awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst_1 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{u_2, v, u, u_1} C FC (CC := CC)
F G : Presheaf.{w, v, u} C X
e : F ⟶ G
U V : Opens (↑X : Type w)
h : U ≤ V
x : ToType.{u_2, v, u, u_1} (FC := FC) (CC := CC) (F.obj (op V))
⊢ Eq (α := CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op U)) : (a : CC (F.obj (op U))) → CC (G.obj (op U)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE h).op) :
              (a : CC (F.obj (op V))) → CC (F.obj (op U)))
            x :
          CC (F.obj (op U))) :
      CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.map (homOfLE h).op) : (a : CC (G.obj (op V))) → CC (G.obj (op U)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V)) : (a : CC (F.obj (op V))) → CC (G.obj (op V))) x :
          CC (G.obj (op V))) :
      CC (G.obj (op U)))
have rw : Eq (α := CC (G.obj (op U)))
  ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE h).op ≫ e.app (op U)) :
        (a : CC (F.obj (op V))) → CC (G.obj (op U)))
      x :
    CC (G.obj (op U)))
  ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.map (homOfLE h).op) : (a : CC (G.obj (op V))) → CC (G.obj (op U)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V)) : (a : CC (F.obj (op V))) → CC (G.obj (op V))) x :
        CC (G.obj (op V))) :
    CC (G.obj (op U)))
Tactic: rw [← ConcreteCategory.comp_apply, NatTrans.naturality, ConcreteCategory.comp_apply]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst_1 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{u_2, v, u, u_1} C FC (CC := CC)
F G : Presheaf.{w, v, u} C X
e : F ⟶ G
U V : Opens (↑X : Type w)
h : U ≤ V
x : ToType.{u_2, v, u, u_1} (FC := FC) (CC := CC) (F.obj (op V))
rw :
  Eq (α := CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE h).op ≫ e.app (op U)) :
          (a : CC (F.obj (op V))) → CC (G.obj (op U)))
        x :
      CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.map (homOfLE h).op) : (a : CC (G.obj (op V))) → CC (G.obj (op U)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V)) : (a : CC (F.obj (op V))) → CC (G.obj (op V))) x :
          CC (G.obj (op V))) :
      CC (G.obj (op U)))
⊢ Eq (α := CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op U)) : (a : CC (F.obj (op U))) → CC (G.obj (op U)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE h).op) :
              (a : CC (F.obj (op V))) → CC (F.obj (op U)))
            x :
          CC (F.obj (op U))) :
      CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.map (homOfLE h).op) : (a : CC (G.obj (op V))) → CC (G.obj (op U)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V)) : (a : CC (F.obj (op V))) → CC (G.obj (op V))) x :
          CC (G.obj (op V))) :
      CC (G.obj (op U)))
have rw₁ : Eq (α := CC (G.obj (op U)))
  ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V) ≫ G.map (homOfLE h).op) :
        (a : CC (F.obj (op V))) → CC (G.obj (op U)))
      x :
    CC (G.obj (op U)))
  ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.map (homOfLE h).op) : (a : CC (G.obj (op V))) → CC (G.obj (op U)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V)) : (a : CC (F.obj (op V))) → CC (G.obj (op V))) x :
        CC (G.obj (op V))) :
    CC (G.obj (op U)))
Tactic: rw [← ConcreteCategory.comp_apply, NatTrans.naturality, ConcreteCategory.comp_apply]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type u_2
inst_1 : (X Y : C) → FunLike.{u_1 + 1, u_2 + 1, u_2 + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory.{u_2, v, u, u_1} C FC (CC := CC)
F G : Presheaf.{w, v, u} C X
e : F ⟶ G
U V : Opens (↑X : Type w)
h : U ≤ V
x : ToType.{u_2, v, u, u_1} (FC := FC) (CC := CC) (F.obj (op V))
rw :
  Eq (α := CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE h).op ≫ e.app (op U)) :
