openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /PopulationCWNonzero.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CanonicalCW | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointPopulationCWNonzero | |
| open MatrixMultiplication.Foundation JointPopulation JointCanonicalization JointCanonicalCW | |
| open CWStrands InheritedMasks | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| variable {H F : Type*} [Fintype H] [DecidableEq H] [Field F] | |
| omit [DecidableEq H] in | |
| theorem class_half_products_nonzero | |
| (counts : H → Shape → ℕ) (leftLength rightLength : H → ℕ) | |
| (parentShape : H → Fin 3 → ℕ) (hleft : ∀ h, leftLength h ≤ 8) | |
| (hsupp : ∀ h u, 0 < counts h u → ∀ s, shapeNat u s ≤ parentShape h s) | |
| (e : Target counts) | |
| (x y z : RawPairs counts (Left leftLength) (Right rightLength)) | |
| (hsource : sourceTensor counts (Left leftLength) (Right rightLength) | |
| (parentTensor (F := F) leftLength rightLength parentShape) x y z ≠ 0) | |
| (hx : coarseMask counts (Left leftLength) (Right rightLength) | |
| (coarse leftLength rightLength hleft) 0 e x) | |
| (hy : coarseMask counts (Left leftLength) (Right rightLength) | |
| (coarse leftLength rightLength hleft) 1 e y) | |
| (hz : coarseMask counts (Left leftLength) (Right rightLength) | |
| (coarse leftLength rightLength hleft) 2 e z) | |
| (h : H) (u : Shape) (hn : 0 < counts h u) : | |
| (∏ i : Class counts e h u, | |
| shapeTensor (F := F) (Fin (leftLength h)) (shapeNat u) | |
| (x h i.val).1 (y h i.val).1 (z h i.val).1) ≠ 0 ∧ | |
| (∏ i : Class counts e h u, | |
| shapeTensor (F := F) (Fin (rightLength h)) (parentShape h - shapeNat u) | |
| (x h i.val).2 (y h i.val).2 (z h i.val).2) ≠ 0 := by | |
| change (∏ h' : H, ∏ i : Positions counts h', | |
| parentTensor (F := F) leftLength rightLength parentShape h' | |
| (x h' i) (y h' i) (z h' i)) ≠ 0 at hsource | |
| have hparent (i : Positions counts h) : | |
| parentTensor (F := F) leftLength rightLength parentShape h | |
| (x h i) (y h i) (z h i) ≠ 0 := | |
| Finset.prod_ne_zero_iff.mp | |
| (Finset.prod_ne_zero_iff.mp hsource h (Finset.mem_univ h)) i (Finset.mem_univ i) | |
| have hhalves (i : Class counts e h u) : | |
| shapeTensor (F := F) (Fin (leftLength h)) (shapeNat u) | |
| (x h i.val).1 (y h i.val).1 (z h i.val).1 * | |
| shapeTensor (F := F) (Fin (rightLength h)) (parentShape h - shapeNat u) | |
| (x h i.val).2 (y h i.val).2 (z h i.val).2 ≠ 0 := by | |
| have hx' : coarse leftLength rightLength hleft 0 h (x h i.val) = u.1 := by | |
| simpa [shapeSide, i.property] using hx h i.val | |
| have hy' : coarse leftLength rightLength hleft 1 h (y h i.val) = u.2.1 := by | |
| simpa [shapeSide, i.property] using hy h i.val | |
| have hz' : coarse leftLength rightLength hleft 2 h (z h i.val) = u.2.2 := by | |
| simpa [shapeSide, i.property] using hz h i.val | |
| have hblock : block (Left leftLength) (Right rightLength) | |
| (parentTensor (F := F) leftLength rightLength parentShape) | |
| (coarse leftLength rightLength hleft) (h, u) | |
| (x h i.val) (y h i.val) (z h i.val) ≠ 0 := by | |
| simpa [block, hx', hy', hz'] using hparent i.val | |
| rw [block_eq_product leftLength rightLength parentShape hleft h u | |
| (hsupp h u hn)] at hblock | |
| exact hblock | |
| constructor | |
| · apply Finset.prod_ne_zero_iff.mpr | |
| intro i _ | |
| exact (mul_ne_zero_iff.mp (hhalves i)).1 | |
| · apply Finset.prod_ne_zero_iff.mpr | |
| intro i _ | |
| exact (mul_ne_zero_iff.mp (hhalves i)).2 | |
| end MatrixMultiplication.JointPopulationCWNonzero | |
| end | |
| end OAI | |