l3lab/ntp-mathlib huggingface dataset:
theorem restrictStalkIso_inv_eq_ofRestrict {U : TopCat} (X : PresheafedSpace.{_, _, v} C)
    {f : U ⟶ (X : TopCat.{v})} (h : OpenEmbedding f) (x : U) :
    (X.restrictStalkIso h x).inv = stalkMap (X.ofRestrict h) x := by
  -- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
  refine colimit.hom_ext fun V => ?_
  induction V with | h V => ?_
  let i : (h.isOpenMap.functorNhds x).obj ((OpenNhds.map f x).obj V) ⟶ V :=
    homOfLE (Set.image_preimage_subset f _)
  erw [Iso.comp_inv_eq, colimit.ι_map_assoc, colimit.ι_map_assoc, colimit.ι_pre]
  simp_rw [Category.assoc]
  

Proof state:
case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
U : TopCat
X : PresheafedSpace C
f : U ⟶ ↑X
h : OpenEmbedding ⇑f
x : ↑U
V : OpenNhds (f x)
i : (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x).obj ((OpenNhds.map f x).obj V) ⟶ V :=
  homOfLE (_ : ⇑f '' (⇑f ⁻¹' V.obj.1) ⊆ V.obj.1)
⊢ colimit.ι (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑↑X)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op).obj X.presheaf)
      (op V) =
    (((whiskeringLeft (OpenNhds ((ofRestrict X h).base x))ᵒᵖ (Opens ↑↑X)ᵒᵖ C).obj
                (OpenNhds.inclusion ((ofRestrict X h).base x)).op).map
            (ofRestrict X h).c).app
        (op V) ≫
      (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (ofRestrict X h).base x).inv) (restrict X h).presheaf).app
          (op V) ≫
        colimit.ι
            (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑↑(restrict X h))ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj
              (restrict X h).presheaf)
            ((OpenNhds.map (ofRestrict X h).base x).op.obj (op V)) ≫
          (restrictStalkIso X h x).hom

Next tactic:
erw [colimit.ι_pre ((OpenNhds.inclusion (f x)).op ⋙ X.presheaf)
      (h.isOpenMap.functorNhds x).op]
theorem restrictStalkIso_inv_eq_ofRestrict {U : TopCat} (X : PresheafedSpace.{_, _, v} C)
    {f : U ⟶ (X : TopCat.{v})} (h : OpenEmbedding f) (x : U) :
    (X.restrictStalkIso h x).inv = stalkMap (X.ofRestrict h) x := by
  -- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
  refine colimit.hom_ext fun V => ?_
  induction V with | h V => ?_
  let i : (h.isOpenMap.functorNhds x).obj ((OpenNhds.map f x).obj V) ⟶ V :=
    homOfLE (Set.image_preimage_subset f _)
  erw [Iso.comp_inv_eq, colimit.ι_map_assoc, colimit.ι_map_assoc, colimit.ι_pre]
  simp_rw [Category.assoc]
  erw [colimit.ι_pre ((OpenNhds.inclusion (f x)).op ⋙ X.presheaf)
      (h.isOpenMap.functorNhds x).op]
  

Proof state:
case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
U : TopCat
X : PresheafedSpace C
f : U ⟶ ↑X
h : OpenEmbedding ⇑f
x : ↑U
V : OpenNhds (f x)
i : (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x).obj ((OpenNhds.map f x).obj V) ⟶ V :=
  homOfLE (_ : ⇑f '' (⇑f ⁻¹' V.obj.1) ⊆ V.obj.1)
⊢ colimit.ι (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑↑X)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op).obj X.presheaf)
      (op V) =
    (((whiskeringLeft (OpenNhds ((ofRestrict X h).base x))ᵒᵖ (Opens ↑↑X)ᵒᵖ C).obj
                (OpenNhds.inclusion ((ofRestrict X h).base x)).op).map
            (ofRestrict X h).c).app
        (op V) ≫
      (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (ofRestrict X h).base x).inv) (restrict X h).presheaf).app
          (op V) ≫
        colimit.ι ((OpenNhds.inclusion (f x)).op ⋙ X.presheaf)
          ((IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x).op.obj ((OpenNhds.map (ofRestrict X h).base x).op.obj (op V)))