          (a : CC (F.obj (op V))) → CC (G.obj (op U)))
        x :
      CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.map (homOfLE h).op) : (a : CC (G.obj (op V))) → CC (G.obj (op U)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V)) : (a : CC (F.obj (op V))) → CC (G.obj (op V))) x :
          CC (G.obj (op V))) :
      CC (G.obj (op U)))
rw₁ :
  Eq (α := CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V) ≫ G.map (homOfLE h).op) :
          (a : CC (F.obj (op V))) → CC (G.obj (op U)))
        x :
      CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.map (homOfLE h).op) : (a : CC (G.obj (op V))) → CC (G.obj (op U)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V)) : (a : CC (F.obj (op V))) → CC (G.obj (op V))) x :
          CC (G.obj (op V))) :
      CC (G.obj (op U)))
⊢ Eq (α := CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op U)) : (a : CC (F.obj (op U))) → CC (G.obj (op U)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE h).op) :
              (a : CC (F.obj (op V))) → CC (F.obj (op U)))
            x :
          CC (F.obj (op U))) :
      CC (G.obj (op U)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.map (homOfLE h).op) : (a : CC (G.obj (op V))) → CC (G.obj (op U)))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V)) : (a : CC (F.obj (op V))) → CC (G.obj (op V))) x :
          CC (G.obj (op V))) :
      CC (G.obj (op U)))
have rw₂ : Eq (α := CC (G.obj (op U)))
  ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.map (homOfLE h).op) : (a : CC (G.obj (op V))) → CC (G.obj (op U)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V)) : (a : CC (F.obj (op V))) → CC (G.obj (op V))) x :
        CC (G.obj (op V))) :
    CC (G.obj (op U)))
  ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.map (homOfLE h).op) : (a : CC (G.obj (op V))) → CC (G.obj (op U)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (e.app (op V)) : (a : CC (F.obj (op V))) → CC (G.obj (op V))) x :
        CC (G.obj (op V))) :
    CC (G.obj (op U)))
Tactic: rw [← ConcreteCategory.comp_apply, NatTrans.naturality, ConcreteCategory.comp_apply]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : TopCat
f g : X ⟶ Y
h : f = g
ℱ : Presheaf.{w, v, u} C X
⊢ obj (pushforward C f) ℱ = obj (pushforward C g) ℱ
have rw : obj (pushforward C g) ℱ = obj (pushforward C g) ℱ
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
C_1 : Type u
inst : Category.{v, u} C_1
X_1 : TopCat
C : Type u
inst_1 : Category.{v, u} C
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)
inst_3 : ConcreteCategory.{u_2, v, u, u_1} C FC (CC := CC)
inst_4 : HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U : Opens (↑X : Type v)
H :
  IsOpen (X := (↑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)) ''
      (↑U : Set (↑X : Type v)))
x : CostructuredArrow (Opens.map f).op (op U) :=
  CostructuredArrow.mk (Y :=
    op
      {
        carrier :=
          (⇑(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)) ''
            (↑U : Set (↑X : Type v)),
        is_open' := H })
    (Quiver.Hom.op (homOfLE ⋯))
s : Cone (C := CostructuredArrow (Opens.map f).op (op U)) (empty (CostructuredArrow (Opens.map f).op (op U)))
⊢ s.pt ⟶ Cone.pt (asEmptyCone x)
have g : s.pt.left ⟶ (Cone.pt (asEmptyCone x)).left
Tactic: fapply CostructuredArrow.homMk
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
C_1 : Type u
inst : Category.{v, u} C_1
X_1 : TopCat
C : Type u
inst_1 : Category.{v, u} C
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)
inst_3 : ConcreteCategory.{u_2, v, u, u_1} C FC (CC := CC)
inst_4 : HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U : Opens (↑X : Type v)
H :
  IsOpen (X := (↑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)) ''
      (↑U : Set (↑X : Type v)))
