openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /CoppersmithWinograd /FieldCW.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.FieldCoefficientExtraction | |
| import Mathlib.Algebra.BigOperators.Option | |
| import Mathlib.Tactic.IntervalCases | |
| namespace OAI | |
| /-! Coppersmith–Winograd tensors, tensor powers and local restrictions. -/ | |
| noncomputable section | |
| open scoped BigOperators | |
| open MatrixMultiplication.Foundation Polynomial | |
| namespace MatrixMultiplication.FieldCW | |
| section Integral | |
| variable (F : Type*) [CommRing F] (q : ℕ) | |
| def zeroForm (x : Fin (q + 2)) : F := if x.val = 0 then 1 else 0 | |
| def interiorForm (i : Fin q) (x : Fin (q + 2)) : F := | |
| if x.val = i.val + 1 then 1 else 0 | |
| def lastForm (x : Fin (q + 2)) : F := if x.val = q + 1 then 1 else 0 | |
| def interiorSum (x : Fin (q + 2)) : F := ∑ i, interiorForm F q i x | |
| def tensor : Tensor F (Fin (q + 2)) (Fin (q + 2)) (Fin (q + 2)) := | |
| fun x y z => | |
| (∑ i : Fin q, | |
| (zeroForm F q x * interiorForm F q i y * interiorForm F q i z + | |
| interiorForm F q i x * zeroForm F q y * interiorForm F q i z + | |
| interiorForm F q i x * interiorForm F q i y * zeroForm F q z)) + | |
| lastForm F q x * zeroForm F q y * zeroForm F q z + | |
| zeroForm F q x * lastForm F q y * zeroForm F q z + | |
| zeroForm F q x * zeroForm F q y * lastForm F q z | |
| theorem interior_pair_sum (x y : Fin (q + 2)) : | |
| (∑ i : Fin q, interiorForm F q i x * interiorForm F q i y) = | |
| if x = y ∧ 1 ≤ x.val ∧ x.val ≤ q then 1 else 0 := by | |
| classical | |
| by_cases hx : 1 ≤ x.val ∧ x.val ≤ q | |
| · let i : Fin q := ⟨x.val - 1, by omega⟩ | |
| rw [Finset.sum_eq_single i] | |
| · simp [interiorForm, i, Nat.sub_add_cancel hx.1, hx.1, hx.2, | |
| Fin.ext_iff, eq_comm] | |
| · intro j hj hji | |
| have hne : x.val ≠ j.val + 1 := by | |
| intro h | |
| apply hji | |
| apply Fin.ext | |
| dsimp [i] | |
| omega | |
| simp [interiorForm, hne] | |
| · simp | |
| · have hsum : (∑ i : Fin q, interiorForm F q i x * interiorForm F q i y) = 0 := by | |
| apply Finset.sum_eq_zero | |
| intro i hi | |
| have hne : x.val ≠ i.val + 1 := by | |
| intro h | |
| apply hx | |
| have := i.isLt | |
| omega | |
| simp [interiorForm, hne] | |
| rw [hsum, ite_eq_right] | |
| intro h | |
| exact hx ⟨h.2.1, h.2.2⟩ | |
| theorem tensor_zero_side (x y z : Fin (q + 2)) (hx : x.val = 0) : | |
| tensor F q x y z = | |
| if (y.val = 0 ∧ z.val = q + 1) ∨ | |
| (y.val = q + 1 ∧ z.val = 0) ∨ | |
| (1 ≤ y.val ∧ y.val ≤ q ∧ y = z) then 1 else 0 := by | |
| have hx0 : zeroForm F q x = 1 := by simp [zeroForm, hx] | |
| have hxlast : lastForm F q x = 0 := by simp [lastForm, hx] | |
| have hxmid (i : Fin q) : interiorForm F q i x = 0 := by | |
| simp [interiorForm, hx] | |
| simp only [tensor, hx0, hxlast, hxmid, one_mul, zero_mul, add_zero] | |
