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

Goal state before:
pp: case refine_1
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
sf : (i : ι) → ToType (F.obj.obj (op (U i)))
h : IsCompatible F.obj U sf
V_eq_supr_U : V = iSup U
gl : ToType (F.obj.obj (op (iSup U)))
gl_spec : IsGluing F.obj U sf gl
gl_uniq : ∀ (y : ToType (F.obj.obj (op (iSup U)))), (fun s ↦ IsGluing F.obj U sf s) y → y = gl
i : ι
⊢ (ConcreteCategory.hom (F.obj.map (iUV i).op)) ((ConcreteCategory.hom (F.obj.map (eqToHom V_eq_supr_U).op)) gl) = sf i

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
case refine_1
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
sf : (i : ι) → ToType (F.obj.obj (op (U i)))
h : IsCompatible F.obj U sf
V_eq_supr_U : V = iSup U
gl : ToType (F.obj.obj (op (iSup U)))
gl_spec : IsGluing F.obj U sf gl
gl_uniq : ∀ (y : ToType (F.obj.obj (op (iSup U)))), (fun s ↦ IsGluing F.obj U sf s) y → y = gl
i : ι
⊢ (ConcreteCategory.hom (F.obj.map (eqToHom V_eq_supr_U).op ≫ F.obj.map (iUV i).op)) gl = sf i
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: case refine_2
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
sf : (i : ι) → ToType (F.obj.obj (op (U i)))
h : IsCompatible F.obj U sf
V_eq_supr_U : V = iSup U
gl : ToType (F.obj.obj (op (iSup U)))
gl_spec : IsGluing F.obj U sf gl
gl_uniq : ∀ (y : ToType (F.obj.obj (op (iSup U)))), (fun s ↦ IsGluing F.obj U sf s) y → y = gl
gl' : ToType (F.obj.obj (op V))
⊢ (fun s ↦ ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) s = sf i) gl' →
    gl' = (ConcreteCategory.hom (F.obj.map (eqToHom V_eq_supr_U).op)) gl

Tactic:
gl'_spec

Goal state after:
case refine_2
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
sf : (i : ι) → ToType (F.obj.obj (op (U i)))
h : IsCompatible F.obj U sf
V_eq_supr_U : V = iSup U
gl : ToType (F.obj.obj (op (iSup U)))
gl_spec : IsGluing F.obj U sf gl
gl_uniq : ∀ (y : ToType (F.obj.obj (op (iSup U)))), (fun s ↦ IsGluing F.obj U sf s) y → y = gl
gl' : ToType (F.obj.obj (op V))
gl'_spec : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) gl' = sf i
⊢ gl' = (ConcreteCategory.hom (F.obj.map (eqToHom V_eq_supr_U).op)) gl
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: case e'_2
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
sf : (i : ι) → ToType (F.obj.obj (op (U i)))
h : IsCompatible F.obj U sf
V_eq_supr_U : V = iSup U
gl : ToType (F.obj.obj (op (iSup U)))
gl_spec : IsGluing F.obj U sf gl
gl_uniq : ∀ (y : ToType (F.obj.obj (op (iSup U)))), (fun s ↦ IsGluing F.obj U sf s) y → y = gl
gl' : ToType (F.obj.obj (op V))
gl'_spec : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) gl' = sf i
⊢ gl' =
    (ConcreteCategory.hom (F.obj.map (eqToHom V_eq_supr_U).op)) ((ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) gl')

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
case e'_2
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
sf : (i : ι) → ToType (F.obj.obj (op (U i)))
h : IsCompatible F.obj U sf
V_eq_supr_U : V = iSup U
gl : ToType (F.obj.obj (op (iSup U)))
gl_spec : IsGluing F.obj U sf gl
gl_uniq : ∀ (y : ToType (F.obj.obj (op (iSup U)))), (fun s ↦ IsGluing F.obj U sf s) y → y = gl
gl' : ToType (F.obj.obj (op V))
gl'_spec : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) gl' = sf i
⊢ gl' = (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op ≫ F.obj.map (eqToHom V_eq_supr_U).op)) gl'
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: case refine_2.convert_3
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
sf : (i : ι) → ToType (F.obj.obj (op (U i)))
h : IsCompatible F.obj U sf
V_eq_supr_U : V = iSup U
gl : ToType (F.obj.obj (op (iSup U)))
gl_spec : IsGluing F.obj U sf gl
gl_uniq : ∀ (y : ToType (F.obj.obj (op (iSup U)))), (fun s ↦ IsGluing F.obj U sf s) y → y = gl
gl' : ToType (F.obj.obj (op V))
gl'_spec : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) gl' = sf i
i : ι
⊢ (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) ((ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) gl') = sf i

