentShape h s) : | |
| block (Left leftLength) (Right rightLength) | |
| (parentTensor leftLength rightLength parentShape) | |
| (coarse leftLength rightLength hleft) (h, u) = | |
| Tensor.product (leftTensor (F := F) leftLength (h, u)) | |
| (rightTensor rightLength parentShape (h, u)) := by | |
| have hadd : shapeNat u + (parentShape h - shapeNat u) = parentShape h := by | |
| funext s | |
| exact Nat.add_sub_of_le (hu s) | |
| funext x y z | |
| have hs := shape_split (F := F) (P := Fin (leftLength h)) | |
| (Q := Fin (rightLength h)) (shapeNat u) (parentShape h - shapeNat u) x y z | |
| rw [hadd] at hs | |
| simpa [block, parentTensor, coarse, CWCoarseIndices.coarseIndex, | |
| Fin.ext_iff, leftTensor, rightTensor, shapeNat, shapeSide, | |
| Fin.isValue, ↓reduceIte] using hs | |
| omit [DecidableEq H] in | |
| theorem canonicalBase_eq_children | |
| (hsupport : ∀ h u, 0 < counts h u → ∀ s, shapeNat u s ≤ parentShape h s) : | |
| canonicalBase counts (Left leftLength) (Right rightLength) | |
| (parentTensor leftLength rightLength parentShape) | |
| (coarse leftLength rightLength hleft) = | |
| pairClassProduct (leftTensor (F := F) leftLength) | |
| (rightTensor rightLength parentShape) := by | |
| funext x y z | |
| unfold canonicalBase | |
| have hf : (∏ c : ClassKey (H := H), ∏ j : ClassPositions counts c, | |
| block (Left leftLength) (Right rightLength) | |
| (parentTensor leftLength rightLength parentShape) | |
| (coarse leftLength rightLength hleft) c | |
| (x.left c j, x.right c j) (y.left c j, y.right c j) (z.left c j, z.right c j)) = | |
| ∏ c : ClassKey (H := H), ∏ j : ClassPositions counts c, | |
| (leftTensor (F := F) leftLength c (x.left c j) (y.left c j) (z.left c j) * | |
| rightTensor rightLength parentShape c (x.right c j) (y.right c j) (z.right c j)) := by | |
| apply Finset.prod_congr rfl | |
| intro c _ | |
| apply Finset.prod_congr rfl | |
| intro j _ | |
| have hc : 0 < counts c.1 c.2 := lt_of_le_of_lt (Nat.zero_le j.val) j.isLt | |
| rw [block_eq_product leftLength rightLength parentShape hleft c.1 c.2 | |
| (hsupport c.1 c.2 hc)] | |
| rfl | |
| rw [hf] | |
| simp only [pairClassProduct, classProduct, Finset.prod_mul_distrib] | |
| abbrev RawStrands := ∀ h, Positions counts h → | |
| ((Fin (leftLength h) ⊕ Fin (rightLength h)) → Fin 7) | |
| def rawParentSource : Tensor F (RawStrands counts leftLength rightLength) | |
| (RawStrands counts leftLength rightLength) (RawStrands counts leftLength rightLength) := | |
| classProduct (fun h => shapeTensor (Fin (leftLength h) ⊕ Fin (rightLength h)) (parentShape h)) | |
| def joinWords (w : RawPairs counts (Left leftLength) (Right rightLength)) : | |
| RawStrands counts leftLength rightLength := | |
| fun h i => Sum.elim (w h i).1 (w h i).2 | |
| def joinMatrix : RawPairs counts (Left leftLength) (Right rightLength) → | |
| RawStrands counts leftLength rightLength → F := | |
| fun w v => if v = joinWords counts leftLength rightLength w then 1 else 0 | |
| theorem parentTensor_restrict : | |
| Tensor.restrict (joinMatrix (F := F) counts leftLength rightLength) | |
| (joinMatrix counts leftLength rightLength) (joinMatrix counts leftLength rightLength) | |
| (rawParentSource counts leftLength rightLength parentShape) = | |
| sourceTensor counts (Left leftLength) (Right rightLength) | |
| (parentTensor leftLength rightLength parentShape) := by | |
| classical | |
| funext x y z | |
| simp [Tensor.restrict, joinMatrix, rawParentSource, sourceTensor, | |
| classProduct, joinWords, parentTensor, ite_mul, mul_ite] | |
| end MatrixMultiplication.JointCanonicalCW | |
| end | |
| end OAI | |