awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Sheaves.CommRingCat (goal)
Goal:
X : TopCat
C : Type u
inst_1 : Category.{v, u} C
F_1 : Presheaf CommRingCat X
G : F_1.SubmonoidPresheaf
F : Sheaf CommRingCat X
U : (Opens (↑X : Type w))ᵒᵖ
m : Submonoid (↑(F.presheaf.obj U) : Type w) :=
  (F.presheaf.submonoidPresheafOfStalk fun (x : (↑X : Type w)) ↦ (↑(F.presheaf.stalk x) : Type w)⁰).obj U
s : (↑(F.presheaf.obj U) : Type w)
hs : s ∈ m
t : (↑(F.presheaf.obj U) : Type w)
e : t * s = 0
x : (↑X : Type w)
hx : x ∈ unop U
inst :
  Eq (α :=
    (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
            CommRingCat.carrier) (Y := F.presheaf.stalk x) (F.presheaf.germ (unop U) x hx) :
          (a : (↑(F.presheaf.obj (op (unop U))) : Type w)) → (↑(F.presheaf.stalk (↑⟨x, hx⟩ : (↑X : Type w))) : Type w))
        (t * s) :
      (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
    (0 :
      (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
⊢ HMul.hMul (α :=
      (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
      (β :=
      (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
              CommRingCat.carrier) (Y := F.presheaf.stalk x) (F.presheaf.germ (unop U) x hx) :
            (a : (↑(F.presheaf.obj (op (unop U))) : Type w)) → (↑(F.presheaf.stalk x) : Type w))
          t :
        (↑(F.presheaf.stalk x) : Type w))
      ((CommRingCat.Hom.hom
              (F.presheaf.germ (unop U) (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))
                ⋯) :
            (a : (↑(F.presheaf.obj U) : Type w)) → (↑(F.presheaf.stalk (↑⟨x, hx⟩ : (↑X : Type w))) : Type w))
          s :
        (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) :
          Type w)) =
    0
have inst₁ : Eq (α :=
  (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
  ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
          CommRingCat.carrier) (Y := F.presheaf.stalk x) (F.presheaf.germ (unop U) x hx) :
        (a : (↑(F.presheaf.obj (op (unop U))) : Type w)) → (↑(F.presheaf.stalk (↑⟨x, hx⟩ : (↑X : Type w))) : Type w))
      0 :
    (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
  (0 : (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
Tactic: rw [← map_mul, e, map_zero]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.CommRingCat (goal)
Goal:
X : TopCat
C : Type u
inst_1 : Category.{v, u} C
F_1 : Presheaf CommRingCat X
G : F_1.SubmonoidPresheaf
F : Sheaf CommRingCat X
U : (Opens (↑X : Type w))ᵒᵖ
m : Submonoid (↑(F.presheaf.obj U) : Type w) :=
  (F.presheaf.submonoidPresheafOfStalk fun (x : (↑X : Type w)) ↦ (↑(F.presheaf.stalk x) : Type w)⁰).obj U
s : (↑(F.presheaf.obj U) : Type w)
hs : s ∈ m
t : (↑(F.presheaf.obj U) : Type w)
e : t * s = 0
x : (↑X : Type w)
hx : x ∈ unop U
inst :
  Eq (α :=
    (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
            CommRingCat.carrier) (Y := F.presheaf.stalk x) (F.presheaf.germ (unop U) x hx) :
          (a : (↑(F.presheaf.obj (op (unop U))) : Type w)) → (↑(F.presheaf.stalk (↑⟨x, hx⟩ : (↑X : Type w))) : Type w))
        (t * s) :
      (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
    (0 :
      (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
inst₁ :
  Eq (α :=
    (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
            CommRingCat.carrier) (Y := F.presheaf.stalk x) (F.presheaf.germ (unop U) x hx) :
          (a : (↑(F.presheaf.obj (op (unop U))) : Type w)) → (↑(F.presheaf.stalk (↑⟨x, hx⟩ : (↑X : Type w))) : Type w))
        0 :
      (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
    (0 :
      (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
⊢ HMul.hMul (α :=
      (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
      (β :=
      (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
