Module: Mathlib.Topology.Sheaves.Flasque

Goal state before:
pp: X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
s : ↑(S.X₃.obj.obj (op U))
t : Under S.g s
ht : ∀ (a : Under S.g s), Nonempty (a ⟶ t) → Nonempty (t ⟶ a)
tle : unop (StructuredArrow.right t).fst ≤ U
tcomp :
  (s |_ unop (StructuredArrow.right t).fst) tle =
    (ConcreteCategory.hom (S.g.hom.app (StructuredArrow.right t).fst)) (StructuredArrow.right t).snd
x : ↑X
hx : x ∈ U
this✝¹ : Presheaf.IsLocallySurjective S.g.hom
W : Opens ↑X
Wle : W ≤ U
hW : x ∈ W
t₁ : ↑(S.X₂.obj.obj (op W))
ht₁ : (ConcreteCategory.hom (S.g.hom.app (op W))) t₁ = (s |_ W) Wle
t₂ : ToType (S.X₂.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W))) :=
  ((StructuredArrow.right t).snd |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ -
    (t₁ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
this✝ : (ConcreteCategory.hom (S.g.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₂ = 0
t₃ : ↑(S.X₁.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W)))
ht₃ : (ConcreteCategory.hom (S.f.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₃ = t₂
t₄ : ↑(S.X₁.obj.obj (op W))
ht₄ : (t₄ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = t₃
f : Fin 2 → Opens ↑X := ![unop (StructuredArrow.right t).fst, W]
sf : (i : Fin 2) → ↑(S.X₂.obj.obj (op (f i))) :=
  fun x ↦
    match x with
    | 0 => (StructuredArrow.right t).snd
    | 1 => t₁ + (ConcreteCategory.hom (S.f.hom.app (op W))) t₄
this : (sf 0 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = (sf 1 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
t₅ : ToType (S.X₂.obj.obj (op (iSup f)))
ht₅ : IsGluing S.X₂.obj f sf t₅
right✝ : ∀ (y : ToType (S.X₂.obj.obj (op (iSup f)))), (fun t₅ ↦ IsGluing S.X₂.obj f sf t₅) y → y = t₅
le : iSup f ≤ U
⊢ (ConcreteCategory.hom (S.g.hom.app (op (![unop (StructuredArrow.right t).fst, W] 1)))) t₁ +
      (ConcreteCategory.hom (S.g.hom.app (op (![unop (StructuredArrow.right t).fst, W] 1))))
        ((ConcreteCategory.hom (S.f.hom.app (op W))) t₄) =
    (ConcreteCategory.hom (S.X₃.obj.map (homOfLE ⋯).op)) s

Tactic:
have : (S.f.hom.app (op W) ≫ S.g.hom.app (op W)) = 0 := by
            rw [← NatTrans.comp_app, ← ObjectProperty.FullSubcategory.comp_hom, S.zero]
            rfl

Goal state after:
X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
s : ↑(S.X₃.obj.obj (op U))
t : Under S.g s
ht : ∀ (a : Under S.g s), Nonempty (a ⟶ t) → Nonempty (t ⟶ a)
tle : unop (StructuredArrow.right t).fst ≤ U
tcomp :
  (s |_ unop (StructuredArrow.right t).fst) tle =
    (ConcreteCategory.hom (S.g.hom.app (StructuredArrow.right t).fst)) (StructuredArrow.right t).snd
x : ↑X
hx : x ∈ U
this✝² : Presheaf.IsLocallySurjective S.g.hom
W : Opens ↑X
Wle : W ≤ U
hW : x ∈ W
t₁ : ↑(S.X₂.obj.obj (op W))
ht₁ : (ConcreteCategory.hom (S.g.hom.app (op W))) t₁ = (s |_ W) Wle
t₂ : ToType (S.X₂.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W))) :=
  ((StructuredArrow.right t).snd |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ -
    (t₁ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
this✝¹ : (ConcreteCategory.hom (S.g.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₂ = 0
t₃ : ↑(S.X₁.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W)))
ht₃ : (ConcreteCategory.hom (S.f.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₃ = t₂
t₄ : ↑(S.X₁.obj.obj (op W))
ht₄ : (t₄ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = t₃
f : Fin 2 → Opens ↑X := ![unop (StructuredArrow.right t).fst, W]
sf : (i : Fin 2) → ↑(S.X₂.obj.obj (op (f i))) :=
  fun x ↦
    match x with
    | 0 => (StructuredArrow.right t).snd
    | 1 => t₁ + (ConcreteCategory.hom (S.f.hom.app (op W))) t₄
this✝ : (sf 0 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = (sf 1 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
t₅ : ToType (S.X₂.obj.obj (op (iSup f)))
ht₅ : IsGluing S.X₂.obj f sf t₅
right✝ : ∀ (y : ToType (S.X₂.obj.obj (op (iSup f)))), (fun t₅ ↦ IsGluing S.X₂.obj f sf t₅) y → y = t₅
le : iSup f ≤ U
this : S.f.hom.app (op W) ≫ S.g.hom.app (op W) = 0
⊢ (ConcreteCategory.hom (S.g.hom.app (op (![unop (StructuredArrow.right t).fst, W] 1)))) t₁ +
      (ConcreteCategory.hom (S.g.hom.app (op (![unop (StructuredArrow.right t).fst, W] 1))))
        ((ConcreteCategory.hom (S.f.hom.app (op W))) t₄) =
    (ConcreteCategory.hom (S.X₃.obj.map (homOfLE ⋯).op)) s