x : CostructuredArrow (Opens.map f).op (op U) :=
  CostructuredArrow.mk (Y :=
    op
      {
        carrier :=
          (⇑(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)) ''
            (↑U : Set (↑X : Type v)),
        is_open' := H })
    (Quiver.Hom.op (homOfLE ⋯))
s : Cone (C := CostructuredArrow (Opens.map f).op (op U)) (empty (CostructuredArrow (Opens.map f).op (op U)))
g : s.pt.left ⟶ (Cone.pt (asEmptyCone x)).left
⊢ s.pt ⟶ Cone.pt (asEmptyCone x)
have w : autoParam ((Opens.map f).op.map (id (homOfLE sorry).op) ≫ (Cone.pt (asEmptyCone x)).hom = s.pt.hom)
  CostructuredArrow.homMk._auto_1
Tactic: fapply CostructuredArrow.homMk
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
C_1 : Type u
inst : Category.{v, u} C_1
X_1 : TopCat
C : Type u
inst_1 : Category.{v, u} C
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)
inst_3 : ConcreteCategory.{u_2, v, u, u_1} C FC (CC := CC)
inst_4 : HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U : Opens (↑X : Type v)
H :
  IsOpen (X := (↑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)) ''
      (↑U : Set (↑X : Type v)))
x : CostructuredArrow (Opens.map f).op (op U) :=
  CostructuredArrow.mk (Y :=
    op
      {
        carrier :=
          (⇑(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)) ''
            (↑U : Set (↑X : Type v)),
        is_open' := H })
    (Quiver.Hom.op (homOfLE ⋯))
s : Cone (C := CostructuredArrow (Opens.map f).op (op U)) (empty (CostructuredArrow (Opens.map f).op (op U)))
⊢ s.pt.left ⟶ (Cone.pt (asEmptyCone x)).left
have g : op (unop s.pt.1) ⟶
  op
    {
      carrier :=
        (⇑(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)) ''
          (↑U : Set (↑X : Type v)),
      is_open' := H : Opens (↑Y : Type v) }
Tactic: change op (unop _) ⟶ op (⟨_, H⟩ : Opens _)
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
C_1 : Type u
inst : Category.{v, u} C_1
X_1 : TopCat
C : Type u
inst_1 : Category.{v, u} C
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)
inst_3 : ConcreteCategory.{u_2, v, u, u_1} C FC (CC := CC)
inst_4 : HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U : Opens (↑X : Type v)
H :
  IsOpen (X := (↑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)) ''
      (↑U : Set (↑X : Type v)))
x : CostructuredArrow (Opens.map f).op (op U) :=
  CostructuredArrow.mk (Y :=
    op
      {
        carrier :=
          (⇑(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)) ''
            (↑U : Set (↑X : Type v)),
        is_open' := H })
    (Quiver.Hom.op (homOfLE ⋯))
s : Cone (C := CostructuredArrow (Opens.map f).op (op U)) (empty (CostructuredArrow (Opens.map f).op (op U)))
⊢ op (unop s.pt.1) ⟶
    op
      {
        carrier :=
          (⇑(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)) ''
            (↑U : Set (↑X : Type v)),
        is_open' := H : Opens (↑Y : Type v) }
have g : {
    carrier :=
      (⇑(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)) ''
        (↑U : Set (↑X : Type v)),
    is_open' := H : Opens (↑Y : Type v) } ≤
  unop s.pt.1
Tactic: refine (homOfLE ?_).op
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
C_1 : Type u
inst : Category.{v, u} C_1
X_1 : TopCat
C : Type u
inst_1 : Category.{v, u} C
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)
inst_3 : ConcreteCategory.{u_2, v, u, u_1} C FC (CC := CC)
inst_4 : HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U : Opens (↑X : Type v)
H :
  IsOpen (X := (↑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)) ''
      (↑U : Set (↑X : Type v)))
x : CostructuredArrow (Opens.map f).op (op U) :=
  CostructuredArrow.mk (Y :=
    op
      {
        carrier :=
          (⇑(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)) ''
            (↑U : Set (↑X : Type v)),