| rw [interior_pair_sum] | |
| by_cases hy0 : y.val = 0 | |
| · simp [zeroForm, lastForm, hy0] | |
| · by_cases hylast : y.val = q + 1 | |
| · have hyinter : ¬ y.val ≤ q := by omega | |
| simp [zeroForm, lastForm, hylast] | |
| · have hylo : 1 ≤ y.val := by omega | |
| have hyhi : y.val ≤ q := by have := y.isLt; omega | |
| simp [zeroForm, lastForm, hy0, hylast, hylo, hyhi] | |
| theorem tensor_cyclic (x y z : Fin (q + 2)) : | |
| tensor F q x y z = tensor F q y z x := by | |
| unfold tensor | |
| have hsum : | |
| (∑ i : Fin q, | |
| (zeroForm F q x * interiorForm F q i y * interiorForm F q i z + | |
| interiorForm F q i x * zeroForm F q y * interiorForm F q i z + | |
| interiorForm F q i x * interiorForm F q i y * zeroForm F q z)) = | |
| ∑ i : Fin q, | |
| (zeroForm F q y * interiorForm F q i z * interiorForm F q i x + | |
| interiorForm F q i y * zeroForm F q z * interiorForm F q i x + | |
| interiorForm F q i y * interiorForm F q i z * zeroForm F q x) := by | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| ring | |
| rw [hsum] | |
| ring | |
| def form (e : ℕ) (u v : F) : Polynomial F := C u + X ^ e * C v | |
| def left : Option (Option (Fin q)) → Fin (q + 2) → Polynomial F | |
| | some (some i), x => X * form F 1 (zeroForm F q x) (interiorForm F q i x) | |
| | some none, x => -form F 2 (zeroForm F q x) (interiorSum F q x) | |
| | none, x => (1 - C (q : F) * X) * form F 3 (zeroForm F q x) (lastForm F q x) | |
| def other : Option (Option (Fin q)) → Fin (q + 2) → Polynomial F | |
| | some (some i), x => form F 1 (zeroForm F q x) (interiorForm F q i x) | |
| | some none, x => form F 2 (zeroForm F q x) (interiorSum F q x) | |
| | none, x => form F 3 (zeroForm F q x) (lastForm F q x) | |
| def polynomial : Tensor (Polynomial F) (Fin (q + 2)) (Fin (q + 2)) (Fin (q + 2)) := | |
| fun x y z => ∑ i, Tensor.rankOne (left F q i) (other F q i) (other F q i) x y z | |
| theorem rank_bound : Tensor.RankAtMost (polynomial F q) (q + 2) := by | |
| have h := Tensor.rankAtMost_sum_rankOne (left F q) (other F q) (other F q) | |
| simp only [Fintype.card_option, Fintype.card_fin, Nat.add_assoc] at h | |
| convert h using 1 | |
| rfl | |
| theorem polynomial_apply (x y z : Fin (q + 2)) : | |
| polynomial F q x y z = | |
| (∑ i : Fin q, X * form F 1 (zeroForm F q x) (interiorForm F q i x) * | |
| form F 1 (zeroForm F q y) (interiorForm F q i y) * | |
| form F 1 (zeroForm F q z) (interiorForm F q i z)) - | |
| form F 2 (zeroForm F q x) (interiorSum F q x) * | |
| form F 2 (zeroForm F q y) (interiorSum F q y) * | |
| form F 2 (zeroForm F q z) (interiorSum F q z) + | |
| (1 - C (q : F) * X) * form F 3 (zeroForm F q x) (lastForm F q x) * | |
| form F 3 (zeroForm F q y) (lastForm F q y) * | |
| form F 3 (zeroForm F q z) (lastForm F q z) := by | |
| simp only [polynomial, Fintype.sum_option, Tensor.rankOne, left, other] | |
| ring | |