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
case refine_2.convert_3
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
sf : (i : ι) → ToType (F.obj.obj (op (U i)))
h : IsCompatible F.obj U sf
V_eq_supr_U : V = iSup U
gl : ToType (F.obj.obj (op (iSup U)))
gl_spec : IsGluing F.obj U sf gl
gl_uniq : ∀ (y : ToType (F.obj.obj (op (iSup U)))), (fun s ↦ IsGluing F.obj U sf s) y → y = gl
gl' : ToType (F.obj.obj (op V))
gl'_spec : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) gl' = sf i
i : ι
⊢ (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op ≫ F.obj.map (leSupr U i).op)) gl' = sf i
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: 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
ι : Type u_4
U : ι → Opens ↑X
s t : ToType (F.obj.obj (op (iSup U)))
h :
  ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) s = (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) t
sf : (i : ι) → ToType (F.obj.obj (op (U i))) := fun i ↦ (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) s
⊢ IsCompatible F.obj U sf

Tactic:
intro i j

Goal state after:
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
ι : Type u_4
U : ι → Opens ↑X
s t : ToType (F.obj.obj (op (iSup U)))
h :
  ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) s = (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) t
sf : (i : ι) → ToType (F.obj.obj (op (U i))) := fun i ↦ (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) s
i j : ι
⊢ (ConcreteCategory.hom (F.obj.map ((U i).infLELeft (U j)).op)) (sf i) =
    (ConcreteCategory.hom (F.obj.map ((U i).infLERight (U j)).op)) (sf j)
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: 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
ι : Type u_4
U : ι → Opens ↑X
s t : ToType (F.obj.obj (op (iSup U)))
h :
  ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) s = (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) t
sf : (i : ι) → ToType (F.obj.obj (op (U i))) := fun i ↦ (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) s
i j : ι
⊢ (ConcreteCategory.hom (F.obj.map ((U i).infLELeft (U j)).op)) ((ConcreteCategory.hom (F.obj.map (leSupr U i).op)) s) =
    (ConcreteCategory.hom (F.obj.map ((U i).infLERight (U j)).op))
      ((ConcreteCategory.hom (F.obj.map (leSupr U j).op)) s)

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
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
ι : Type u_4
U : ι → Opens ↑X
s t : ToType (F.obj.obj (op (iSup U)))
h :
  ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) s = (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) t
sf : (i : ι) → ToType (F.obj.obj (op (U i))) := fun i ↦ (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) s
i j : ι
⊢ (ConcreteCategory.hom (F.obj.map (leSupr U i).op ≫ F.obj.map ((U i).infLELeft (U j)).op)) s =
    (ConcreteCategory.hom (F.obj.map (leSupr U j).op ≫ F.obj.map ((U i).infLERight (U j)).op)) s
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
F : Presheaf C X
U V : Opens ↑X
i : U ⟶ V
x : ↑X
hx : x ∈ U
inst✝ : ConcreteCategory C FC
s : CC (F.obj (op V))
⊢ (ConcreteCategory.hom (F.germ U x hx)) ((ConcreteCategory.hom (F.map i.op)) s) =
    (ConcreteCategory.hom (F.germ V x ⋯)) s

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
F : Presheaf C X
U V : Opens ↑X
i : U ⟶ V
x : ↑X
hx : x ∈ U
inst✝ : ConcreteCategory C FC
s : CC (F.obj (op V))
⊢ (ConcreteCategory.hom (F.map i.op ≫ F.germ U x hx)) s = (ConcreteCategory.hom (F.germ V x ⋯)) s
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
F : Presheaf C X
U V : Opens ↑X
i : op V ⟶ op U
x : ↑X
hx : x ∈ U
inst✝ : ConcreteCategory C FC
s : CC (F.obj (op V))
⊢ (ConcreteCategory.hom (F.germ U x hx)) ((ConcreteCategory.hom (F.map i)) s) = (ConcreteCategory.hom (F.germ V x ⋯)) s

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
F : Presheaf C X
U V : Opens ↑X
i : op V ⟶ op U
x : ↑X
hx : x ∈ U
inst✝ : ConcreteCategory C FC
s : CC (F.obj (op V))
⊢ (ConcreteCategory.hom (F.map i ≫ F.germ U x hx)) s = (ConcreteCategory.hom (F.germ V x ⋯)) s
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: case e'_2
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
s t : ToType (F.obj.obj (op V))
h : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) s = (ConcreteCategory.hom (F.obj.map (iUV i).op)) t
V_eq_supr_U : V = iSup U
this : (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) s = (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) t
⊢ s = (ConcreteCategory.hom (F.obj.map (eqToHom V_eq_supr_U).op)) ((ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) s)

