mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case mpr
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 := ⋯
x✝ : ↑X
⊢ (∃ i, x✝ ∈ ↑(WalkingPair.casesOn i.down U V)) → x✝ ∈ ↑(U ⊔ V)

Tactic:
·

Goal state after:
case mpr
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 := ⋯
x✝ : ↑X
⊢ (∃ i, x✝ ∈ ↑(WalkingPair.casesOn i.down U V)) → x✝ ∈ ↑(U ⊔ V)
Module: Mathlib.Topology.Sheaves.SheafCondition.PairwiseIntersections

Goal state before:
pp: case refine_1
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
⊢ (X_1 : (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) →
    ((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).obj X_1 ⟶
      ((Pairwise.diagram ι).op ⋙ F.presheaf).obj X_1

pp: case refine_2
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
⊢ ∀ ⦃X_1 Y : (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ⦄ (f : X_1 ⟶ Y),
    ((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).map f ≫ ?refine_1 Y =
      ?refine_1 X_1 ≫ ((Pairwise.diagram ι).op ⋙ F.presheaf).map f

Tactic:
·

Goal state after:
case refine_1
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
⊢ (X_1 : (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ) →
    ((Functor.const (CategoryTheory.Pairwise (ULift.{w, 0} WalkingPair))ᵒᵖ).obj s.pt).obj X_1 ⟶
      ((Pairwise.diagram ι).op ⋙ F.presheaf).obj X_1
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: case iUV
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ (i : Bool) → (if i = true then U₁ else U₂) ⟶ V

pp: case hcover
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ V ≤ ⨆ t, if t = true then U₁ else U₂

pp: case h
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ ∀ (i : Bool),
    (ConcreteCategory.hom (F.obj.map (Quiver.Hom.op (?iUV i)))) s =
      (ConcreteCategory.hom (F.obj.map (Quiver.Hom.op (?iUV i)))) t

Tactic:
·

Goal state after:
case iUV
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ (i : Bool) → (if i = true then U₁ else U₂) ⟶ V
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasColimits C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst✝ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U : (OpenNhds y)ᵒᵖ
⊢ (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F).obj U ⟶
    ((Functor.const (OpenNhds y)ᵒᵖ).obj ((stalkFunctor C x).1 F)).obj U

pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasColimits C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst✝ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : ↑X
h : x ⤳ y
⊢ ∀ ⦃X_1 Y : (OpenNhds y)ᵒᵖ⦄ (f : X_1 ⟶ Y),
    (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F).map f ≫ ?refine_1 =
      ?refine_1 ≫ ((Functor.const (OpenNhds y)ᵒᵖ).obj ((stalkFunctor C x).1 F)).map f

Tactic:
·

Goal state after:
case refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasColimits C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst✝ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : ↑X
h : x ⤳ y
U : (OpenNhds y)ᵒᵖ
⊢ (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F).obj U ⟶
    ((Functor.const (OpenNhds y)ᵒᵖ).obj ((stalkFunctor C x).1 F)).obj U
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasColimits C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst✝ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : ↑X
h : x ⤳ y
⊢ ∀ ⦃X_1 Y : (OpenNhds y)ᵒᵖ⦄ (f : X_1 ⟶ Y),
    (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F).map f ≫
        colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op ⟨↑(unop Y), ⋯⟩) =
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op ⟨↑(unop X_1), ⋯⟩) ≫
        ((Functor.const (OpenNhds y)ᵒᵖ).obj ((stalkFunctor C x).1 F)).map f

Tactic:
·

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasColimits C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst✝ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
F : Presheaf C X
x y : ↑X
h : x ⤳ y
⊢ ∀ ⦃X_1 Y : (OpenNhds y)ᵒᵖ⦄ (f : X_1 ⟶ Y),
    (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F).map f ≫
        colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op ⟨↑(unop Y), ⋯⟩) =
      colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op ⟨↑(unop X_1), ⋯⟩) ≫
        ((Functor.const (OpenNhds y)ᵒᵖ).obj ((stalkFunctor C x).1 F)).map f
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: case hcover
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ V ≤ ⨆ t, if t = true then U₁ else U₂

pp: case h
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ ∀ (i : Bool),
    (ConcreteCategory.hom (F.obj.map (if h : i = true then eqToHom ⋯ ≫ i₁ else eqToHom ⋯ ≫ i₂).op)) s =
      (ConcreteCategory.hom (F.obj.map (if h : i = true then eqToHom ⋯ ≫ i₁ else eqToHom ⋯ ≫ i₂).op)) t

Tactic:
·

Goal state after:
case hcover
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ V ≤ ⨆ t, if t = true then U₁ else U₂
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: case hcover.left
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ U₁ ≤ ⨆ t, if t = true then U₁ else U₂

pp: case hcover.right
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ U₂ ≤ ⨆ t, if t = true then U₁ else U₂

Tactic:
·

Goal state after:
case hcover.left
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ U₁ ≤ ⨆ t, if t = true then U₁ else U₂
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: case hcover.right
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ U₂ ≤ ⨆ t, if t = true then U₁ else U₂