Next tactic:
erw [← X.presheaf.map_comp_assoc]
theorem restrictStalkIso_inv_eq_ofRestrict {U : TopCat} (X : PresheafedSpace.{_, _, v} C)
    {f : U ⟶ (X : TopCat.{v})} (h : OpenEmbedding f) (x : U) :
    (X.restrictStalkIso h x).inv = stalkMap (X.ofRestrict h) x := by
  -- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
  refine colimit.hom_ext fun V => ?_
  induction V with | h V => ?_
  let i : (h.isOpenMap.functorNhds x).obj ((OpenNhds.map f x).obj V) ⟶ V :=
    homOfLE (Set.image_preimage_subset f _)
  erw [Iso.comp_inv_eq, colimit.ι_map_assoc, colimit.ι_map_assoc, colimit.ι_pre]
  simp_rw [Category.assoc]
  erw [colimit.ι_pre ((OpenNhds.inclusion (f x)).op ⋙ X.presheaf)
      (h.isOpenMap.functorNhds x).op]
  erw [← X.presheaf.map_comp_assoc]
  

Proof state:
case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
U : TopCat
X : PresheafedSpace C
f : U ⟶ ↑X
h : OpenEmbedding ⇑f
x : ↑U
V : OpenNhds (f x)
i : (IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x).obj ((OpenNhds.map f x).obj V) ⟶ V :=
  homOfLE (_ : ⇑f '' (⇑f ⁻¹' V.obj.1) ⊆ V.obj.1)
⊢ colimit.ι (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑↑X)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op).obj X.presheaf)
      (op V) =
    X.presheaf.map
        (((IsOpenMap.adjunction (_ : IsOpenMap ⇑f)).counit.app
              ((OpenNhds.inclusion ((ofRestrict X h).base x)).op.obj (op V)).unop).op ≫
          (IsOpenMap.functor (_ : IsOpenMap ⇑f)).op.map
            ((NatTrans.op (OpenNhds.inclusionMapIso (ofRestrict X h).base x).inv).app (op V))) ≫
      colimit.ι ((OpenNhds.inclusion (f x)).op ⋙ X.presheaf)
        ((IsOpenMap.functorNhds (_ : IsOpenMap ⇑f) x).op.obj ((OpenNhds.map (ofRestrict X h).base x).op.obj (op V)))

Next tactic:
exact (colimit.w ((OpenNhds.inclusion (f x)).op ⋙ X.presheaf) i.op).symm
instance ofRestrict_stalkMap_isIso {U : TopCat} (X : PresheafedSpace.{_, _, v} C)
    {f : U ⟶ (X : TopCat.{v})} (h : OpenEmbedding f) (x : U) :
    IsIso (stalkMap (X.ofRestrict h) x) := by
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
U : TopCat
X : PresheafedSpace C
f : U ⟶ ↑X
h : OpenEmbedding ⇑f
x : ↑U
⊢ IsIso (stalkMap (ofRestrict X h) x)

Next tactic:
rw [← restrictStalkIso_inv_eq_ofRestrict]
instance ofRestrict_stalkMap_isIso {U : TopCat} (X : PresheafedSpace.{_, _, v} C)
    {f : U ⟶ (X : TopCat.{v})} (h : OpenEmbedding f) (x : U) :
    IsIso (stalkMap (X.ofRestrict h) x) := by
  rw [← restrictStalkIso_inv_eq_ofRestrict]; 

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
U : TopCat
X : PresheafedSpace C
f : U ⟶ ↑X
h : OpenEmbedding ⇑f
x : ↑U
⊢ IsIso (restrictStalkIso X h x).inv

Next tactic:
infer_instance
@[simp]
theorem id (X : PresheafedSpace.{_, _, v} C) (x : X) :
    stalkMap (𝟙 X) x = 𝟙 (X.stalk x) := by
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
⊢ stalkMap (𝟙 X) x = 𝟙 (stalk X x)

