awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
hf :
  IsInducing (X := (↑X : Type v)) (Y := (↑Y : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f) :
      (a : (↑X : Type v)) → (↑Y : Type v))
F : Presheaf C X
x : (↑X : Type v)
this : (OpenNhds.map f x).Initial
e_3 :
  (obj (pushforward C f) F).stalk
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x) =
    colimit ((OpenNhds.map f x).op ⋙ (OpenNhds.inclusion x).op ⋙ F)
e_4 : F.stalk x = colimit ((OpenNhds.inclusion x).op ⋙ F)
U : Opens (↑Y : Type v)
hU :
  Membership.mem (α := (↑Y : Type v)) U
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x :
      (↑Y : Type v))
⊢ Eq (α := (obj (pushforward C f) F).obj (op U) ⟶ F.stalk x) (F.germ ((Opens.map f).obj U) x hU)
    (CategoryStruct.comp (Y :=
      (obj (pushforward C f) F).stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x))
      ((obj (pushforward C f) F).germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU)
      (Iso.hom (Final.colimitIso (OpenNhds.map f x).op ((OpenNhds.inclusion x).op ⋙ F))))
have h : Eq (α := (obj (pushforward C f) F).obj (op U) ⟶ F.stalk x)
  (CategoryStruct.comp (Y :=
      (obj (pushforward C f) F).stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x))
      ((obj (pushforward C f) F).germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU)
      (Iso.hom (Final.colimitIso (OpenNhds.map f x).op ((OpenNhds.inclusion x).op ⋙ F))) :
    (obj (pushforward C f) F).obj (op U) ⟶ F.stalk x)
  (F.germ ((Opens.map f).obj U) x hU : F.obj (op ((Opens.map f).obj U)) ⟶ F.stalk x)
Tactic: symm
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y : TopCat
Z : C
f : X ⟶ Y
F : Presheaf C Y
U : (Opens (↑X : Type v))ᵒᵖ
φ ψ : (obj (pullback C f) F).obj U ⟶ Z
h :
  ∀ (V : Opens (↑Y : Type v)) (hV : unop U ≤ (Opens.map f).obj V),
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
        (obj (pullback C f) F).map (homOfLE hV).op ≫ φ =
      NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
        (obj (pullback C f) F).map (homOfLE hV).op ≫ ψ
⊢ φ = ψ
have apply : ∀ (j : CostructuredArrow (Opens.map f).op U),
  ((LeftExtension.mk ((Opens.map f).op.leftKanExtension F) ((Opens.map f).op.leftKanExtensionUnit F)).coconeAt U).ι.app
        j ≫
      φ =
    ((LeftExtension.mk ((Opens.map f).op.leftKanExtension F) ((Opens.map f).op.leftKanExtensionUnit F)).coconeAt
              U).ι.app
        j ≫
      ψ
Tactic: apply ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit F _).hom_ext
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
F : Presheaf C Y
U : Opens (↑X : Type v)
x : (↑X : Type v)
hx : x ∈ U
⊢ germToPullbackStalk C f F U x hx ≫ stalkPullbackHom C f F x = (obj (pullback C f) F).germ U x hx
have h : ∀ (V : Opens (↑Y : Type v)) (hV : unop (op U) ≤ (Opens.map f).obj V),
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
      (obj (pullback C f) F).map (homOfLE hV).op ≫ germToPullbackStalk C f F U x hx ≫ stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V) ≫
      (obj (pullback C f) F).map (homOfLE hV).op ≫ (obj (pullback C f) F).germ U x hx
Tactic: ext V hV
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
⊢ ((OpenNhds.inclusion x).op ⋙ obj (pullback C f) F).map i ≫
      germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x ⋯ ≫
      map
        (obj (Functor.const (OpenNhds x)ᵒᵖ)
          (F.stalk
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
                (a : (↑X : Type v)) → (↑Y : Type v))
              x)))
        i
have dsimp : (obj (pullback C f) F).map ((OpenNhds.inclusion x).map i.unop).op ≫
    germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x sorry =
  germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x sorry ≫
    𝟙
      (F.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
⊢ (obj (pullback C f) F).map ((OpenNhds.inclusion x).map i.unop).op ≫
      germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x ⋯ ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))
have h : ∀ (W : Opens (↑Y : Type v)) (hW : unop (op ((OpenNhds.inclusion x).obj (unop U))) ≤ (Opens.map f).obj W),
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        (obj (pullback C f) F).map ((OpenNhds.inclusion x).map i.unop).op ≫
          germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x sorry =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x sorry ≫
          𝟙
            (F.stalk
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                    f :
                  (a : (↑X : Type v)) → (↑Y : Type v))