Tactic:
·

Goal state after:
case hcover.right
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ U₂ ≤ ⨆ t, if t = true then U₁ else U₂
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: case h
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ ∀ (i : Bool),
    (ConcreteCategory.hom (F.obj.map (if h : i = true then eqToHom ⋯ ≫ i₁ else eqToHom ⋯ ≫ i₂).op)) s =
      (ConcreteCategory.hom (F.obj.map (if h : i = true then eqToHom ⋯ ≫ i₁ else eqToHom ⋯ ≫ i₂).op)) t

Tactic:
·

Goal state after:
case h
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory C FC
inst✝² : HasLimitsOfSize.{x, x, v_1, u_1} C
inst✝¹ : (CategoryTheory.forget C).ReflectsIsomorphisms
inst✝ : PreservesLimitsOfSize.{x, x, v_1, u_3, u_1, u_3 + 1} (CategoryTheory.forget C)
X : TopCat
F : Sheaf C X
U₁ U₂ V : Opens ↑X
i₁ : U₁ ⟶ V
i₂ : U₂ ⟶ V
hcover : V ≤ U₁ ⊔ U₂
s t : ToType (F.obj.obj (op V))
h₁ : (ConcreteCategory.hom (F.obj.map i₁.op)) s = (ConcreteCategory.hom (F.obj.map i₁.op)) t
h₂ : (ConcreteCategory.hom (F.obj.map i₂.op)) s = (ConcreteCategory.hom (F.obj.map i₂.op)) t
⊢ ∀ (i : Bool),
    (ConcreteCategory.hom (F.obj.map (if h : i = true then eqToHom ⋯ ≫ i₁ else eqToHom ⋯ ≫ i₂).op)) s =
      (ConcreteCategory.hom (F.obj.map (if h : i = true then eqToHom ⋯ ≫ i₁ else eqToHom ⋯ ≫ i₂).op)) t
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: case hcover
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).ReflectsIsomorphisms
F : Sheaf C X
U : Opens ↑X
s t : ToType (F.obj.obj (op U))
h :
  ∀ (x : ↑X) (hx : x ∈ U),
    (ConcreteCategory.hom (F.presheaf.germ U x hx)) s = (ConcreteCategory.hom (F.presheaf.germ U x hx)) t
V : ↥U → Opens ↑X
m : ∀ (x : ↥U), ↑x ∈ V x
i₁ i₂ : (x : ↥U) → V x ⟶ U
heq :
  ∀ (x : ↥U), (ConcreteCategory.hom (F.presheaf.map (i₁ x).op)) s = (ConcreteCategory.hom (F.presheaf.map (i₂ x).op)) t
⊢ U ≤ iSup V

pp: case h
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).ReflectsIsomorphisms
F : Sheaf C X
U : Opens ↑X
s t : ToType (F.obj.obj (op U))
h :
  ∀ (x : ↑X) (hx : x ∈ U),
    (ConcreteCategory.hom (F.presheaf.germ U x hx)) s = (ConcreteCategory.hom (F.presheaf.germ U x hx)) t
V : ↥U → Opens ↑X
m : ∀ (x : ↥U), ↑x ∈ V x
i₁ i₂ : (x : ↥U) → V x ⟶ U
heq :
  ∀ (x : ↥U), (ConcreteCategory.hom (F.presheaf.map (i₁ x).op)) s = (ConcreteCategory.hom (F.presheaf.map (i₂ x).op)) t
⊢ ∀ (i : ↥U), (ConcreteCategory.hom (F.obj.map (i₁ i).op)) s = (ConcreteCategory.hom (F.obj.map (i₁ i).op)) t

Tactic:
·

Goal state after:
case hcover
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).ReflectsIsomorphisms
F : Sheaf C X
U : Opens ↑X
s t : ToType (F.obj.obj (op U))
h :
  ∀ (x : ↑X) (hx : x ∈ U),
    (ConcreteCategory.hom (F.presheaf.germ U x hx)) s = (ConcreteCategory.hom (F.presheaf.germ U x hx)) t
V : ↥U → Opens ↑X
m : ∀ (x : ↥U), ↑x ∈ V x
i₁ i₂ : (x : ↥U) → V x ⟶ U
heq :
  ∀ (x : ↥U), (ConcreteCategory.hom (F.presheaf.map (i₁ x).op)) s = (ConcreteCategory.hom (F.presheaf.map (i₂ x).op)) t
⊢ U ≤ iSup V
Module: Mathlib.Topology.Algebra.Module.Basic

Goal state before:
pp: case refine_1
R : Type u_1
M : Type u_2
inst✝¹⁰ : Ring R
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : AddCommGroup M
inst✝⁶ : ContinuousAdd M
inst✝⁵ : Module R M
inst✝⁴ : ContinuousSMul R M
inst✝³ : IsDomain R
inst✝² : Nontrivial M
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : IsTorsionFree R M
x y : M
hy : y ≠ 0
⊢ Tendsto (fun c ↦ x + c • y) (𝓝 0) (𝓝 x)

pp: case refine_2
R : Type u_1
M : Type u_2
inst✝¹⁰ : Ring R
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : AddCommGroup M
inst✝⁶ : ContinuousAdd M
inst✝⁵ : Module R M
inst✝⁴ : ContinuousSMul R M
inst✝³ : IsDomain R
inst✝² : Nontrivial M
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : IsTorsionFree R M
x y : M
hy : y ≠ 0
⊢ ∀ a ∈ {0}ᶜ, x + a • y ∈ {x}ᶜ

