mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Limits.Final.Type

Goal state before:
pp: case refine_1
C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
P : D ⥤ Type w
inst✝ : F.Initial
s₁ s₂ : ↑P.sections
h : F.sectionsPrecomp s₁ = F.sectionsPrecomp s₂
Y : D
X : CostructuredArrow F Y := Classical.arbitrary (CostructuredArrow F Y)
this : ↑s₁ (F.obj X.left) = ↑s₂ (F.obj X.left)
h₁ : (ConcreteCategory.hom (P.map X.hom)) (↑s₁ (F.obj X.left)) = ↑s₁ Y
h₂ : (ConcreteCategory.hom (P.map X.hom)) (↑s₂ (F.obj X.left)) = ↑s₂ Y
⊢ ↑s₁ Y = ↑s₂ Y

Tactic:
← h₁,

Goal state after:
case refine_1
C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
P : D ⥤ Type w
inst✝ : F.Initial
s₁ s₂ : ↑P.sections
h : F.sectionsPrecomp s₁ = F.sectionsPrecomp s₂
Y : D
X : CostructuredArrow F Y := Classical.arbitrary (CostructuredArrow F Y)
this : ↑s₁ (F.obj X.left) = ↑s₂ (F.obj X.left)
h₁ : (ConcreteCategory.hom (P.map X.hom)) (↑s₁ (F.obj X.left)) = ↑s₁ Y
h₂ : (ConcreteCategory.hom (P.map X.hom)) (↑s₂ (F.obj X.left)) = ↑s₂ Y
⊢ (ConcreteCategory.hom (P.map X.hom)) (↑s₁ (F.obj X.left)) = ↑s₂ Y
Module: Mathlib.CategoryTheory.Simple

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
h : ∀ {Z : C} (f : X ⟶ Z) [Epi f], IsIso f ↔ f ≠ 0
Y : C
f : Y ⟶ X
I : Mono f
⊢ IsIso f ↔ f ≠ 0

Tactic:
by
    fconstructor
    · intros
      have hx := cokernel.π_of_epi f
      by_contra h
      subst h
      exact (h _).mp inferInstance hx
    · intro hf
      suffices Epi f by exact isIso_of_mono_of_epi _
      apply Preadditive.epi_of_cokernel_zero
      by_contra h'
      exact cokernel_not_iso_of_nonzero hf ((h _).mpr h')

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.single.left.single.left
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
g : Pairwise.single { down := WalkingPair.left } ⟶ Pairwise.single { down := WalkingPair.left }
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map g.op ≫
      Opposite.casesOn (op (Pairwise.single { down := WalkingPair.left })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.single { down := WalkingPair.left })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map g.op

Tactic:
rcases g with ⟨⟩

Goal state after:
case refine_2.single.left.single.left.id_single
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map
        (Quiver.Hom.op (Pairwise.Hom.id_single { down := WalkingPair.left })) ≫
      Opposite.casesOn (op (Pairwise.single { down := WalkingPair.left })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.single { down := WalkingPair.left })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map (Quiver.Hom.op (Pairwise.Hom.id_single { down := WalkingPair.left }))
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.single.left.single.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
g : Pairwise.single { down := WalkingPair.right } ⟶ Pairwise.single { down := WalkingPair.left }
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map g.op ≫
      Opposite.casesOn (op (Pairwise.single { down := WalkingPair.right })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.single { down := WalkingPair.left })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map g.op

Tactic:
rcases g with ⟨⟩

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.single.left.pair.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
g : Pairwise.pair { down := WalkingPair.right } { down := down✝ } ⟶ Pairwise.single { down := WalkingPair.left }
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map g.op ≫
      Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.right } { down := down✝ })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.single { down := WalkingPair.left })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map g.op

Tactic:
rcases g with ⟨⟩

Goal state after:
case refine_2.single.left.pair.right.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map
        (Quiver.Hom.op (Pairwise.Hom.right { down := WalkingPair.right } { down := WalkingPair.left })) ≫
      Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.right } { down := WalkingPair.left })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.single { down := WalkingPair.left })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map
        (Quiver.Hom.op (Pairwise.Hom.right { down := WalkingPair.right } { down := WalkingPair.left }))
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.single.right.single.left
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
g : Pairwise.single { down := WalkingPair.left } ⟶ Pairwise.single { down := WalkingPair.right }
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map g.op ≫
      Opposite.casesOn (op (Pairwise.single { down := WalkingPair.left })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.single { down := WalkingPair.right })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map g.op

Tactic:
rcases g with ⟨⟩

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.single.right.single.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
g : Pairwise.single { down := WalkingPair.right } ⟶ Pairwise.single { down := WalkingPair.right }
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map g.op ≫
      Opposite.casesOn (op (Pairwise.single { down := WalkingPair.right })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.single { down := WalkingPair.right })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map g.op

