import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Convolution.Rank
import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Arithmetic.RankExponent

namespace OAI

/-!
# Exact rank of polynomial convolution

Each output coefficient of polynomial multiplication occurs in some product
of two input monomials. Choosing one such pair for each output gives an
identity minor in the output flattening. Together with evaluation and
interpolation, this proves the exact rank of `C(a,b)` for positive dimensions.
-/

noncomputable section

namespace MatrixMultiplication.AuxiliarySeparation

open MatrixMultiplication.Foundation

/-- Choosing one product for every output monomial gives an identity minor
of the output flattening. -/
theorem convolution_identity_minor {a b : ℕ} (ha : 0 < a) (hb : 0 < b) :
    ∃ (x : Fin (a + b - 1) → Fin a) (y : Fin (a + b - 1) → Fin b),
      ∀ k l, convolution a b (x k) (y k) l = if k = l then 1 else 0 := by
  choose x y hxy using fun k : Fin (a + b - 1) => convolution_output_supported ha hb k
  have hsum (k : Fin (a + b - 1)) : (x k).val + (y k).val = k.val :=
    (convolution_eq_one_iff _ _ _).mp (hxy k)
  refine ⟨x, y, ?_⟩
  intro k l
  simp only [convolution_apply, hsum, Fin.val_inj]

/-- Every rank decomposition of convolution has at least one term per output
coefficient. -/
theorem convolution_rank_lower {a b r : ℕ} (ha : 0 < a) (hb : 0 < b)
    (hRank : Tensor.RankAtMost (convolution a b) r) : a + b - 1 ≤ r := by
  obtain ⟨x, y, hxy⟩ := convolution_identity_minor ha hb
  simpa using hRank.card_le_of_identity x y id hxy

/-- Polynomial multiplication has exact rank `a+b-1` when both inputs have
positive dimension. -/
theorem exactRank_convolution {a b : ℕ} (ha : 0 < a) (hb : 0 < b) :
    exactRank (convolution a b) = a + b - 1 := by
  apply le_antisymm (exactRank_le (convolution_rankAtMost a b))
  exact convolution_rank_lower ha hb (exactRank_spec _)

end MatrixMultiplication.AuxiliarySeparation

end

end OAI