        is_open' := H })
    (Quiver.Hom.op (homOfLE ⋯))
s : Cone (C := CostructuredArrow (Opens.map f).op (op U)) (empty (CostructuredArrow (Opens.map f).op (op U)))
⊢ {
      carrier :=
        (⇑(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)) ''
          (↑U : Set (↑X : Type v)),
      is_open' := H : Opens (↑Y : Type v) } ≤
    unop s.pt.1
have g : (⇑(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)) ''
    (↑(unop ((Opens.map f).op.obj s.pt.left)) : Set (↑X : Type v)) ⊆
  (↑(unop s.pt.1) : Set (↑Y : Type v))
Tactic: apply (Set.image_mono s.pt.hom.unop.le).trans
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
C_1 : Type u
inst : Category.{v, u} C_1
X_1 : TopCat
C : Type u
inst_1 : Category.{v, u} C
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)
inst_3 : ConcreteCategory.{u_2, v, u, u_1} C FC (CC := CC)
inst_4 : HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
ℱ : Presheaf C Y
U : Opens (↑X : Type v)
H :
  IsOpen (X := (↑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)) ''
      (↑U : Set (↑X : Type v)))
x : CostructuredArrow (Opens.map f).op (op U) :=
  CostructuredArrow.mk (Y :=
    op
      {
        carrier :=
          (⇑(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)) ''
            (↑U : Set (↑X : Type v)),
        is_open' := H })
    (Quiver.Hom.op (homOfLE ⋯))
⊢ (obj (pullback C f) ℱ).obj (op U) ≅
    ℱ.obj
      (op
        {
          carrier :=
            (⇑(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)) ''
              (↑U : Set (↑X : Type v)),
          is_open' := H })
have have : (hx : IsTerminal x) →
  (obj (pullback C f) ℱ).obj (op U) ≅
    ℱ.obj
      (op
        {
          carrier :=
            (⇑(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)) ''
              (↑U : Set (↑X : Type v)),
          is_open' := H })
Tactic: have hx : IsTerminal x :=
    { lift := fun s ↦ by
        fapply CostructuredArrow.homMk
        · change op (unop _) ⟶ op (⟨_, H⟩ : Opens _)
          refine (homOfLE ?_).op
          apply (Set.image_mono s.pt.hom.unop.le).trans
          exact Set.image_preimage.l_u_le (SetLike.coe s.pt.left.unop)
        · simp [eq_iff_true_of_subsingleton] }
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (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 V : Opens (↑X : Type v)
HU :
  IsOpen (X := (↑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)) ''
      (↑U : Set (↑X : Type v)))
HV :
  IsOpen (X := (↑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)) ''
      (↑V : Set (↑X : Type v)))
le : U ≤ V
⊢ (obj (pullback C f) ℱ).map (homOfLE le).op ≫ Iso.hom (pullbackObjObjOfImageOpen f ℱ U HU) =
    Iso.hom (pullbackObjObjOfImageOpen f ℱ V HV) ≫ ℱ.map (IsOpenMap.functorMap HU HV le).op
have dsimp : (obj (pullback C f) ℱ).map (homOfLE le).op ≫
    Iso.hom
      (IsColimit.coconePointUniqueUpToIso ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op U))
        (colimitOfDiagramTerminal
          {
            lift :=
              fun
                (s :
                  Cone (C := CostructuredArrow (Opens.map f).op (op U))
                    (empty (CostructuredArrow (Opens.map f).op (op U)))) ↦
              CostructuredArrow.homMk (homOfLE sorry).op sorry,
            fac := sorry, uniq := sorry :
            IsLimit (C := CostructuredArrow (Opens.map f).op (op U))
              (asEmptyCone
                (CostructuredArrow.mk (Y :=
                  op
                    {
                      carrier :=
                        (⇑(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)) ''
                          (↑U : Set (↑X : Type v)),
                      is_open' := HU : Opens (↑Y : Type v) })
                  (homOfLE sorry).op)) }
