openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /CoppersmithWinograd /CWCompleteStatistics.lean
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWWindowedLeaves | |
| namespace OAI | |
| /-! Coppersmith–Winograd tensors, tensor powers and local restrictions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CWCompleteStatistics | |
| open CWWindowedLeaves CWLeafStatistics CWStrands AllFieldParameters | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| def pairEquiv : Raw 2 ≃ (Fin 7 × Fin 7) where | |
| toFun w := (w 0, w 1) | |
| invFun u := ![u.1, u.2] | |
| left_inv w := by funext i; fin_cases i <;> rfl | |
| right_inv u := by cases u; rfl | |
| def orderedEquiv : Raw 4 ≃ ((Fin 7 × Fin 7) × (Fin 7 × Fin 7)) where | |
| toFun w := ((w 0, w 1), (w 2, w 3)) | |
| invFun u := ![u.1.1, u.1.2, u.2.1, u.2.2] | |
| left_inv w := by funext i; fin_cases i <;> rfl | |
| right_inv u := by cases u with | mk u v => cases u; cases v; rfl | |
| def twoStatistic (w : Raw 2) : Fin 6 := pairStatistic (pairEquiv w) | |
| def fourStatistic (w : Raw 4) : Fin 6 × Fin 6 := | |
| (pairStatistic (orderedEquiv w).1, pairStatistic (orderedEquiv w).2) | |
| def complementStatistic : Fin 6 ≃ Fin 6 := | |
| { toFun := kappa | |
| invFun := kappa | |
| left_inv := kappa_involution | |
| right_inv := kappa_involution } | |
| def orderedComplementStatistic : (Fin 6 × Fin 6) ≃ (Fin 6 × Fin 6) := | |
| Equiv.prodCongr complementStatistic complementStatistic | |
| theorem two_weight (w : Raw 2) : | |
| CWStrands.weight w = statisticWeight (twoStatistic w) := by | |
| rw [twoStatistic, statistic_weight] | |
| simp [CWStrands.weight, pairEquiv, Fin.sum_univ_succ] | |
| theorem four_weight (w : Raw 4) : | |
| CWStrands.weight w = statisticWeight (fourStatistic w).1 + | |
| statisticWeight (fourStatistic w).2 := by | |
| simp only [fourStatistic, statistic_weight] | |
| simp [CWStrands.weight, orderedEquiv, Fin.sum_univ_succ, Nat.add_assoc] | |
| theorem two_multiplicity (s : Fin 6) : | |
| Fintype.card {w : Raw 2 // twoStatistic w = s} = AllFieldParameters.multiplicity s := by | |
| let e : {w : Raw 2 // twoStatistic w = s} ≃ | |
| {u : Fin 7 × Fin 7 // pairStatistic u = s} := | |
| { toFun := fun w => ⟨pairEquiv w.val, w.property⟩ | |
| invFun := fun u => ⟨pairEquiv.symm u.val, by simpa [twoStatistic] using u.property⟩ | |
| left_inv := by intro w; apply Subtype.ext; simp | |
| right_inv := by intro u; apply Subtype.ext; simp } | |
| rw [Fintype.card_congr e, pair_multiplicity] | |
| theorem four_multiplicity (s : Fin 6 × Fin 6) : | |
| Fintype.card {w : Raw 4 // fourStatistic w = s} = | |
| AllFieldParameters.multiplicity s.1 * AllFieldParameters.multiplicity s.2 := by | |
| let e : {w : Raw 4 // fourStatistic w = s} ≃ | |
| {u : (Fin 7 × Fin 7) × (Fin 7 × Fin 7) // | |
| pairStatistic u.1 = s.1 ∧ pairStatistic u.2 = s.2} := | |
| { toFun := fun w => ⟨orderedEquiv w.val, | |
| congrArg Prod.fst w.property, congrArg Prod.snd w.property⟩ | |
| invFun := fun u => ⟨orderedEquiv.symm u.val, by | |
| apply Prod.ext | |
| · simpa [fourStatistic] using u.property.1 | |
| · simpa [fourStatistic] using u.property.2⟩ | |
| left_inv := by intro w; apply Subtype.ext; simp | |
| right_inv := by intro u; apply Subtype.ext; simp } | |
| rw [Fintype.card_congr e, ordered_pair_multiplicity] | |
| theorem two_complement (w : Raw 2) : | |
| twoStatistic (complementWord w) = complementStatistic (twoStatistic w) := by | |
| exact complement_statistic (pairEquiv w) | |
| theorem four_complement (w : Raw 4) : | |
| fourStatistic (complementWord w) = orderedComplementStatistic (fourStatistic w) := by | |
| apply Prod.ext | |
| · exact complement_statistic (orderedEquiv w).1 | |
| · exact complement_statistic (orderedEquiv w).2 | |
| theorem two_fixedWeight_multiplicity (a : ℕ) (s : Fin 6) (hs : statisticWeight s = a) : | |
| Fintype.card {w : Alphabet 2 a // twoStatistic w.val = s} = | |
| AllFieldParameters.multiplicity s := by | |
| rw [ExactStatisticWords.card_fixedWeight_fiber twoStatistic CWStrands.weight | |
| statisticWeight two_weight a s hs, two_multiplicity] | |
| theorem four_fixedWeight_multiplicity (a : ℕ) (s : Fin 6 × Fin 6) | |
| (hs : statisticWeight s.1 + statisticWeight s.2 = a) : | |
| Fintype.card {w : Alphabet 4 a // fourStatistic w.val = s} = | |
| AllFieldParameters.multiplicity s.1 * AllFieldParameters.multiplicity s.2 := by | |
| rw [ExactStatisticWords.card_fixedWeight_fiber fourStatistic CWStrands.weight | |
| (fun s => statisticWeight s.1 + statisticWeight s.2) four_weight a s hs, | |
| four_multiplicity] | |
| end MatrixMultiplication.CWCompleteStatistics | |
| end | |
| end OAI | |