              CommRingCat.carrier) (Y := F.presheaf.stalk x) (F.presheaf.germ (unop U) x hx) :
            (a : (↑(F.presheaf.obj (op (unop U))) : Type w)) → (↑(F.presheaf.stalk x) : Type w))
          t :
        (↑(F.presheaf.stalk x) : Type w))
      ((CommRingCat.Hom.hom
              (F.presheaf.germ (unop U) (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))
                ⋯) :
            (a : (↑(F.presheaf.obj U) : Type w)) → (↑(F.presheaf.stalk (↑⟨x, hx⟩ : (↑X : Type w))) : Type w))
          s :
        (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) :
          Type w)) =
    0
have inst₂ : Eq.{w + 1} (α :=
  (↑(F.presheaf.stalk (↑(Subtype.mk (p := fun (x : (↑X : Type w)) ↦ x ∈ unop U) x hx) : (↑X : Type w))) : Type w)) 0 0
Tactic: rw [← map_mul, e, map_zero]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.CommRingCat (goal)
Goal:
X_1 : TopCat
R_1 : TopCommRingCat
R : TopCommRingCat
X : TopCatᵒᵖ
⊢ continuousFunctions.pullback (𝟙 X) R = 𝟙 (continuousFunctions X R)
have hf : ∀ (x : (↑(continuousFunctions X R) : Type u)),
  Eq (α := (↑(continuousFunctions X R) : Type u))
    ((CommRingCat.Hom.hom (continuousFunctions.pullback (𝟙 X) R) :
          (a : (↑(continuousFunctions X R) : Type u)) → (↑(continuousFunctions X R) : Type u))
        x :
      (↑(continuousFunctions X R) : Type u))
    ((CommRingCat.Hom.hom (𝟙 (continuousFunctions X R)) :
          (a : (↑(continuousFunctions X R) : Type u)) → (↑(continuousFunctions X R) : Type u))
        x :
      (↑(continuousFunctions X R) : Type u))
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
g : F.EtaleSpace (CC := CC) (FC := FC)
U : Opens (↑X : Type v)
h : g.base (CC := CC) (FC := FC) ∈ U
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
hs :
  (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (g.base (CC := CC) (FC := FC)) h) :
        (a : CC (F.obj (op U))) → CC (F.stalk g.base))
      s =
    g.germ (CC := CC) (FC := FC)
⊢ ∀ᶠ (g' : F.EtaleSpace (CC := CC) (FC := FC)) in 𝓝 g,
    ∃ (hgU : g'.base (CC := CC) (FC := FC) ∈ U),
      g'.germ (CC := CC) (FC := FC) =
        (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (g'.base (CC := CC) (FC := FC)) hgU) :
            (a : CC (F.obj (op U))) → CC (F.stalk g'.base))
          s
have simp : Membership.mem (γ := Filter (F.EtaleSpace (CC := CC) (FC := FC)))
  (⨅ (s : Set (F.EtaleSpace (CC := CC) (FC := FC))),
    ⨅ (_ : g ∈ s),
      ⨅ (i : Opens (↑X : Type v)),
        ⨅ (i_1 : ToType (FC := FC) (CC := CC) (F.obj (op i))),
          ⨅ (_ :
            {g : F.EtaleSpace (CC := CC) (FC := FC) |
                ∃ (h : g.base (CC := CC) (FC := FC) ∈ i),
                  g.germ (CC := CC) (FC := FC) =
                    (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ i (g.base (CC := CC) (FC := FC)) h) :
                        (a : CC (F.obj (op i))) → CC (F.stalk g.base))
                      i_1} =
              s),
            𝓟 s)
  {x : F.EtaleSpace (CC := CC) (FC := FC) |
    ∃ (hgU : x.base (CC := CC) (FC := FC) ∈ U),
      x.germ (CC := CC) (FC := FC) =
        (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (x.base (CC := CC) (FC := FC)) hgU) :
            (a : CC (F.obj (op U))) → CC (F.stalk x.base))
          s}
Tactic: simp only [nhds_generateFrom, Filter.Eventually, mem_setOf_eq, iInf_and, iInf_exists]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
g : F.EtaleSpace (CC := CC) (FC := FC)
U : Opens (↑X : Type v)
h : g.base (CC := CC) (FC := FC) ∈ U
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
hs :
  (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (g.base (CC := CC) (FC := FC)) h) :
        (a : CC (F.obj (op U))) → CC (F.stalk g.base))
      s =
    g.germ (CC := CC) (FC := FC)
⊢ Membership.mem (γ := Filter (F.EtaleSpace (CC := CC) (FC := FC)))