Next tactic:
dsimp [stalkMap]
@[simp]
theorem id (X : PresheafedSpace.{_, _, v} C) (x : X) :
    stalkMap (𝟙 X) x = 𝟙 (X.stalk x) := by
  dsimp [stalkMap]
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
⊢ (stalkFunctor C ((𝟙 ↑X) x)).map (𝟙 X).c ≫ stalkPushforward C (𝟙 ↑X) X.presheaf x = 𝟙 (stalk X x)

Next tactic:
simp only [stalkPushforward.id]
@[simp]
theorem id (X : PresheafedSpace.{_, _, v} C) (x : X) :
    stalkMap (𝟙 X) x = 𝟙 (X.stalk x) := by
  dsimp [stalkMap]
  simp only [stalkPushforward.id]
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
⊢ (stalkFunctor C ((𝟙 ↑X) x)).map (𝟙 X).c ≫ (stalkFunctor C x).map (Pushforward.id X.presheaf).hom = 𝟙 (stalk X x)

Next tactic:
erw [← map_comp]
@[simp]
theorem id (X : PresheafedSpace.{_, _, v} C) (x : X) :
    stalkMap (𝟙 X) x = 𝟙 (X.stalk x) := by
  dsimp [stalkMap]
  simp only [stalkPushforward.id]
  erw [← map_comp]
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
⊢ (stalkFunctor C ((𝟙 ↑X) x)).map ((𝟙 X).c ≫ (Pushforward.id X.presheaf).hom) = 𝟙 (stalk X x)

Next tactic:
convert (stalkFunctor C x).map_id X.presheaf
@[simp]
theorem id (X : PresheafedSpace.{_, _, v} C) (x : X) :
    stalkMap (𝟙 X) x = 𝟙 (X.stalk x) := by
  dsimp [stalkMap]
  simp only [stalkPushforward.id]
  erw [← map_comp]
  convert (stalkFunctor C x).map_id X.presheaf
  

Proof state:
case h.e'_2.h.h.e'_8
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
e_1✝ : (stalk X ((𝟙 ↑X) x) ⟶ stalk X x) = ((stalkFunctor C x).obj X.presheaf ⟶ (stalkFunctor C x).obj X.presheaf)
⊢ (𝟙 X).c ≫ (Pushforward.id X.presheaf).hom = 𝟙 X.presheaf

Next tactic:
ext
@[simp]
theorem id (X : PresheafedSpace.{_, _, v} C) (x : X) :
    stalkMap (𝟙 X) x = 𝟙 (X.stalk x) := by
  dsimp [stalkMap]
  simp only [stalkPushforward.id]
  erw [← map_comp]
  convert (stalkFunctor C x).map_id X.presheaf
  ext
  

Proof state:
case h.e'_2.h.h.e'_8.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
e_1✝ : (stalk X ((𝟙 ↑X) x) ⟶ stalk X x) = ((stalkFunctor C x).obj X.presheaf ⟶ (stalkFunctor C x).obj X.presheaf)
U✝ : Opens ↑↑X
⊢ ((𝟙 X).c ≫ (Pushforward.id X.presheaf).hom).app (op U✝) = (𝟙 X.presheaf).app (op U✝)

Next tactic:
simp only [id_c, id_comp, Pushforward.id_hom_app, op_obj, eqToHom_refl, map_id]
@[simp]
theorem id (X : PresheafedSpace.{_, _, v} C) (x : X) :
    stalkMap (𝟙 X) x = 𝟙 (X.stalk x) := by
  dsimp [stalkMap]
  simp only [stalkPushforward.id]
  erw [← map_comp]
  convert (stalkFunctor C x).map_id X.presheaf
  ext
  simp only [id_c, id_comp, Pushforward.id_hom_app, op_obj, eqToHom_refl, map_id]
  

Proof state:
case h.e'_2.h.h.e'_8.w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
e_1✝ : (stalk X ((𝟙 ↑X) x) ⟶ stalk X x) = ((stalkFunctor C x).obj X.presheaf ⟶ (stalkFunctor C x).obj X.presheaf)
U✝ : Opens ↑↑X
⊢ 𝟙 (X.presheaf.obj (op ((Opens.map (𝟙 ↑X)).obj (op U✝).unop))) = (𝟙 X.presheaf).app (op U✝)