| private theorem coeff_mul_two (a b : Polynomial F) : | |
| (a * b).coeff 2 = a.coeff 0 * b.coeff 2 + a.coeff 1 * b.coeff 1 + | |
| a.coeff 2 * b.coeff 0 := by | |
| rw [Polynomial.coeff_mul, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] | |
| simp [Finset.sum_range_succ, add_assoc] | |
| private theorem coeff_mul_three (a b : Polynomial F) : | |
| (a * b).coeff 3 = a.coeff 0 * b.coeff 3 + a.coeff 1 * b.coeff 2 + | |
| a.coeff 2 * b.coeff 1 + a.coeff 3 * b.coeff 0 := by | |
| rw [Polynomial.coeff_mul, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] | |
| simp [Finset.sum_range_succ, add_assoc] | |
| theorem vanishes (x y z : Fin (q + 2)) (k : ℕ) (hk : k < 3) : | |
| (polynomial F q x y z).coeff k = 0 := by | |
| interval_cases k <;> | |
| simp [polynomial_apply, Polynomial.mul_coeff_zero, | |
| Polynomial.mul_coeff_one, coeff_mul_two, form, Polynomial.coeff_add, | |
| Polynomial.coeff_sub, interiorSum, | |
| Polynomial.coeff_one, Polynomial.coeff_X, | |
| Finset.sum_add_distrib, Finset.sum_mul, Finset.mul_sum] <;> | |
| simp only [mul_assoc, mul_comm] <;> ring_nf | |
| simp only [Finset.sum_add_distrib, Finset.mul_sum, mul_assoc, mul_comm] | |
| ring | |
| theorem leading (x y z : Fin (q + 2)) : | |
| (polynomial F q x y z).coeff 3 = tensor F q x y z := by | |
| simp [polynomial_apply, Polynomial.mul_coeff_zero, | |
| Polynomial.mul_coeff_one, coeff_mul_two, coeff_mul_three, form, | |
| Polynomial.coeff_add, Polynomial.coeff_sub, | |
| tensor, Polynomial.coeff_one, Polynomial.coeff_X, | |
| Finset.sum_add_distrib] | |
| simp only [mul_assoc, mul_comm, mul_left_comm] | |
| ring_nf | |
| simp only [Finset.sum_add_distrib, mul_assoc, mul_comm, mul_left_comm] | |
| ring | |
| def degreeBound : ℕ := Finset.univ.sup fun xyz : | |
| Fin (q + 2) × Fin (q + 2) × Fin (q + 2) => | |
| (polynomial F q xyz.1 xyz.2.1 xyz.2.2).natDegree | |
| theorem degree_bound (x y z : Fin (q + 2)) : | |
| (polynomial F q x y z).degree ≤ degreeBound F q := by | |
| let f : (Fin (q + 2) × Fin (q + 2) × Fin (q + 2)) → ℕ := | |
| fun xyz => (polynomial F q xyz.1 xyz.2.1 xyz.2.2).natDegree | |
| have h := Finset.le_sup (s := Finset.univ) (f := f) (b := (x, y, z)) | |
| (Finset.mem_univ (x, y, z)) | |
| exact Polynomial.degree_le_of_natDegree_le h | |
| end Integral | |
| variable (F : Type*) [Field F] (q : ℕ) | |
| def approximation : Tensor.PolynomialApproximation (tensor F q) (q + 2) 3 | |
| (degreeBound F q) where | |
| polynomial := polynomial F q | |
| rank_bound := rank_bound F q | |
| vanishes := vanishes F q | |
| leading := leading F q | |
| degree_bound := degree_bound F q | |
| theorem rank_power (n : ℕ) : | |
| Tensor.RankAtMost (Tensor.power (tensor F q) n) ((3 * n + 1) ^ 2 * (q + 2) ^ n) := | |
| FieldCoefficientExtraction.rank_power (approximation F q) n | |
| end MatrixMultiplication.FieldCW | |
| end | |
| end OAI | |