                x))
Tactic: ext W hW
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
W : Opens (↑Y : Type v)
hW : unop (op ((OpenNhds.inclusion x).obj (unop U))) ≤ (Opens.map f).obj W
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        (obj (pullback C f) F).map ((OpenNhds.inclusion x).map i.unop).op ≫
          germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x ⋯ ≫
          𝟙
            (F.stalk
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                    f :
                  (a : (↑X : Type v)) → (↑Y : Type v))
                x))
have h : NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
    (obj (pullback C f) F).map (homOfLE hW).op ≫
      (obj (pullback C f) F).map (sorry.functor.map i.unop).op ≫
        germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x sorry =
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
    (obj (pullback C f) F).map (homOfLE hW).op ≫
      germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x sorry ≫
        𝟙
          (F.stalk
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
                (a : (↑X : Type v)) → (↑Y : Type v))
              x))
Tactic: dsimp [OpenNhds.inclusion]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
W : Opens (↑Y : Type v)
hW : unop (op ((OpenNhds.inclusion x).obj (unop U))) ≤ (Opens.map f).obj W
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        (obj (pullback C f) F).map (⋯.functor.map i.unop).op ≫
          germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map (homOfLE hW).op ≫
        germToPullbackStalk C f F (↑(unop U) : Opens (↑X : Type v)) x ⋯ ≫
          𝟙
            (F.stalk
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier)
                    f :
                  (a : (↑X : Type v)) → (↑Y : Type v))
                x))
have h : NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
    (obj (pullback C f) F).map ((homOfLE hW).op ≫ (sorry.functor.map i.unop).op) ≫
      germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x sorry =
  F.germ W
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
        (a : (↑X : Type v)) → (↑Y : Type v))
      x)
    sorry
Tactic: rw [Category.comp_id, ← Functor.map_comp_assoc,
              pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
U V : (OpenNhds x)ᵒᵖ
i : U ⟶ V
W : Opens (↑Y : Type v)
hW : unop (op ((OpenNhds.inclusion x).obj (unop U))) ≤ (Opens.map f).obj W
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op W) ≫
      (obj (pullback C f) F).map ((homOfLE hW).op ≫ (⋯.functor.map i.unop).op) ≫
        germToPullbackStalk C f F (↑(unop V) : Opens (↑X : Type v)) x ⋯ =
    F.germ W
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x)
      ⋯
have h : Eq (α :=
  F.obj (op W) ⟶
    F.stalk
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x))
  (F.germ W
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
        (a : (↑X : Type v)) → (↑Y : Type v))
      x)
    sorry)
  (F.germ W
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
        (a : (↑X : Type v)) → (↑Y : Type v))
      x)
    sorry)
Tactic: erw [pullbackPushforwardAdjunction_unit_pullback_map_germToPullbackStalk]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
⊢ stalkPullbackHom C f F x ≫ stalkPullbackInv C f F x =
    𝟙
      (F.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x))
have ih : ∀ (U : Opens (↑Y : Type v))
  (hU :
    Membership.mem (α := (↑Y : Type v)) U
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x :
        (↑Y : Type v))),
  F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      stalkPullbackHom C f F x ≫ stalkPullbackInv C f F x =
    F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))
Tactic: ext U hU
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
U : Opens (↑Y : Type v)
hU :
  Membership.mem (α := (↑Y : Type v)) U
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x :
      (↑Y : Type v))
⊢ F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      stalkPullbackHom C f F x ≫ stalkPullbackInv C f F x =
    F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))
have ih : NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op U) ≫
    (obj (pullback C f) F).germ ((Opens.map f).obj U) x hU ≫ stalkPullbackInv C f F x =
  F.germ U
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x)
      hU ≫
    𝟙
      (F.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x))
Tactic: rw [germ_stalkPullbackHom_assoc, germ_stalkPullbackInv, Category.comp_id,