Next tactic:
rfl
@[simp]
theorem comp {X Y Z : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (β : Y ⟶ Z) (x : X) :
    stalkMap (α ≫ β) x =
      (stalkMap β (α.base x) : Z.stalk (β.base (α.base x)) ⟶ Y.stalk (α.base x)) ≫
        (stalkMap α x : Y.stalk (α.base x) ⟶ X.stalk x) := by
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : PresheafedSpace C
α : X ⟶ Y
β : Y ⟶ Z
x : ↑↑X
⊢ stalkMap (α ≫ β) x = stalkMap β (α.base x) ≫ stalkMap α x

Next tactic:
dsimp [stalkMap, stalkFunctor, stalkPushforward]
@[simp]
theorem comp {X Y Z : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (β : Y ⟶ Z) (x : X) :
    stalkMap (α ≫ β) x =
      (stalkMap β (α.base x) : Z.stalk (β.base (α.base x)) ⟶ Y.stalk (α.base x)) ≫
        (stalkMap α x : Y.stalk (α.base x) ⟶ X.stalk x) := by
  dsimp [stalkMap, stalkFunctor, stalkPushforward]
  -- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : PresheafedSpace C
α : X ⟶ Y
β : Y ⟶ Z
x : ↑↑X
⊢ colimMap (whiskerLeft (OpenNhds.inclusion ((α.base ≫ β.base) x)).op (α ≫ β).c) ≫
      colimMap (whiskerRight (𝟙 (OpenNhds.map (α.base ≫ β.base) x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
        colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map (α.base ≫ β.base) x).op =
    (colimMap (whiskerLeft (OpenNhds.inclusion (β.base (α.base x))).op β.c) ≫
        colimMap (whiskerRight (𝟙 (OpenNhds.map β.base (α.base x) ⋙ OpenNhds.inclusion (α.base x)).op) Y.presheaf) ≫
          colimit.pre ((OpenNhds.inclusion (α.base x)).op ⋙ Y.presheaf) (OpenNhds.map β.base (α.base x)).op) ≫
      colimMap (whiskerLeft (OpenNhds.inclusion (α.base x)).op α.c) ≫
        colimMap (whiskerRight (𝟙 (OpenNhds.map α.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
          colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map α.base x).op

Next tactic:
refine colimit.hom_ext fun U => ?_
@[simp]
theorem comp {X Y Z : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (β : Y ⟶ Z) (x : X) :
    stalkMap (α ≫ β) x =
      (stalkMap β (α.base x) : Z.stalk (β.base (α.base x)) ⟶ Y.stalk (α.base x)) ≫
        (stalkMap α x : Y.stalk (α.base x) ⟶ X.stalk x) := by
  dsimp [stalkMap, stalkFunctor, stalkPushforward]
  -- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
  refine colimit.hom_ext fun U => ?_
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : PresheafedSpace C
α : X ⟶ Y
β : Y ⟶ Z
x : ↑↑X
U : (OpenNhds ((α.base ≫ β.base) x))ᵒᵖ
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds ((α.base ≫ β.base) x))ᵒᵖ (Opens ↑↑Z)ᵒᵖ C).obj
              (OpenNhds.inclusion ((α.base ≫ β.base) x)).op).obj
          Z.presheaf)
        U ≫
      colimMap (whiskerLeft (OpenNhds.inclusion ((α.base ≫ β.base) x)).op (α ≫ β).c) ≫
        colimMap (whiskerRight (𝟙 (OpenNhds.map (α.base ≫ β.base) x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
          colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map (α.base ≫ β.base) x).op =
    colimit.ι
        (((whiskeringLeft (OpenNhds ((α.base ≫ β.base) x))ᵒᵖ (Opens ↑↑Z)ᵒᵖ C).obj
              (OpenNhds.inclusion ((α.base ≫ β.base) x)).op).obj
          Z.presheaf)
        U ≫
      (colimMap (whiskerLeft (OpenNhds.inclusion (β.base (α.base x))).op β.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map β.base (α.base x) ⋙ OpenNhds.inclusion (α.base x)).op) Y.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion (α.base x)).op ⋙ Y.presheaf) (OpenNhds.map β.base (α.base x)).op) ≫
        colimMap (whiskerLeft (OpenNhds.inclusion (α.base x)).op α.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map α.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map α.base x).op