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
M : Type u_2
inst✝¹⁰ : Ring R
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : AddCommGroup M
inst✝⁶ : ContinuousAdd M
inst✝⁵ : Module R M
inst✝⁴ : ContinuousSMul R M
inst✝³ : IsDomain R
inst✝² : Nontrivial M
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : IsTorsionFree R M
x y : M
hy : y ≠ 0
⊢ Tendsto (fun c ↦ x + c • y) (𝓝 0) (𝓝 x)
Module: Mathlib.Topology.Algebra.Module.Basic

Goal state before:
pp: case refine_2
R : Type u_1
M : Type u_2
inst✝¹⁰ : Ring R
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : AddCommGroup M
inst✝⁶ : ContinuousAdd M
inst✝⁵ : Module R M
inst✝⁴ : ContinuousSMul R M
inst✝³ : IsDomain R
inst✝² : Nontrivial M
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : IsTorsionFree R M
x y : M
hy : y ≠ 0
⊢ ∀ a ∈ {0}ᶜ, x + a • y ∈ {x}ᶜ

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
M : Type u_2
inst✝¹⁰ : Ring R
inst✝⁹ : TopologicalSpace R
inst✝⁸ : TopologicalSpace M
inst✝⁷ : AddCommGroup M
inst✝⁶ : ContinuousAdd M
inst✝⁵ : Module R M
inst✝⁴ : ContinuousSMul R M
inst✝³ : IsDomain R
inst✝² : Nontrivial M
inst✝¹ : (𝓝[≠] 0).NeBot
inst✝ : IsTorsionFree R M
x y : M
hy : y ≠ 0
⊢ ∀ a ∈ {0}ᶜ, x + a • y ∈ {x}ᶜ
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: case h
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).ReflectsIsomorphisms
F : Sheaf C X
U : Opens ↑X
s t : ToType (F.obj.obj (op U))
h :
  ∀ (x : ↑X) (hx : x ∈ U),
    (ConcreteCategory.hom (F.presheaf.germ U x hx)) s = (ConcreteCategory.hom (F.presheaf.germ U x hx)) t
V : ↥U → Opens ↑X
m : ∀ (x : ↥U), ↑x ∈ V x
i₁ i₂ : (x : ↥U) → V x ⟶ U
heq :
  ∀ (x : ↥U), (ConcreteCategory.hom (F.presheaf.map (i₁ x).op)) s = (ConcreteCategory.hom (F.presheaf.map (i₂ x).op)) t
⊢ ∀ (i : ↥U), (ConcreteCategory.hom (F.obj.map (i₁ i).op)) s = (ConcreteCategory.hom (F.obj.map (i₁ i).op)) t

Tactic:
·

Goal state after:
case h
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).ReflectsIsomorphisms
F : Sheaf C X
U : Opens ↑X
s t : ToType (F.obj.obj (op U))
h :
  ∀ (x : ↑X) (hx : x ∈ U),
    (ConcreteCategory.hom (F.presheaf.germ U x hx)) s = (ConcreteCategory.hom (F.presheaf.germ U x hx)) t
V : ↥U → Opens ↑X
m : ∀ (x : ↥U), ↑x ∈ V x
i₁ i₂ : (x : ↥U) → V x ⟶ U
heq :
  ∀ (x : ↥U), (ConcreteCategory.hom (F.presheaf.map (i₁ x).op)) s = (ConcreteCategory.hom (F.presheaf.map (i₂ x).op)) t
⊢ ∀ (i : ↥U), (ConcreteCategory.hom (F.obj.map (i₁ i).op)) s = (ConcreteCategory.hom (F.obj.map (i₁ i).op)) t
Module: Mathlib.Topology.Algebra.Module.Basic

Goal state before:
pp: case refine_1
R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
⊢ IsSeparable {f | ∀ (i : Fin n), (f i).2 ∈ s}

pp: case refine_2
R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
⊢ Continuous[Pi.topologicalSpace, inst✝⁴] fun f ↦ ∑ i, (f i).1 • (f i).2

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
⊢ IsSeparable {f | ∀ (i : Fin n), (f i).2 ∈ s}
Module: Mathlib.Topology.Algebra.Module.Basic

Goal state before:
pp: case refine_2
R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
⊢ Continuous[Pi.topologicalSpace, inst✝⁴] fun f ↦ ∑ i, (f i).1 • (f i).2

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
M : Type u_2
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Semiring R
inst✝⁵ : Module R M
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace R
inst✝² : SeparableSpace R
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousSMul R M
s : Set M
hs : IsSeparable s
n : ℕ
⊢ Continuous[Pi.topologicalSpace, inst✝⁴] fun f ↦ ∑ i, (f i).1 • (f i).2