    (⨅ (s : Set (F.EtaleSpace (CC := CC) (FC := FC))),
      ⨅ (_ : g ∈ s),
        ⨅ (i : Opens (↑X : Type v)),
          ⨅ (i_1 : ToType (FC := FC) (CC := CC) (F.obj (op i))),
            ⨅ (_ :
              {g : F.EtaleSpace (CC := CC) (FC := FC) |
                  ∃ (h : g.base (CC := CC) (FC := FC) ∈ i),
                    g.germ (CC := CC) (FC := FC) =
                      (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ i (g.base (CC := CC) (FC := FC)) h) :
                          (a : CC (F.obj (op i))) → CC (F.stalk g.base))
                        i_1} =
                s),
              𝓟 s)
    {x : F.EtaleSpace (CC := CC) (FC := FC) |
      ∃ (hgU : x.base (CC := CC) (FC := FC) ∈ U),
        x.germ (CC := CC) (FC := FC) =
          (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (x.base (CC := CC) (FC := FC)) hgU) :
              (a : CC (F.obj (op U))) → CC (F.stalk x.base))
            s}
have refine : Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
  {g : F.EtaleSpace (CC := CC) (FC := FC) |
    ∃ (h : g.base (CC := CC) (FC := FC) ∈ U),
      g.germ (CC := CC) (FC := FC) =
        (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (g.base (CC := CC) (FC := FC)) h) :
            (a : CC (F.obj (op U))) → CC (F.stalk g.base))
          s}
  g
Tactic: refine mem_iInf_of_mem _ <| mem_iInf_of_mem ?_ <| mem_iInf_of_mem U <| mem_iInf_of_mem s <|
    mem_iInf_of_mem rfl <| mem_principal_self _
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
⊢ Continuous (X := F.EtaleSpace (CC := CC) (FC := FC)) (base (CC := CC) (FC := FC))
have rw : ∀ (x : F.EtaleSpace (CC := CC) (FC := FC)), ContinuousAt (base (CC := CC) (FC := FC)) x
Tactic: rw [continuous_iff_continuousAt]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
x : F.EtaleSpace (CC := CC) (FC := FC)
⊢ ContinuousAt (base (CC := CC) (FC := FC)) x
have rw : ∀ (i : Set (↑X : Type v)),
  x.base (CC := CC) (FC := FC) ∈ i ∧ IsOpen i →
    ∀ᶠ (x : F.EtaleSpace (CC := CC) (FC := FC)) in 𝓝 x, x.base (CC := CC) (FC := FC) ∈ i
Tactic: rw [ContinuousAt, (nhds_basis_opens _).tendsto_right_iff]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
x : F.EtaleSpace (CC := CC) (FC := FC)
U : Set (↑X : Type v)
hxU : x.base (CC := CC) (FC := FC) ∈ U
hUo : IsOpen U
⊢ ∀ᶠ (x : F.EtaleSpace (CC := CC) (FC := FC)) in 𝓝 x, x.base (CC := CC) (FC := FC) ∈ U
have lift : ∀ (U : Opens (↑X : Type v)) (hxU : x.base (CC := CC) (FC := FC) ∈ (↑U : Set (↑X : Type v))),
  ∀ᶠ (x : F.EtaleSpace (CC := CC) (FC := FC)) in 𝓝 x, x.base (CC := CC) (FC := FC) ∈ (↑U : Set (↑X : Type v))
Tactic: lift U to Opens X using hUo
Cleared hypotheses: U, hxU, hUo
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
x : F.EtaleSpace (CC := CC) (FC := FC)
U : Opens (↑X : Type v)
hxU : x.base (CC := CC) (FC := FC) ∈ (↑U : Set (↑X : Type v))
⊢ ∀ᶠ (x : F.EtaleSpace (CC := CC) (FC := FC)) in 𝓝 x, x.base (CC := CC) (FC := FC) ∈ (↑U : Set (↑X : Type v))
have rcases : ∀ (V : Opens (↑X : Type v)) (hVU : V ≤ U) (hxV : x.base (CC := CC) (FC := FC) ∈ V)
  (f : ToType (FC := FC) (CC := CC) (F.obj (op V)))
  (hf :
    (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V (x.base (CC := CC) (FC := FC)) hxV) :
          (a : CC (F.obj (op V))) → CC (F.stalk x.base))
        f =
      x.germ (CC := CC) (FC := FC)),
  ∀ᶠ (x : F.EtaleSpace (CC := CC) (FC := FC)) in 𝓝 x, x.base (CC := CC) (FC := FC) ∈ (↑U : Set (↑X : Type v))
Tactic: rcases F.exists_le_germ_eq x.germ hxU with ⟨V, hVU, hxV, f, hf⟩
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
x : F.EtaleSpace (CC := CC) (FC := FC)
U : Opens (↑X : Type v)
hxU : x.base (CC := CC) (FC := FC) ∈ (↑U : Set (↑X : Type v))
V : Opens (↑X : Type v)
hVU : V ≤ U
hxV : x.base (CC := CC) (FC := FC) ∈ V
f : ToType (FC := FC) (CC := CC) (F.obj (op V))
hf :
