| 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 |
|
|
