openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Tensor /FieldCoefficientExtraction.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialApproximation | |
| import Mathlib.Algebra.BigOperators.NatAntidiagonal | |
| import Lean.Elab.Tactic.Omega | |
| namespace OAI | |
| /-! Finite coefficient tensors and their algebraic transformations. -/ | |
| noncomputable section | |
| open scoped BigOperators | |
| open MatrixMultiplication.Foundation | |
| namespace MatrixMultiplication.FieldCoefficientExtraction | |
| variable {F X Y Z : Type*} | |
| theorem coeff_triple [CommSemiring F] (a b c : Polynomial F) (d : ℕ) : | |
| (a * b * c).coeff d = | |
| ∑ i : Fin (d + 1), ∑ j : Fin (d + 1), | |
| (if j.val ≤ d - i.val then a.coeff (d - i.val - j.val) else 0) * | |
| b.coeff j.val * c.coeff i.val := by | |
| classical | |
| rw [show a * b * c = c * (b * a) by ring] | |
| rw [Polynomial.coeff_mul, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] | |
| rw [← Fin.sum_univ_eq_sum_range] | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| rw [Polynomial.coeff_mul, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] | |
| rw [Finset.mul_sum] | |
| rw [Fin.sum_univ_eq_sum_range (fun j : ℕ => | |
| (if j ≤ d - i.val then a.coeff (d - i.val - j) else 0) * | |
| b.coeff j * c.coeff i.val) (d + 1)] | |
| symm | |
| calc | |
| (∑ j ∈ Finset.range (d + 1), | |
| (if j ≤ d - i.val then a.coeff (d - i.val - j) else 0) * | |
| b.coeff j * c.coeff i.val) = | |
| ∑ j ∈ Finset.range (d - i.val + 1), | |
| (if j ≤ d - i.val then a.coeff (d - i.val - j) else 0) * | |
| b.coeff j * c.coeff i.val := by | |
| symm | |
| apply Finset.sum_subset (Finset.range_mono (by omega)) | |
| intro j hj hjnot | |
| have h : ¬ j ≤ d - i.val := by | |
| simpa only [Finset.mem_range, Nat.lt_succ_iff] using hjnot | |
| simp [h] | |
| _ = ∑ j ∈ Finset.range (d - i.val + 1), | |
| c.coeff i.val * (b.coeff j * a.coeff (d - i.val - j)) := by | |
| apply Finset.sum_congr rfl | |
| intro j hj | |
| rw [ite_eq_left (Nat.le_of_lt_succ (Finset.mem_range.mp hj))] | |
| ring | |
| theorem rankAtMost_coeff [CommSemiring F] | |
| {P : Tensor (Polynomial F) X Y Z} {r : ℕ} | |
| (hP : Tensor.RankAtMost P r) (d : ℕ) : | |
| Tensor.RankAtMost (fun x y z => (P x y z).coeff d) ((d + 1) ^ 2 * r) := by | |
| classical | |
| rcases hP with ⟨a, b, c, rfl⟩ | |
| let aa : (Fin r × Fin (d + 1) × Fin (d + 1)) → X → F := | |
| fun v x => if v.2.2.val ≤ d - v.2.1.val then | |
| (a v.1 x).coeff (d - v.2.1.val - v.2.2.val) else 0 | |
| let bb : (Fin r × Fin (d + 1) × Fin (d + 1)) → Y → F := | |
| fun v y => (b v.1 y).coeff v.2.2.val | |
| let cc : (Fin r × Fin (d + 1) × Fin (d + 1)) → Z → F := | |
| fun v z => (c v.1 z).coeff v.2.1.val | |
| have heq : (fun x y z => (∑ j, Tensor.rankOne (a j) (b j) (c j) x y z).coeff d) = | |
| fun x y z => ∑ v, Tensor.rankOne (aa v) (bb v) (cc v) x y z := by | |
| funext x y z | |
| simp only [Polynomial.finsetSum_coeff, Tensor.rankOne, Fintype.sum_prod_type, | |
| coeff_triple, aa, bb, cc] | |
| rw [heq] | |
| simpa only [Fintype.card_prod, Fintype.card_fin, pow_two, Nat.mul_comm, | |
| Nat.mul_left_comm, Nat.mul_assoc] using Tensor.rankAtMost_sum_rankOne aa bb cc | |
| theorem rank_power [Field F] {T : Tensor F X Y Z} {r d D : ℕ} | |
| (A : Tensor.PolynomialApproximation T r d D) (n : ℕ) : | |
| Tensor.RankAtMost (Tensor.power T n) ((d * n + 1) ^ 2 * r ^ n) := by | |
| have h := rankAtMost_coeff (A.power n).rank_bound (d * n) | |
| have heq : (fun x y z => ((A.power n).polynomial x y z).coeff (d * n)) = | |
| Tensor.power T n := by | |
| funext x y z | |
| exact (A.power n).leading x y z | |
| rw [heq] at h | |
| exact h | |
| end MatrixMultiplication.FieldCoefficientExtraction | |
| end | |
| end OAI | |