Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Pullbacks

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
⊢ ∀ {X_1 Y_1 : WalkingSpan} (f_1 : X_1 ⟶ Y_1),
    (span f.op g.op).map f_1 ≫
        (match Y_1 with
          | none => Iso.refl ((span f.op g.op).obj none)
          | some WalkingPair.left => Iso.refl ((span f.op g.op).obj WalkingSpan.left)
          | some WalkingPair.right => Iso.refl ((span f.op g.op).obj WalkingSpan.right)).hom =
      (match X_1 with
          | none => Iso.refl ((span f.op g.op).obj none)
          | some WalkingPair.left => Iso.refl ((span f.op g.op).obj WalkingSpan.left)
          | some WalkingPair.right => Iso.refl ((span f.op g.op).obj WalkingSpan.right)).hom ≫
        (walkingCospanOpEquiv.inverse ⋙ (cospan f g).op).map f_1

Tactic:
rintro (_ | _ | _) (_ | _ | _) f

Goal state after:
case none.none
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : C
f✝ : X ⟶ Z
g : Y ⟶ Z
f : none ⟶ none
⊢ (span f✝.op g.op).map f ≫
      (match none with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom =
    (match none with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom ≫
      (walkingCospanOpEquiv.inverse ⋙ (cospan f✝ g).op).map f

case none.some.left
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : C
f✝ : X ⟶ Z
g : Y ⟶ Z
f : none ⟶ some WalkingPair.left
⊢ (span f✝.op g.op).map f ≫
      (match some WalkingPair.left with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom =
    (match none with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom ≫
      (walkingCospanOpEquiv.inverse ⋙ (cospan f✝ g).op).map f

case none.some.right
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : C
f✝ : X ⟶ Z
g : Y ⟶ Z
f : none ⟶ some WalkingPair.right
⊢ (span f✝.op g.op).map f ≫
      (match some WalkingPair.right with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom =
    (match none with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom ≫
      (walkingCospanOpEquiv.inverse ⋙ (cospan f✝ g).op).map f

case some.left.none
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : C
f✝ : X ⟶ Z
g : Y ⟶ Z
f : some WalkingPair.left ⟶ none
⊢ (span f✝.op g.op).map f ≫
      (match none with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom =
    (match some WalkingPair.left with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom ≫
      (walkingCospanOpEquiv.inverse ⋙ (cospan f✝ g).op).map f

case some.left.some.left
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : C
f✝ : X ⟶ Z
g : Y ⟶ Z
f : some WalkingPair.left ⟶ some WalkingPair.left
⊢ (span f✝.op g.op).map f ≫
      (match some WalkingPair.left with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom =
    (match some WalkingPair.left with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom ≫
      (walkingCospanOpEquiv.inverse ⋙ (cospan f✝ g).op).map f

case some.left.some.right
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : C
f✝ : X ⟶ Z
g : Y ⟶ Z
f : some WalkingPair.left ⟶ some WalkingPair.right
⊢ (span f✝.op g.op).map f ≫
      (match some WalkingPair.right with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom =
    (match some WalkingPair.left with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom ≫
      (walkingCospanOpEquiv.inverse ⋙ (cospan f✝ g).op).map f

case some.right.none
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : C
f✝ : X ⟶ Z
g : Y ⟶ Z
f : some WalkingPair.right ⟶ none
⊢ (span f✝.op g.op).map f ≫
      (match none with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom =
    (match some WalkingPair.right with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom ≫
      (walkingCospanOpEquiv.inverse ⋙ (cospan f✝ g).op).map f

case some.right.some.left
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : C
f✝ : X ⟶ Z
g : Y ⟶ Z
f : some WalkingPair.right ⟶ some WalkingPair.left
⊢ (span f✝.op g.op).map f ≫
      (match some WalkingPair.left with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom =
    (match some WalkingPair.right with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom ≫
      (walkingCospanOpEquiv.inverse ⋙ (cospan f✝ g).op).map f

case some.right.some.right
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
J : Type u₂
inst✝ : Category.{v₂, u₂} J
X Y Z : C
f✝ : X ⟶ Z
g : Y ⟶ Z
f : some WalkingPair.right ⟶ some WalkingPair.right
⊢ (span f✝.op g.op).map f ≫
      (match some WalkingPair.right with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom =
    (match some WalkingPair.right with
        | none => Iso.refl ((span f✝.op g.op).obj none)
        | some WalkingPair.left => Iso.refl ((span f✝.op g.op).obj WalkingSpan.left)
        | some WalkingPair.right => Iso.refl ((span f✝.op g.op).obj WalkingSpan.right)).hom ≫
      (walkingCospanOpEquiv.inverse ⋙ (cospan f✝ g).op).map f