Module: Mathlib.CategoryTheory.Limits.Constructions.Over.Products

Goal state before:
pp: case none
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X Y✝ Z B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
Y : WidePullbackShape J
f : none ⟶ Y
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) Y =
    Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) none ≫
      WidePullbackShape.Hom.rec (motive := fun a a_1 x ↦
        none = a →
          Y = a_1 →
            f ≍ x →
              (Option.rec B (fun val ↦ (F.obj { as := val }).left) none ⟶
                Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
        (fun X h ↦
          Eq.rec (motive := fun x x_1 ↦
            Y = x →
              f ≍ 𝟙 x →
                (Option.rec B (fun val ↦ (F.obj { as := val }).left) none ⟶
                  Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
            (fun h ↦
              Eq.rec (motive := fun x x_1 ↦
                (f : none ⟶ x) →
                  f ≍ 𝟙 none →
                    (Option.rec B (fun val ↦ (F.obj { as := val }).left) none ⟶
                      Option.rec B (fun val ↦ (F.obj { as := val }).left) x))
                (fun f h ↦ 𝟙 (Option.rec B (fun val ↦ (F.obj { as := val }).left) none)) ⋯ f)
            h)
        (fun j h ↦
          Eq.rec (motive := fun x x_1 ↦
            (f : x ⟶ Y) →
              Y = none →
                f ≍ WidePullbackShape.Hom.term j →
                  (Option.rec B (fun val ↦ (F.obj { as := val }).left) x ⟶
                    Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
            (fun f h ↦
              Eq.rec (motive := fun x x_1 ↦
                (f : some j ⟶ x) →
                  f ≍ WidePullbackShape.Hom.term j →
                    ((F.obj { as := j }).left ⟶ Option.rec B (fun val ↦ (F.obj { as := val }).left) x))
                (fun f h ↦ (F.obj { as := j }).hom) ⋯ f)
            ⋯ f)
        f ⋯ ⋯ ⋯

pp: case some
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X Y✝ Z B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
Y : WidePullbackShape J
val✝ : J
f : some val✝ ⟶ Y
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) Y =
    Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) (some val✝) ≫
      WidePullbackShape.Hom.rec (motive := fun a a_1 x ↦
        some val✝ = a →
          Y = a_1 →
            f ≍ x →
              (Option.rec B (fun val ↦ (F.obj { as := val }).left) (some val✝) ⟶
                Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
        (fun X h ↦
          Eq.rec (motive := fun x x_1 ↦
            Y = x →
              f ≍ 𝟙 x →
                (Option.rec B (fun val ↦ (F.obj { as := val }).left) (some val✝) ⟶
                  Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
            (fun h ↦
              Eq.rec (motive := fun x x_1 ↦
                (f : some val✝ ⟶ x) →
                  f ≍ 𝟙 (some val✝) →
                    (Option.rec B (fun val ↦ (F.obj { as := val }).left) (some val✝) ⟶
                      Option.rec B (fun val ↦ (F.obj { as := val }).left) x))
                (fun f h ↦ 𝟙 (Option.rec B (fun val ↦ (F.obj { as := val }).left) (some val✝))) ⋯ f)
            h)
        (fun j h ↦
          Eq.rec (motive := fun x x_1 ↦
            (f : x ⟶ Y) →
              Y = none →
                f ≍ WidePullbackShape.Hom.term j →
                  (Option.rec B (fun val ↦ (F.obj { as := val }).left) x ⟶
                    Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
            (fun f h ↦
              Eq.rec (motive := fun x x_1 ↦
                (f : some j ⟶ x) →
                  f ≍ WidePullbackShape.Hom.term j →
                    ((F.obj { as := j }).left ⟶ Option.rec B (fun val ↦ (F.obj { as := val }).left) x))
                (fun f h ↦ (F.obj { as := j }).hom) ⋯ f)
            ⋯ f)
        f ⋯ ⋯ ⋯

Tactic:
<;>

Goal state after:
case none
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X Y✝ Z B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
Y : WidePullbackShape J
f : none ⟶ Y
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) Y =
    Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) none ≫
      WidePullbackShape.Hom.rec (motive := fun a a_1 x ↦
        none = a →
          Y = a_1 →
            f ≍ x →
              (Option.rec B (fun val ↦ (F.obj { as := val }).left) none ⟶
                Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
        (fun X h ↦
          Eq.rec (motive := fun x x_1 ↦
            Y = x →
              f ≍ 𝟙 x →
                (Option.rec B (fun val ↦ (F.obj { as := val }).left) none ⟶
                  Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
            (fun h ↦
              Eq.rec (motive := fun x x_1 ↦
                (f : none ⟶ x) →
                  f ≍ 𝟙 none →
                    (Option.rec B (fun val ↦ (F.obj { as := val }).left) none ⟶
                      Option.rec B (fun val ↦ (F.obj { as := val }).left) x))
                (fun f h ↦ 𝟙 (Option.rec B (fun val ↦ (F.obj { as := val }).left) none)) ⋯ f)
            h)
        (fun j h ↦
          Eq.rec (motive := fun x x_1 ↦
            (f : x ⟶ Y) →
              Y = none →
                f ≍ WidePullbackShape.Hom.term j →
                  (Option.rec B (fun val ↦ (F.obj { as := val }).left) x ⟶
                    Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
            (fun f h ↦
              Eq.rec (motive := fun x x_1 ↦
                (f : some j ⟶ x) →
                  f ≍ WidePullbackShape.Hom.term j →
                    ((F.obj { as := j }).left ⟶ Option.rec B (fun val ↦ (F.obj { as := val }).left) x))
                (fun f h ↦ (F.obj { as := j }).hom) ⋯ f)
            ⋯ f)
        f ⋯ ⋯ ⋯

case some
J✝ : Type w
C : Type u
inst✝ : Category.{v, u} C
X Y✝ Z B : C
J : Type w
F : Discrete J ⥤ Over B
c : Cone F
Y : WidePullbackShape J
val✝ : J
f : some val✝ ⟶ Y
⊢ 𝟙 c.pt.left ≫ Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) Y =
    Option.rec c.pt.hom (fun val ↦ Hom.left (c.π.app { as := val })) (some val✝) ≫
      WidePullbackShape.Hom.rec (motive := fun a a_1 x ↦
        some val✝ = a →
          Y = a_1 →
            f ≍ x →
              (Option.rec B (fun val ↦ (F.obj { as := val }).left) (some val✝) ⟶
                Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
        (fun X h ↦
          Eq.rec (motive := fun x x_1 ↦
            Y = x →
              f ≍ 𝟙 x →
                (Option.rec B (fun val ↦ (F.obj { as := val }).left) (some val✝) ⟶
                  Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
            (fun h ↦
              Eq.rec (motive := fun x x_1 ↦
                (f : some val✝ ⟶ x) →
                  f ≍ 𝟙 (some val✝) →
                    (Option.rec B (fun val ↦ (F.obj { as := val }).left) (some val✝) ⟶
                      Option.rec B (fun val ↦ (F.obj { as := val }).left) x))
                (fun f h ↦ 𝟙 (Option.rec B (fun val ↦ (F.obj { as := val }).left) (some val✝))) ⋯ f)
            h)
        (fun j h ↦
          Eq.rec (motive := fun x x_1 ↦
            (f : x ⟶ Y) →
              Y = none →
                f ≍ WidePullbackShape.Hom.term j →
                  (Option.rec B (fun val ↦ (F.obj { as := val }).left) x ⟶
                    Option.rec B (fun val ↦ (F.obj { as := val }).left) Y))
            (fun f h ↦
              Eq.rec (motive := fun x x_1 ↦
                (f : some j ⟶ x) →
                  f ≍ WidePullbackShape.Hom.term j →
                    ((F.obj { as := j }).left ⟶ Option.rec B (fun val ↦ (F.obj { as := val }).left) x))
                (fun f h ↦ (F.obj { as := j }).hom) ⋯ f)
            ⋯ f)
        f ⋯ ⋯ ⋯