      pullbackPushforwardAdjunction_unit_app_app_germToPullbackStalk]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
U : Opens (↑Y : Type v)
hU :
  Membership.mem (α := (↑Y : Type v)) U
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x :
      (↑Y : Type v))
ih :
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op U) ≫
      (obj (pullback C f) F).germ ((Opens.map f).obj U) x hU ≫ stalkPullbackInv C f F x =
    F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))
⊢ F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      stalkPullbackHom C f F x ≫ stalkPullbackInv C f F x =
    F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))
have ih₁ : NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op U) ≫
    germToPullbackStalk C f F ((Opens.map f).obj U) x hU =
  F.germ U
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x)
      hU ≫
    𝟙
      (F.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x))
Tactic: rw [germ_stalkPullbackHom_assoc, germ_stalkPullbackInv, Category.comp_id,
      pullbackPushforwardAdjunction_unit_app_app_germToPullbackStalk]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
U : Opens (↑Y : Type v)
hU :
  Membership.mem (α := (↑Y : Type v)) U
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x :
      (↑Y : Type v))
ih :
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op U) ≫
      (obj (pullback C f) F).germ ((Opens.map f).obj U) x hU ≫ stalkPullbackInv C f F x =
    F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))
ih₁ :
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op U) ≫
      germToPullbackStalk C f F ((Opens.map f).obj U) x hU =
    F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))
⊢ F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      stalkPullbackHom C f F x ≫ stalkPullbackInv C f F x =
    F.germ U
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
            (a : (↑X : Type v)) → (↑Y : Type v))
          x)
        hU ≫
      𝟙
        (F.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
              (a : (↑X : Type v)) → (↑Y : Type v))
            x))
have ih₂ : Eq (α :=
  F.obj (op U) ⟶
    F.stalk
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
          (a : (↑X : Type v)) → (↑Y : Type v))
        x))
  (NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op U) ≫
    germToPullbackStalk C f F ((Opens.map f).obj U) x hU)
  (F.germ U
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := carrier) f :
        (a : (↑X : Type v)) → (↑Y : Type v))
      x)
    hU)
Tactic: rw [germ_stalkPullbackHom_assoc, germ_stalkPullbackInv, Category.comp_id,
      pullbackPushforwardAdjunction_unit_app_app_germToPullbackStalk]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
⊢ stalkPullbackInv C f F x ≫ stalkPullbackHom C f F x = 𝟙 ((obj (pullback C f) F).stalk x)
have ih : ∀ (V : Opens (↑X : Type v)) (hV : x ∈ V) (V_1 : Opens (↑Y : Type v)) (hV_1 : unop (op V) ≤ (Opens.map f).obj V_1),
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫
        (obj (pullback C f) F).germ V x hV ≫ stalkPullbackInv C f F x ≫ stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫
        (obj (pullback C f) F).germ V x hV ≫ 𝟙 ((obj (pullback C f) F).stalk x)
Tactic: ext V hV
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Stalks (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasColimits.{v, u} C
X Y Z : TopCat
FC : C → C → Type u_1
CC : C → Type u_2
inst_2 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
f : X ⟶ Y
F : Presheaf C Y
x : (↑X : Type v)
V : Opens (↑X : Type v)
hV : x ∈ V
V_1 : Opens (↑Y : Type v)
hV_1 : unop (op V) ≤ (Opens.map f).obj V_1
⊢ NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫
        (obj (pullback C f) F).germ V x hV ≫ stalkPullbackInv C f F x ≫ stalkPullbackHom C f F x =
    NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
      (obj (pullback C f) F).map (homOfLE hV_1).op ≫
        (obj (pullback C f) F).germ V x hV ≫ 𝟙 ((obj (pullback C f) F).stalk x)
have ih : NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
    (obj (pullback C f) F).map (homOfLE hV_1).op ≫ germToPullbackStalk C f F V x hV ≫ stalkPullbackHom C f F x =
  NatTrans.app (NatTrans.app (Adjunction.unit (pullbackPushforwardAdjunction C f)) F) (op V_1) ≫
    (obj (pullback C f) F).map (homOfLE hV_1).op ≫
      (obj (pullback C f) F).germ V x hV ≫ 𝟙 ((obj (pullback C f) F).stalk x)
Tactic: rw [germ_stalkPullbackInv_assoc, Category.comp_id, germToPullbackStalk_stalkPullbackHom]