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
case e'_2
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
s t : ToType (F.obj.obj (op V))
h : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) s = (ConcreteCategory.hom (F.obj.map (iUV i).op)) t
V_eq_supr_U : V = iSup U
this : (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) s = (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) t
⊢ s = (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op ≫ F.obj.map (eqToHom V_eq_supr_U).op)) s
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
F G : Presheaf C X
U : Opens ↑X
x : ↑X
hx : x ∈ U
f : F ⟶ G
s : CC (F.obj (op U))
⊢ (ConcreteCategory.hom ((stalkFunctor C x).map f)) ((ConcreteCategory.hom (F.germ U x hx)) s) =
    (ConcreteCategory.hom (G.germ U x hx)) ((ConcreteCategory.hom (f.app (op U))) s)

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
F G : Presheaf C X
U : Opens ↑X
x : ↑X
hx : x ∈ U
f : F ⟶ G
s : CC (F.obj (op U))
⊢ (ConcreteCategory.hom ((stalkFunctor C x).map f)) ((ConcreteCategory.hom (F.germ U x hx)) s) =
    (ConcreteCategory.hom (f.app (op U) ≫ G.germ U x hx)) s
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: case e'_3
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
s t : ToType (F.obj.obj (op V))
h : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) s = (ConcreteCategory.hom (F.obj.map (iUV i).op)) t
V_eq_supr_U : V = iSup U
this : (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) s = (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) t
⊢ t = (ConcreteCategory.hom (F.obj.map (eqToHom V_eq_supr_U).op)) ((ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) t)

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
case e'_3
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
s t : ToType (F.obj.obj (op V))
h : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) s = (ConcreteCategory.hom (F.obj.map (iUV i).op)) t
V_eq_supr_U : V = iSup U
this : (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) s = (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) t
⊢ t = (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op ≫ F.obj.map (eqToHom V_eq_supr_U).op)) t
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: 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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
s t : ToType (F.obj.obj (op V))
h : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) s = (ConcreteCategory.hom (F.obj.map (iUV i).op)) t
V_eq_supr_U : V = iSup U
i : ι
⊢ (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) ((ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) s) =
    (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) ((ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) t)

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
s t : ToType (F.obj.obj (op V))
h : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) s = (ConcreteCategory.hom (F.obj.map (iUV i).op)) t
V_eq_supr_U : V = iSup U
i : ι
⊢ (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op ≫ F.obj.map (leSupr U i).op)) s =
    (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) ((ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) t)
Module: Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing

Goal state before:
pp: 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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
s t : ToType (F.obj.obj (op V))
h : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) s = (ConcreteCategory.hom (F.obj.map (iUV i).op)) t
V_eq_supr_U : V = iSup U
i : ι
⊢ (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op ≫ F.obj.map (leSupr U i).op)) s =
    (ConcreteCategory.hom (F.obj.map (leSupr U i).op)) ((ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op)) t)

Tactic:
← ConcreteCategory.comp_apply,

Goal state after:
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
ι : Type u_4
U : ι → Opens ↑X
V : Opens ↑X
iUV : (i : ι) → U i ⟶ V
hcover : V ≤ iSup U
s t : ToType (F.obj.obj (op V))
h : ∀ (i : ι), (ConcreteCategory.hom (F.obj.map (iUV i).op)) s = (ConcreteCategory.hom (F.obj.map (iUV i).op)) t
V_eq_supr_U : V = iSup U
i : ι
⊢ (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op ≫ F.obj.map (leSupr U i).op)) s =
    (ConcreteCategory.hom (F.obj.map (eqToHom ⋯).op ≫ F.obj.map (leSupr U i).op)) t
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : TopCat
F : Presheaf C X
x : ↑X
⊢ F.stalkSpecializes ⋯ = 𝟙 (F.stalk x)

Tactic:
by
  ext
  simp

Goal state after:
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : TopCat
F : Presheaf C X
x y z : ↑X
h : x ⤳ y
h' : y ⤳ z
⊢ F.stalkSpecializes h' ≫ F.stalkSpecializes h = F.stalkSpecializes ⋯

Tactic:
by
  ext
  simp

Goal state after:
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : TopCat
F G : Presheaf C X
f : F ⟶ G
x y : ↑X
h : x ⤳ y
⊢ F.stalkSpecializes h ≫ (stalkFunctor C x).map f = (stalkFunctor C y).map f ≫ G.stalkSpecializes h

Tactic:
by
  ext
  simp

Goal state after:
Module: Mathlib.Topology.Sheaves.Stalks

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : TopCat
f : X ⟶ Y
F : Presheaf C X
x y : ↑X
h : x ⤳ y
⊢ ((pushforward C f).obj F).stalkSpecializes ⋯ ≫ stalkPushforward C f F x =
    stalkPushforward C f F y ≫ F.stalkSpecializes h

Tactic:
by
  ext
  simp

Goal state after: