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f18e7de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | 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
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