  (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V (x.base (CC := CC) (FC := FC)) hxV) :
        (a : CC (F.obj (op V))) → CC (F.stalk x.base))
      f =
    x.germ (CC := CC) (FC := FC)
⊢ ∀ᶠ (x : F.EtaleSpace (CC := CC) (FC := FC)) in 𝓝 x, x.base (CC := CC) (FC := FC) ∈ (↑U : Set (↑X : Type v))
have refine : ∀ (x : F.EtaleSpace (CC := CC) (FC := FC)),
  (∃ (hgU : x.base (CC := CC) (FC := FC) ∈ V),
      x.germ (CC := CC) (FC := FC) =
        (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V (x.base (CC := CC) (FC := FC)) hgU) :
            (a : CC (F.obj (op V))) → CC (F.stalk x.base))
          f) →
    x.base (CC := CC) (FC := FC) ∈ (↑U : Set (↑X : Type v))
Tactic: refine x.eventually_nhds hxV f hf |>.mono ?_
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
α : Type u_1
l : Filter α
g : α → F.EtaleSpace (CC := CC) (FC := FC)
g₀ : F.EtaleSpace (CC := CC) (FC := FC)
h : Tendsto g l (𝓝 g₀)
⊢ ∃ (U : Opens (↑X : Type v)),
    g₀.base (CC := CC) (FC := FC) ∈ U ∧
      ∃ (f : ToType (FC := FC) (CC := CC) (F.obj (op U))),
        ∀ᶠ (a : α) in l,
          ∃ (ha : (g a).base (CC := CC) (FC := FC) ∈ U),
            (g a).germ (CC := CC) (FC := FC) =
              (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U ((g a).base (CC := CC) (FC := FC)) ha) :
                  (a_1 : CC (F.obj (op U))) → CC (F.stalk (g a).base))
                f
have rcases : ∀ (U : Opens (↑X : Type v)) (hU : g₀.base (CC := CC) (FC := FC) ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (op U)))
  (hs :
    (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (g₀.base (CC := CC) (FC := FC)) hU) :
          (a : CC (F.obj (op U))) → CC (F.stalk g₀.base))
        s =
      g₀.germ (CC := CC) (FC := FC)),
  ∃ (U : Opens (↑X : Type v)),
    g₀.base (CC := CC) (FC := FC) ∈ U ∧
      ∃ (f : ToType (FC := FC) (CC := CC) (F.obj (op U))),
        ∀ᶠ (a : α) in l,
          ∃ (ha : (g a).base (CC := CC) (FC := FC) ∈ U),
            (g a).germ (CC := CC) (FC := FC) =
              (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U ((g a).base (CC := CC) (FC := FC)) ha) :
                  (a_1 : CC (F.obj (op U))) → CC (F.stalk (g a).base))
                f
Tactic: rcases F.exists_germ_eq g₀.germ with ⟨U, hU, s, hs⟩
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
α : Type u_1
l : Filter α
g : α → F.EtaleSpace (CC := CC) (FC := FC)
g₀ : F.EtaleSpace (CC := CC) (FC := FC)
h : Tendsto g l (𝓝 g₀)
U : Opens (↑X : Type v)
hU : g₀.base (CC := CC) (FC := FC) ∈ U
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
hs :
  (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U (g₀.base (CC := CC) (FC := FC)) hU) :
        (a : CC (F.obj (op U))) → CC (F.stalk g₀.base))
      s =
    g₀.germ (CC := CC) (FC := FC)
⊢ ∃ (U : Opens (↑X : Type v)),
    g₀.base (CC := CC) (FC := FC) ∈ U ∧
      ∃ (f : ToType (FC := FC) (CC := CC) (F.obj (op U))),
        ∀ᶠ (a : α) in l,
          ∃ (ha : (g a).base (CC := CC) (FC := FC) ∈ U),
            (g a).germ (CC := CC) (FC := FC) =
              (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U ((g a).base (CC := CC) (FC := FC)) ha) :
                  (a_1 : CC (F.obj (op U))) → CC (F.stalk (g a).base))
                f
have h₁ : ∀ᶠ (a : α) in l,
  ∃ (ha : (g a).base (CC := CC) (FC := FC) ∈ U),
    (g a).germ (CC := CC) (FC := FC) =
      (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U ((g a).base (CC := CC) (FC := FC)) ha) :
          (a_1 : CC (F.obj (op U))) → CC (F.stalk (g a).base))
        s
Tactic: use U, hU, s
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
U : Opens (↑X : Type v)
hF_bij :
  ∀ (x : (↑X : Type v)) (hx : x ∈ U),
    Bijective.{v + 1, v + 1} (α := CC (F.obj (op U))) (β := CC (F.stalk x))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx)) : (a : CC (F.obj (op U))) → CC (F.stalk x))