          (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ))) =
  Iso.hom
      (IsColimit.coconePointUniqueUpToIso ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op V))
        (colimitOfDiagramTerminal
          {
            lift :=
              fun
                (s :
                  Cone (C := CostructuredArrow (Opens.map f).op (op V))
                    (empty (CostructuredArrow (Opens.map f).op (op V)))) ↦
              CostructuredArrow.homMk (homOfLE sorry).op sorry,
            fac := sorry, uniq := sorry }
          (CostructuredArrow.proj (Opens.map f).op (op V) ⋙ ℱ))) ≫
    ℱ.map (IsOpenMap.functorMap HU HV le).op
Tactic: dsimp [pullbackObjObjOfImageOpen]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Limits.HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
hf :
  IsOpenMap (X := (↑X : Type v)) (Y := (↑Y : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type v)) → (↑Y : Type v))
ℱ 𝒢 : TopCat.Presheaf C Y
u : ℱ ⟶ 𝒢
⊢ Eq (α := obj (pullback C f) ℱ ⟶ hf.functor.op ⋙ 𝒢)
    (CategoryTheory.Functor.map (pullback C f) u ≫ Iso.hom (hf.pullbackObjIso 𝒢))
    (Iso.hom (hf.pullbackObjIso ℱ) ≫ hf.functor.op.whiskerLeft u)
have w : ∀ (U : Opens (↑X : Type v)),
  Eq (α := (obj (pullback C f) ℱ).obj (op U) ⟶ (hf.functor.op ⋙ 𝒢).obj (op U))
    (NatTrans.app (CategoryTheory.Functor.map (pullback C f) u ≫ Iso.hom (hf.pullbackObjIso 𝒢)) (op U))
    (NatTrans.app (Iso.hom (hf.pullbackObjIso ℱ) ≫ hf.functor.op.whiskerLeft u) (op U))
Tactic: ext U
-- Lean 4 proof step from Mathlib.Topology.Sheaves.Presheaf (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Limits.HasColimits.{v, u} C
X Y : TopCat
f : X ⟶ Y
hf :
  IsOpenMap (X := (↑X : Type v)) (Y := (↑Y : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) f) :
      (a : (↑X : Type v)) → (↑Y : Type v))
ℱ 𝒢 : TopCat.Presheaf C Y
u : ℱ ⟶ 𝒢
U : Opens (↑X : Type v)
⊢ Eq (α := (obj (pullback C f) ℱ).obj (op U) ⟶ (hf.functor.op ⋙ 𝒢).obj (op U))
    (NatTrans.app (CategoryTheory.Functor.map (pullback C f) u ≫ Iso.hom (hf.pullbackObjIso 𝒢)) (op U))
    (NatTrans.app (Iso.hom (hf.pullbackObjIso ℱ) ≫ hf.functor.op.whiskerLeft u) (op U))
have w : Eq (α := (obj (pullback C f) ℱ).obj (op U) ⟶ 𝒢.obj (op (hf.functor.obj U)))
  (NatTrans.app (CategoryTheory.Functor.map (pullback C f) u) (op U) ≫
    Iso.hom
      (Limits.IsColimit.coconePointUniqueUpToIso
        ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit 𝒢 (op U))
        (Limits.colimitOfDiagramTerminal
          {
            lift :=
              fun
                (s :
                  Limits.Cone (C := CostructuredArrow (Opens.map f).op (op U))
                    (empty (CostructuredArrow (Opens.map f).op (op U)))) ↦
              CostructuredArrow.homMk (homOfLE sorry).op sorry,
            fac := sorry, uniq := sorry }
          (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ 𝒢))))
  (CategoryStruct.comp (Y := ℱ.obj (op (hf.functor.obj U)))
    (Iso.hom
        (Limits.IsColimit.coconePointUniqueUpToIso
          ((Opens.map f).op.isPointwiseLeftKanExtensionLeftKanExtensionUnit ℱ (op U))
          (Limits.colimitOfDiagramTerminal
            {
              lift :=
                fun
                  (s :
                    Limits.Cone (C := CostructuredArrow (Opens.map f).op (op U))
                      (empty (CostructuredArrow (Opens.map f).op (op U)))) ↦
                CostructuredArrow.homMk (homOfLE sorry).op sorry,
              fac := sorry, uniq := sorry }
            (CostructuredArrow.proj (Opens.map f).op (op U) ⋙ ℱ))) :
      (obj (pullback C f) ℱ).obj (op U) ⟶ ℱ.obj (op (hf.functor.obj U)))
    (u.app (op (hf.functor.obj U)) : ℱ.obj (op (hf.functor.obj U)) ⟶ 𝒢.obj (op (hf.functor.obj U))))
Tactic: dsimp [pullbackObjIso, pullbackObjObjOfImageOpen]