gaussian' — Mathlib

Lean4
/-- If `f` is a morphism `X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism,
then we can construct isomorphisms `L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂` and `R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂`
so that `L.hom ≫ g ≫ R.hom` is diagonal (with `X₁ ⟶ Y₁` component still `f`),
via Gaussian elimination.

(This is the version of `Biprod.gaussian` written in terms of components.)
-/
def gaussian' [IsIso f₁₁] :
    Σ' (L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂) (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂) (g₂₂ : X₂ ⟶ Y₂),
      L.hom ≫ Biprod.ofComponents f₁₁ f₁₂ f₂₁ f₂₂ ≫ R.hom = biprod.map f₁₁ g₂₂ :=
  ⟨Biprod.unipotentLower (-f₂₁ ≫ inv f₁₁), Biprod.unipotentUpper (-inv f₁₁ ≫ f₁₂), f₂₂ - f₂₁ ≫ inv f₁₁ ≫ f₁₂, by
    ext <;> simp; abel⟩
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