Next tactic:
induction U with | h U => ?_
@[simp]
theorem comp {X Y Z : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (β : Y ⟶ Z) (x : X) :
    stalkMap (α ≫ β) x =
      (stalkMap β (α.base x) : Z.stalk (β.base (α.base x)) ⟶ Y.stalk (α.base x)) ≫
        (stalkMap α x : Y.stalk (α.base x) ⟶ X.stalk x) := by
  dsimp [stalkMap, stalkFunctor, stalkPushforward]
  -- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
  refine colimit.hom_ext fun U => ?_
  induction U with | h U => ?_
  

Proof state:
case h
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : PresheafedSpace C
α : X ⟶ Y
β : Y ⟶ Z
x : ↑↑X
U : OpenNhds ((α.base ≫ β.base) x)
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds ((α.base ≫ β.base) x))ᵒᵖ (Opens ↑↑Z)ᵒᵖ C).obj
              (OpenNhds.inclusion ((α.base ≫ β.base) x)).op).obj
          Z.presheaf)
        (op U) ≫
      colimMap (whiskerLeft (OpenNhds.inclusion ((α.base ≫ β.base) x)).op (α ≫ β).c) ≫
        colimMap (whiskerRight (𝟙 (OpenNhds.map (α.base ≫ β.base) x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
          colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map (α.base ≫ β.base) x).op =
    colimit.ι
        (((whiskeringLeft (OpenNhds ((α.base ≫ β.base) x))ᵒᵖ (Opens ↑↑Z)ᵒᵖ C).obj
              (OpenNhds.inclusion ((α.base ≫ β.base) x)).op).obj
          Z.presheaf)
        (op U) ≫
      (colimMap (whiskerLeft (OpenNhds.inclusion (β.base (α.base x))).op β.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map β.base (α.base x) ⋙ OpenNhds.inclusion (α.base x)).op) Y.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion (α.base x)).op ⋙ Y.presheaf) (OpenNhds.map β.base (α.base x)).op) ≫
        colimMap (whiskerLeft (OpenNhds.inclusion (α.base x)).op α.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map α.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map α.base x).op

Next tactic:
cases U
@[simp]
theorem comp {X Y Z : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (β : Y ⟶ Z) (x : X) :
    stalkMap (α ≫ β) x =
      (stalkMap β (α.base x) : Z.stalk (β.base (α.base x)) ⟶ Y.stalk (α.base x)) ≫
        (stalkMap α x : Y.stalk (α.base x) ⟶ X.stalk x) := by
  dsimp [stalkMap, stalkFunctor, stalkPushforward]
  -- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
  refine colimit.hom_ext fun U => ?_
  induction U with | h U => ?_
  cases U
  

Proof state:
case h.mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : PresheafedSpace C
α : X ⟶ Y
β : Y ⟶ Z
x : ↑↑X
obj✝ : Opens ↑↑Z
property✝ : (α.base ≫ β.base) x ∈ obj✝
⊢ colimit.ι
        (((whiskeringLeft (OpenNhds ((α.base ≫ β.base) x))ᵒᵖ (Opens ↑↑Z)ᵒᵖ C).obj
              (OpenNhds.inclusion ((α.base ≫ β.base) x)).op).obj
          Z.presheaf)
        (op { obj := obj✝, property := property✝ }) ≫
      colimMap (whiskerLeft (OpenNhds.inclusion ((α.base ≫ β.base) x)).op (α ≫ β).c) ≫
        colimMap (whiskerRight (𝟙 (OpenNhds.map (α.base ≫ β.base) x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
          colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map (α.base ≫ β.base) x).op =
    colimit.ι
        (((whiskeringLeft (OpenNhds ((α.base ≫ β.base) x))ᵒᵖ (Opens ↑↑Z)ᵒᵖ C).obj
              (OpenNhds.inclusion ((α.base ≫ β.base) x)).op).obj
          Z.presheaf)
        (op { obj := obj✝, property := property✝ }) ≫
      (colimMap (whiskerLeft (OpenNhds.inclusion (β.base (α.base x))).op β.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map β.base (α.base x) ⋙ OpenNhds.inclusion (α.base x)).op) Y.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion (α.base x)).op ⋙ Y.presheaf) (OpenNhds.map β.base (α.base x)).op) ≫
        colimMap (whiskerLeft (OpenNhds.inclusion (α.base x)).op α.c) ≫
          colimMap (whiskerRight (𝟙 (OpenNhds.map α.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫
            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map α.base x).op

