Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.Cospan

Goal state before:
pp: case none.none.id
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X Y Z X' Y' Z' : C
iX : X ≅ X'
iY : Y ≅ Y'
iZ : Z ≅ Z'
F G : WalkingSpan ⥤ C
z : F.obj WalkingSpan.zero ≅ G.obj WalkingSpan.zero
l : F.obj WalkingSpan.left ≅ G.obj WalkingSpan.left
r : F.obj WalkingSpan.right ≅ G.obj WalkingSpan.right
hl : F.map fst ≫ l.hom = z.hom ≫ G.map fst
hr : F.map snd ≫ r.hom = z.hom ≫ G.map snd
⊢ F.map (WidePushoutShape.Hom.id none) ≫ (Option.casesOn none z fun val ↦ WalkingPair.casesOn val l r).hom =
    (Option.casesOn none z fun val ↦ WalkingPair.casesOn val l r).hom ≫ G.map (WidePushoutShape.Hom.id none)

pp: case none.some.left.init
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X Y Z X' Y' Z' : C
iX : X ≅ X'
iY : Y ≅ Y'
iZ : Z ≅ Z'
F G : WalkingSpan ⥤ C
z : F.obj WalkingSpan.zero ≅ G.obj WalkingSpan.zero
l : F.obj WalkingSpan.left ≅ G.obj WalkingSpan.left
r : F.obj WalkingSpan.right ≅ G.obj WalkingSpan.right
hl : F.map fst ≫ l.hom = z.hom ≫ G.map fst
hr : F.map snd ≫ r.hom = z.hom ≫ G.map snd
⊢ F.map (init WalkingPair.left) ≫ (Option.casesOn (some WalkingPair.left) z fun val ↦ WalkingPair.casesOn val l r).hom =
    (Option.casesOn none z fun val ↦ WalkingPair.casesOn val l r).hom ≫ G.map (init WalkingPair.left)

pp: case none.some.right.init
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X Y Z X' Y' Z' : C
iX : X ≅ X'
iY : Y ≅ Y'
iZ : Z ≅ Z'
F G : WalkingSpan ⥤ C
z : F.obj WalkingSpan.zero ≅ G.obj WalkingSpan.zero
l : F.obj WalkingSpan.left ≅ G.obj WalkingSpan.left
r : F.obj WalkingSpan.right ≅ G.obj WalkingSpan.right
hl : F.map fst ≫ l.hom = z.hom ≫ G.map fst
hr : F.map snd ≫ r.hom = z.hom ≫ G.map snd
⊢ F.map (init WalkingPair.right) ≫
      (Option.casesOn (some WalkingPair.right) z fun val ↦ WalkingPair.casesOn val l r).hom =
    (Option.casesOn none z fun val ↦ WalkingPair.casesOn val l r).hom ≫ G.map (init WalkingPair.right)

pp: case some.left.some.left.id
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X Y Z X' Y' Z' : C
iX : X ≅ X'
iY : Y ≅ Y'
iZ : Z ≅ Z'
F G : WalkingSpan ⥤ C
z : F.obj WalkingSpan.zero ≅ G.obj WalkingSpan.zero
l : F.obj WalkingSpan.left ≅ G.obj WalkingSpan.left
r : F.obj WalkingSpan.right ≅ G.obj WalkingSpan.right
hl : F.map fst ≫ l.hom = z.hom ≫ G.map fst
hr : F.map snd ≫ r.hom = z.hom ≫ G.map snd
⊢ F.map (WidePushoutShape.Hom.id (some WalkingPair.left)) ≫
      (Option.casesOn (some WalkingPair.left) z fun val ↦ WalkingPair.casesOn val l r).hom =
    (Option.casesOn (some WalkingPair.left) z fun val ↦ WalkingPair.casesOn val l r).hom ≫
      G.map (WidePushoutShape.Hom.id (some WalkingPair.left))

pp: case some.right.some.right.id
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X Y Z X' Y' Z' : C
iX : X ≅ X'
iY : Y ≅ Y'
iZ : Z ≅ Z'
F G : WalkingSpan ⥤ C
z : F.obj WalkingSpan.zero ≅ G.obj WalkingSpan.zero
l : F.obj WalkingSpan.left ≅ G.obj WalkingSpan.left
r : F.obj WalkingSpan.right ≅ G.obj WalkingSpan.right
hl : F.map fst ≫ l.hom = z.hom ≫ G.map fst
hr : F.map snd ≫ r.hom = z.hom ≫ G.map snd
⊢ F.map (WidePushoutShape.Hom.id (some WalkingPair.right)) ≫
      (Option.casesOn (some WalkingPair.right) z fun val ↦ WalkingPair.casesOn val l r).hom =
    (Option.casesOn (some WalkingPair.right) z fun val ↦ WalkingPair.casesOn val l r).hom ≫
      G.map (WidePushoutShape.Hom.id (some WalkingPair.right))

