openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /Source.lean
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.FieldCW | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.Witness | |
| import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialRestriction | |
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexTensorBatching | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| open scoped BigOperators | |
| open MatrixMultiplication.Foundation | |
| namespace MatrixMultiplication.AllFieldSource | |
| abbrev Word (K N : ℕ) := Fin (8 * K * N) → Fin 7 | |
| abbrev Index (Copies : Type*) (K N : ℕ) := Copies × Word K N | |
| def tensor (F Copies : Type*) [CommRing F] [DecidableEq Copies] (K N : ℕ) : | |
| Tensor F (Index Copies K N) (Index Copies K N) (Index Copies K N) := | |
| Tensor.directSum (fun _ : Copies => Tensor.power (FieldCW.tensor F 5) (8 * K * N)) | |
| def rankBudget (Copies : Type*) [Fintype Copies] (K N : ℕ) : ℕ := | |
| Fintype.card Copies * ((3 * (8 * K * N) + 1) ^ 2 * 7 ^ (8 * K * N)) | |
| theorem rank_bound (F Copies : Type*) [Field F] | |
| [Fintype Copies] [DecidableEq Copies] (K N : ℕ) : | |
| Tensor.RankAtMost (tensor F Copies K N) (rankBudget Copies K N) := by | |
| have h := (Tensor.diagonal_rankAtMost (K := F) Copies).product | |
| (FieldCW.rank_power F 5 (8 * K * N)) | |
| convert h using 1 | |
| funext x y z | |
| simp [tensor, Tensor.directSum, Tensor.product, Tensor.diagonal, ite_mul] | |
| rfl | |
| theorem rankBudget_pos (Copies : Type*) [Fintype Copies] (K N : ℕ) | |
| (hcopies : 0 < Fintype.card Copies) : 0 < rankBudget Copies K N := by | |
| unfold rankBudget | |
| exact Nat.mul_pos hcopies | |
| (Nat.mul_pos (pow_pos (Nat.succ_pos _) _) (pow_pos (by decide) _)) | |
| theorem rankBudget_prod (Slots Copies : Type*) [Fintype Slots] [Fintype Copies] | |
| (K N : ℕ) : | |
| rankBudget (Slots × Copies) K N = Fintype.card Slots * rankBudget Copies K N := by | |
| simp only [rankBudget, Fintype.card_prod, Nat.mul_assoc] | |
| def matrixTarget (F : Type*) [CommSemiring F] (a b c L : ℕ) := | |
| Tensor.directSum (fun _ : Fin L => | |
| Tensor.matrixCoefficients (K := F) (Fin a) (Fin b) (Fin c)) | |
| section Consumer | |
| variable {F Copies : Type*} [Field F] [Fintype Copies] [DecidableEq Copies] | |
| variable {K N a b c L : ℕ} | |
| theorem restricted_rank_bound | |
| (A : (Fin L × (Fin a × Fin b)) → Index Copies K N → F) | |
| (B : (Fin L × (Fin b × Fin c)) → Index Copies K N → F) | |
| (C : (Fin L × (Fin c × Fin a)) → Index Copies K N → F) | |
| (hcoeff : ∀ x y z, | |
| Tensor.restrict A B C (tensor F Copies K N) x y z = matrixTarget F a b c L x y z) : | |
| Tensor.RankAtMost (matrixTarget F a b c L) (rankBudget Copies K N) := by | |
| have heq : Tensor.restrict A B C (tensor F Copies K N) = matrixTarget F a b c L := | |
| funext fun x => funext fun y => funext fun z => hcoeff x y z | |
| rw [← heq] | |
| exact (rank_bound F Copies K N).restrict A B C | |
| def restrictedApproximation | |
| (A : (Fin L × (Fin a × Fin b)) → Index Copies K N → F) | |
| (B : (Fin L × (Fin b × Fin c)) → Index Copies K N → F) | |
| (C : (Fin L × (Fin c × Fin a)) → Index Copies K N → F) | |
| (hcoeff : ∀ x y z, | |
| Tensor.restrict A B C (tensor F Copies K N) x y z = matrixTarget F a b c L x y z) : | |
| Tensor.PolynomialApproximation (matrixTarget F a b c L) (rankBudget Copies K N) 0 0 := | |
| Tensor.PolynomialApproximation.ofRankAtMost _ _ (restricted_rank_bound A B C hcoeff) | |
| def ofRestriction | |
| (A : (Fin L × (Fin a × Fin b)) → Index Copies K N → F) | |
| (B : (Fin L × (Fin b × Fin c)) → Index Copies K N → F) | |
| (C : (Fin L × (Fin c × Fin a)) → Index Copies K N → F) | |
| (hcoeff : ∀ x y z, | |
| Tensor.restrict A B C (tensor F Copies K N) x y z = matrixTarget F a b c L x y z) | |
| (hcopies : 0 < Fintype.card Copies) (ha : 0 < a) (hb : 0 < b) | |
| (hc : 0 < c) (hL : 0 < L) : AllFieldWitness F where | |
| rows := a | |
| inner := b | |
| columns := c | |
| multiplicity := L | |
| rankBound := rankBudget Copies K N | |
| order := 0 | |
| degree := 0 | |
| rows_pos := ha | |
| inner_pos := hb | |
| columns_pos := hc | |
| multiplicity_pos := hL | |
| rankBound_pos := rankBudget_pos Copies K N hcopies | |
| approximation := restrictedApproximation A B C hcoeff | |
| theorem ofRestriction_polynomial | |
| (A : (Fin L × (Fin a × Fin b)) → Index Copies K N → F) | |
| (B : (Fin L × (Fin b × Fin c)) → Index Copies K N → F) | |
| (C : (Fin L × (Fin c × Fin a)) → Index Copies K N → F) | |
| (hcoeff : ∀ x y z, | |
| Tensor.restrict A B C (tensor F Copies K N) x y z = matrixTarget F a b c L x y z) | |
| (hcopies : 0 < Fintype.card Copies) (ha : 0 < a) (hb : 0 < b) | |
| (hc : 0 < c) (hL : 0 < L) : | |
| (ofRestriction A B C hcoeff hcopies ha hb hc hL).approximation.polynomial = | |
| fun x y z => Polynomial.C (matrixTarget F a b c L x y z) := rfl | |
| end Consumer | |
| end MatrixMultiplication.AllFieldSource | |
| end | |
| end OAI | |