Next tactic:
simp only [whiskeringLeft_obj_obj, comp_obj, op_obj, unop_op, OpenNhds.inclusion_obj,
    ι_colimMap_assoc, pushforwardObj_obj, Opens.map_comp_obj, whiskerLeft_app, comp_c_app,
    OpenNhds.map_obj, whiskerRight_app, NatTrans.id_app, map_id, colimit.ι_pre, id_comp, assoc,
    colimit.ι_pre_assoc]
/-- If `α = β` and `x = x'`, we would like to say that `stalk_map α x = stalk_map β x'`.
Unfortunately, this equality is not well-formed, as their types are not _definitionally_ the same.
To get a proper congruence lemma, we therefore have to introduce these `eq_to_hom` arrows on
either side of the equality.
-/
theorem congr {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y)
    (h₁ : α = β) (x x' : X) (h₂ : x = x') :
    stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by 

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
α β : X ⟶ Y
h₁ : α = β
x x' : ↑↑X
h₂ : x = x'
⊢ stalk X x = stalk X x'

Next tactic:
rw [h₂]
/-- If `α = β` and `x = x'`, we would like to say that `stalk_map α x = stalk_map β x'`.
Unfortunately, this equality is not well-formed, as their types are not _definitionally_ the same.
To get a proper congruence lemma, we therefore have to introduce these `eq_to_hom` arrows on
either side of the equality.
-/
theorem congr {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y)
    (h₁ : α = β) (x x' : X) (h₂ : x = x') :
    stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by rw [h₂]) =
      eqToHom (show Y.stalk (α.base x) = Y.stalk (β.base x') by 

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
α β : X ⟶ Y
h₁ : α = β
x x' : ↑↑X
h₂ : x = x'
⊢ stalk Y (α.base x) = stalk Y (β.base x')

Next tactic:
rw [h₁, h₂]
/-- If `α = β` and `x = x'`, we would like to say that `stalk_map α x = stalk_map β x'`.
Unfortunately, this equality is not well-formed, as their types are not _definitionally_ the same.
To get a proper congruence lemma, we therefore have to introduce these `eq_to_hom` arrows on
either side of the equality.
-/
theorem congr {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y)
    (h₁ : α = β) (x x' : X) (h₂ : x = x') :
    stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by rw [h₂]) =
      eqToHom (show Y.stalk (α.base x) = Y.stalk (β.base x') by rw [h₁, h₂]) ≫ stalkMap β x' := by
  

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
α β : X ⟶ Y
h₁ : α = β
x x' : ↑↑X
h₂ : x = x'
⊢ stalkMap α x ≫ eqToHom (_ : stalk X x = stalk X x') =
    eqToHom (_ : stalk Y (α.base x) = stalk Y (β.base x')) ≫ stalkMap β x'

Next tactic:
ext
/-- If `α = β` and `x = x'`, we would like to say that `stalk_map α x = stalk_map β x'`.
Unfortunately, this equality is not well-formed, as their types are not _definitionally_ the same.
To get a proper congruence lemma, we therefore have to introduce these `eq_to_hom` arrows on
either side of the equality.
-/
theorem congr {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y)
    (h₁ : α = β) (x x' : X) (h₂ : x = x') :
    stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by rw [h₂]) =
      eqToHom (show Y.stalk (α.base x) = Y.stalk (β.base x') by rw [h₁, h₂]) ≫ stalkMap β x' := by
  ext
  