Tactic:
;

Goal state after:
case none.none.id
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X Y Z X' Y' Z' : C
iX : X ≅ X'
iY : Y ≅ Y'
iZ : Z ≅ Z'
F G : WalkingSpan ⥤ C
z : F.obj WalkingSpan.zero ≅ G.obj WalkingSpan.zero
l : F.obj WalkingSpan.left ≅ G.obj WalkingSpan.left
r : F.obj WalkingSpan.right ≅ G.obj WalkingSpan.right
hl : F.map fst ≫ l.hom = z.hom ≫ G.map fst
hr : F.map snd ≫ r.hom = z.hom ≫ G.map snd
⊢ F.map (WidePushoutShape.Hom.id none) ≫ (Option.casesOn none z fun val ↦ WalkingPair.casesOn val l r).hom =
    (Option.casesOn none z fun val ↦ WalkingPair.casesOn val l r).hom ≫ G.map (WidePushoutShape.Hom.id none)

case none.some.left.init
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X Y Z X' Y' Z' : C
iX : X ≅ X'
iY : Y ≅ Y'
iZ : Z ≅ Z'
F G : WalkingSpan ⥤ C
z : F.obj WalkingSpan.zero ≅ G.obj WalkingSpan.zero
l : F.obj WalkingSpan.left ≅ G.obj WalkingSpan.left
r : F.obj WalkingSpan.right ≅ G.obj WalkingSpan.right
hl : F.map fst ≫ l.hom = z.hom ≫ G.map fst
hr : F.map snd ≫ r.hom = z.hom ≫ G.map snd
⊢ F.map (init WalkingPair.left) ≫ (Option.casesOn (some WalkingPair.left) z fun val ↦ WalkingPair.casesOn val l r).hom =
    (Option.casesOn none z fun val ↦ WalkingPair.casesOn val l r).hom ≫ G.map (init WalkingPair.left)

case none.some.right.init
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X Y Z X' Y' Z' : C
iX : X ≅ X'
iY : Y ≅ Y'
iZ : Z ≅ Z'
F G : WalkingSpan ⥤ C
z : F.obj WalkingSpan.zero ≅ G.obj WalkingSpan.zero
l : F.obj WalkingSpan.left ≅ G.obj WalkingSpan.left
r : F.obj WalkingSpan.right ≅ G.obj WalkingSpan.right
hl : F.map fst ≫ l.hom = z.hom ≫ G.map fst
hr : F.map snd ≫ r.hom = z.hom ≫ G.map snd
⊢ F.map (init WalkingPair.right) ≫
      (Option.casesOn (some WalkingPair.right) z fun val ↦ WalkingPair.casesOn val l r).hom =
    (Option.casesOn none z fun val ↦ WalkingPair.casesOn val l r).hom ≫ G.map (init WalkingPair.right)

case some.left.some.left.id
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X Y Z X' Y' Z' : C
iX : X ≅ X'
iY : Y ≅ Y'
iZ : Z ≅ Z'
F G : WalkingSpan ⥤ C
z : F.obj WalkingSpan.zero ≅ G.obj WalkingSpan.zero
l : F.obj WalkingSpan.left ≅ G.obj WalkingSpan.left
r : F.obj WalkingSpan.right ≅ G.obj WalkingSpan.right
hl : F.map fst ≫ l.hom = z.hom ≫ G.map fst
hr : F.map snd ≫ r.hom = z.hom ≫ G.map snd
⊢ F.map (WidePushoutShape.Hom.id (some WalkingPair.left)) ≫
      (Option.casesOn (some WalkingPair.left) z fun val ↦ WalkingPair.casesOn val l r).hom =
    (Option.casesOn (some WalkingPair.left) z fun val ↦ WalkingPair.casesOn val l r).hom ≫
      G.map (WidePushoutShape.Hom.id (some WalkingPair.left))

case some.right.some.right.id
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X Y Z X' Y' Z' : C
iX : X ≅ X'
iY : Y ≅ Y'
iZ : Z ≅ Z'
F G : WalkingSpan ⥤ C
z : F.obj WalkingSpan.zero ≅ G.obj WalkingSpan.zero
l : F.obj WalkingSpan.left ≅ G.obj WalkingSpan.left
r : F.obj WalkingSpan.right ≅ G.obj WalkingSpan.right
hl : F.map fst ≫ l.hom = z.hom ≫ G.map fst
hr : F.map snd ≫ r.hom = z.hom ≫ G.map snd
⊢ F.map (WidePushoutShape.Hom.id (some WalkingPair.right)) ≫
      (Option.casesOn (some WalkingPair.right) z fun val ↦ WalkingPair.casesOn val l r).hom =
    (Option.casesOn (some WalkingPair.right) z fun val ↦ WalkingPair.casesOn val l r).hom ≫
      G.map (WidePushoutShape.Hom.id (some WalkingPair.right))