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ae00908 | 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 | import OAI.Algebra.FormulaHitting.Minors
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
namespace OAI
noncomputable section
namespace NCHitting
open Matrix
variable {R : Type*} [CommRing R] [IsDomain R]
variable {K I : Type*} [Fintype K] [DecidableEq K] [Fintype I] [DecidableEq I]
theorem exists_nonzero_divisible_minor
(Z H : Matrix K I R) (G : Matrix I I R) (D W : Matrix K K R)
(C : Matrix K I R) (E : Matrix I K R)
(hG : IsUnit G) (hD : D.det ≠ 0) (hW : W.det ≠ 0)
(hZG : Z * G = D * H) (hHC : H = W * C) (hHE : W = H * E) :
∃ f : K → I, (Z.submatrix id f).det ≠ 0 ∧ W.det ∣ (Z.submatrix id f).det := by
classical
have hprod : Z * (G * E) = D * W := by
rw [← Matrix.mul_assoc, hZG, Matrix.mul_assoc, ← hHE]
have hprod0 : (Z * (G * E)).det ≠ 0 := by
rw [hprod, Matrix.det_mul]
exact mul_ne_zero hD hW
obtain ⟨f, hf⟩ := exists_minor_ne_zero_of_mul_det_ne_zero Z (G * E) hprod0
refine ⟨f, hf, ?_⟩
have hGdet : IsUnit G.det := (Matrix.isUnit_iff_isUnit_det G).mp hG
have hZ : Z = (D * W) * (C * G⁻¹) := by
calc
Z = (Z * G) * G⁻¹ := (Matrix.mul_nonsing_inv_cancel_right G Z hGdet).symm
_ = (D * W) * (C * G⁻¹) := by rw [hZG, hHC]; simp only [Matrix.mul_assoc]
have hsub : Z.submatrix id f = (D * W) * ((C * G⁻¹).submatrix id f) := by
rw [hZ]
rfl
rw [hsub, Matrix.det_mul, Matrix.det_mul]
exact dvd_mul_of_dvd_left (dvd_mul_left _ _) _
end NCHitting
end
end OAI
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