x : (↑X : Type v)
hx : x ∈ U
base : (↑X : Type v)
s : ToType (FC := FC) (CC := CC) (F.stalk base)
hs :
  Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
    (EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) (mk (CC := CC) (FC := FC) base s)
⊢ Subtype.mk (p := fun (x : F.EtaleSpace (CC := CC) (FC := FC)) ↦
      Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
        (EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) x)
      (mk (CC := CC) (FC := FC) base
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U base ⋯) : (a : CC (F.obj (op U))) → CC (F.stalk base))
          (surjInv ⋯
            ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx) : (a : CC (F.obj (op U))) → CC (F.stalk x))
              (surjInv ⋯ s)))))
      ⋯ =
    Subtype.mk (p := fun (x : F.EtaleSpace (CC := CC) (FC := FC)) ↦
      Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
        (EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) x)
      (mk (CC := CC) (FC := FC) base s) hs
have e_val : (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U base sorry) : (a : CC (F.obj (op U))) → CC (F.stalk base))
    (surjInv sorry
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx) : (a : CC (F.obj (op U))) → CC (F.stalk x))
        (surjInv sorry s))) =
  s
Tactic: congr 2
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
U : Opens (↑X : Type v)
hF_bij :
  ∀ (x : (↑X : Type v)) (hx : x ∈ U),
    Bijective.{v + 1, v + 1} (α := CC (F.obj (op U))) (β := CC (F.stalk x))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx)) : (a : CC (F.obj (op U))) → CC (F.stalk x))
x : (↑X : Type v)
hx : x ∈ U
base : (↑X : Type v)
s : ToType (FC := FC) (CC := CC) (F.stalk base)
hs :
  Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
    (EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) (mk (CC := CC) (FC := FC) base s)
⊢ (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U base ⋯) : (a : CC (F.obj (op U))) → CC (F.stalk base))
      (surjInv ⋯
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx) : (a : CC (F.obj (op U))) → CC (F.stalk x))
          (surjInv ⋯ s))) =
    s
have e_val : (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U base sorry) : (a : CC (F.obj (op U))) → CC (F.stalk base))
    (surjInv sorry s) =
  s
Tactic: rw [leftInverse_surjInv (hF_bij _ _), surjInv_eq (hF_bij _ _).surjective]
-- Lean 4 proof step from Mathlib.Topology.Sheaves.EtaleSpace (goal)
Goal:
X : TopCat
C : Type u
inst : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst_1 : (X Y : C) → FunLike.{w + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
inst_2 : ConcreteCategory C FC (CC := CC)
inst_3 : Limits.HasColimits.{v, u} C
F : Presheaf C X
inst_4 : Limits.PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
U : Opens (↑X : Type v)
hF_bij :
  ∀ (x : (↑X : Type v)) (hx : x ∈ U),
    Bijective.{v + 1, v + 1} (α := CC (F.obj (op U))) (β := CC (F.stalk x))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx)) : (a : CC (F.obj (op U))) → CC (F.stalk x))
x : (↑X : Type v)
hx : x ∈ U
base : (↑X : Type v)
s : ToType (FC := FC) (CC := CC) (F.stalk base)
hs :
  Membership.mem (γ := Set (F.EtaleSpace (CC := CC) (FC := FC)))
    (EtaleSpace.base (CC := CC) (FC := FC) ⁻¹' (↑U : Set (↑X : Type v))) (mk (CC := CC) (FC := FC) base s)
e_val :
  (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U base ⋯) : (a : CC (F.obj (op U))) → CC (F.stalk base))
      (surjInv ⋯ s) =
    s
⊢ (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U base ⋯) : (a : CC (F.obj (op U))) → CC (F.stalk base))
      (surjInv ⋯
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hx) : (a : CC (F.obj (op U))) → CC (F.stalk x))
          (surjInv ⋯ s))) =
    s
have e_val₁ : s = s
Tactic: rw [leftInverse_surjInv (hF_bij _ _), surjInv_eq (hF_bij _ _).surjective]