Proof state:
case ih
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
α β : X ⟶ Y
h₁ : α = β
x x' : ↑↑X
h₂ : x = x'
U✝ : Opens ↑↑Y
hxU✝ : α.base x ∈ U✝
⊢ germ Y.presheaf { val := α.base x, property := hxU✝ } ≫ stalkMap α x ≫ eqToHom (_ : stalk X x = stalk X x') =
    germ Y.presheaf { val := α.base x, property := hxU✝ } ≫
      eqToHom (_ : stalk Y (α.base x) = stalk Y (β.base x')) ≫ stalkMap β x'

Next tactic:
substs h₁ h₂
/-- If `α = β` and `x = x'`, we would like to say that `stalk_map α x = stalk_map β x'`.
Unfortunately, this equality is not well-formed, as their types are not _definitionally_ the same.
To get a proper congruence lemma, we therefore have to introduce these `eq_to_hom` arrows on
either side of the equality.
-/
theorem congr {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y)
    (h₁ : α = β) (x x' : X) (h₂ : x = x') :
    stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by rw [h₂]) =
      eqToHom (show Y.stalk (α.base x) = Y.stalk (β.base x') by rw [h₁, h₂]) ≫ stalkMap β x' := by
  ext
  substs h₁ h₂
  

Proof state:
case ih
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
α : X ⟶ Y
x : ↑↑X
U✝ : Opens ↑↑Y
hxU✝ : α.base x ∈ U✝
⊢ germ Y.presheaf { val := α.base x, property := hxU✝ } ≫ stalkMap α x ≫ eqToHom (_ : stalk X x = stalk X x) =
    germ Y.presheaf { val := α.base x, property := hxU✝ } ≫
      eqToHom (_ : stalk Y (α.base x) = stalk Y (α.base x)) ≫ stalkMap α x

Next tactic:
simp
theorem congr_hom {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y) (h : α = β) (x : X) :
    stalkMap α x =
      eqToHom (show Y.stalk (α.base x) = Y.stalk (β.base x) by 

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
α β : X ⟶ Y
h : α = β
x : ↑↑X
⊢ stalk Y (α.base x) = stalk Y (β.base x)

Next tactic:
rw [h]
theorem congr_hom {X Y : PresheafedSpace.{_, _, v} C} (α β : X ⟶ Y) (h : α = β) (x : X) :
    stalkMap α x =
      eqToHom (show Y.stalk (α.base x) = Y.stalk (β.base x) by rw [h]) ≫ stalkMap β x :=
  by 

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
α β : X ⟶ Y
h : α = β
x : ↑↑X
⊢ stalkMap α x = eqToHom (_ : stalk Y (α.base x) = stalk Y (β.base x)) ≫ stalkMap β x

Next tactic:
rw [← stalkMap.congr α β h x x rfl, eqToHom_refl, Category.comp_id]
theorem congr_point {X Y : PresheafedSpace.{_, _, v} C}
    (α : X ⟶ Y) (x x' : X) (h : x = x') :
    stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by 

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
α : X ⟶ Y
x x' : ↑↑X
h : x = x'
⊢ stalk X x = stalk X x'

Next tactic:
rw [h]
theorem congr_point {X Y : PresheafedSpace.{_, _, v} C}
    (α : X ⟶ Y) (x x' : X) (h : x = x') :
    stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by rw [h]) =
      eqToHom (show Y.stalk (α.base x) = Y.stalk (α.base x') by 

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
α : X ⟶ Y
x x' : ↑↑X
h : x = x'
⊢ stalk Y (α.base x) = stalk Y (α.base x')

Next tactic:
rw [h]
theorem congr_point {X Y : PresheafedSpace.{_, _, v} C}
    (α : X ⟶ Y) (x x' : X) (h : x = x') :
    stalkMap α x ≫ eqToHom (show X.stalk x = X.stalk x' by rw [h]) =
      eqToHom (show Y.stalk (α.base x) = Y.stalk (α.base x') by rw [h]) ≫ stalkMap α x' :=
  by 

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : PresheafedSpace C
α : X ⟶ Y
x x' : ↑↑X
h : x = x'
⊢ stalkMap α x ≫ eqToHom (_ : stalk X x = stalk X x') =
    eqToHom (_ : stalk Y (α.base x) = stalk Y (α.base x')) ≫ stalkMap α x'

Next tactic:
rw [stalkMap.congr α α rfl x x' h]