Tactic:
rcases g with ⟨⟩

Goal state after:
case refine_2.single.right.single.right.id_single
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map
        (Quiver.Hom.op (Pairwise.Hom.id_single { down := WalkingPair.right })) ≫
      Opposite.casesOn (op (Pairwise.single { down := WalkingPair.right })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.single { down := WalkingPair.right })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map (Quiver.Hom.op (Pairwise.Hom.id_single { down := WalkingPair.right }))
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.single.right.pair.left
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
g : Pairwise.pair { down := WalkingPair.left } { down := down✝ } ⟶ Pairwise.single { down := WalkingPair.right }
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map g.op ≫
      Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.left } { down := down✝ })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.single { down := WalkingPair.right })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map g.op

Tactic:
rcases g with ⟨⟩

Goal state after:
case refine_2.single.right.pair.left.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map
        (Quiver.Hom.op (Pairwise.Hom.right { down := WalkingPair.left } { down := WalkingPair.right })) ≫
      Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.left } { down := WalkingPair.right })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.single { down := WalkingPair.right })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map
        (Quiver.Hom.op (Pairwise.Hom.right { down := WalkingPair.left } { down := WalkingPair.right }))
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.pair.left.single.left
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
g : Pairwise.single { down := WalkingPair.left } ⟶ Pairwise.pair { down := WalkingPair.left } { down := down✝ }
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map g.op ≫
      Opposite.casesOn (op (Pairwise.single { down := WalkingPair.left })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.left } { down := down✝ })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map g.op

Tactic:
rcases g with ⟨⟩

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.pair.left.single.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
g : Pairwise.single { down := WalkingPair.right } ⟶ Pairwise.pair { down := WalkingPair.left } { down := down✝ }
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map g.op ≫
      Opposite.casesOn (op (Pairwise.single { down := WalkingPair.right })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.left } { down := down✝ })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map g.op

Tactic:
rcases g with ⟨⟩

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.pair.left.pair.left
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝¹ down✝ : WalkingPair
g :
  Pairwise.pair { down := WalkingPair.left } { down := down✝ } ⟶
    Pairwise.pair { down := WalkingPair.left } { down := down✝¹ }
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map g.op ≫
      Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.left } { down := down✝ })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.left } { down := down✝¹ })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map g.op

Tactic:
rcases g with ⟨⟩

Goal state after:
case refine_2.pair.left.pair.left.id_pair
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map
        (Quiver.Hom.op (Pairwise.Hom.id_pair { down := WalkingPair.left } { down := down✝ })) ≫
      Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.left } { down := down✝ })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.left } { down := down✝ })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map
        (Quiver.Hom.op (Pairwise.Hom.id_pair { down := WalkingPair.left } { down := down✝ }))
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.pair.left.pair.right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝¹ down✝ : WalkingPair
g :
  Pairwise.pair { down := WalkingPair.right } { down := down✝ } ⟶
    Pairwise.pair { down := WalkingPair.left } { down := down✝¹ }
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map g.op ≫
      Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.right } { down := down✝ })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.left } { down := down✝¹ })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map g.op

Tactic:
rcases g with ⟨⟩

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_2.pair.right.single.left
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : TopCat
F : Sheaf C X
U V : Opens ↑X
s : PullbackCone (F.obj.map (homOfLE ⋯).op) (F.obj.map (homOfLE ⋯).op)
ι : ULift.{w, 0} WalkingPair → Opens ↑X := fun j ↦ WalkingPair.casesOn j.down U V
hι : U ⊔ V = iSup ι
down✝ : WalkingPair
g : Pairwise.single { down := WalkingPair.left } ⟶ Pairwise.pair { down := WalkingPair.right } { down := down✝ }
⊢ (((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map g.op ≫
      Opposite.casesOn (op (Pairwise.single { down := WalkingPair.left })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) =
    (Opposite.casesOn (op (Pairwise.pair { down := WalkingPair.right } { down := down✝ })) fun unop ↦
        Pairwise.casesOn unop (fun a ↦ ULift.casesOn a fun down ↦ WalkingPair.casesOn down s.fst s.snd) fun a a_1 ↦
          ULift.casesOn a fun down ↦
            WalkingPair.casesOn down (s.fst ≫ F.obj.map (homOfLE ⋯).op) (s.snd ≫ F.obj.map (homOfLE ⋯).op)) ≫
      ((Pairwise.diagram ι).op ⋙ F.presheaf).map g.op

Tactic:
rcases g with ⟨⟩

Goal state after: