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 : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
F G : Presheaf C X
f : F ⟶ G
h :
  ∀ (U : Opens (↑X : Type v)),
    Function.Injective.{v + 1, v + 1} (α := CC (F.obj (op U))) (β := CC (G.obj (op U)))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (f.app (op U))) : (a : CC (F.obj (op U))) → CC (G.obj (op U)))
x : (↑X : Type v)
U₁ : Opens (↑X : Type v)
hxU₁ : x ∈ U₁
s : ToType (FC := FC) (CC := CC) (F.obj (op U₁))
U₂ : Opens (↑X : Type v)
hxU₂ : x ∈ U₂
t : ToType (FC := FC) (CC := CC) (F.obj (op U₂))
hst :
  Eq (α := CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.germ U₁ x hxU₁) : (a : CC (G.obj (op U₁))) → CC (G.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.app (op U₁)) : (a : CC (F.obj (op U₁))) → CC (G.obj (op U₁)))
            s :
          CC (G.obj (op U₁))) :
      CC (G.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (G.germ U₂ x hxU₂) : (a : CC (G.obj (op U₂))) → CC (G.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (f.app (op U₂)) : (a : CC (F.obj (op U₂))) → CC (G.obj (op U₂)))
            t :
          CC (G.obj (op U₂))) :
      CC (G.stalk x))
W : Opens (↑X : Type v)
hxW : x ∈ W
iWU₁ : W ⟶ U₁
iWU₂ : W ⟶ U₂
heq :
  Eq (α := CC (F.obj (op W)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map iWU₁.op) : (a : CC (F.obj (op U₁))) → CC (F.obj (op W))) s :
      CC (F.obj (op W)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map iWU₂.op) : (a : CC (F.obj (op U₂))) → CC (F.obj (op W))) t :
      CC (F.obj (op W)))
⊢ Eq (α := CC (obj (stalkFunctor C x) F))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₁ x hxU₁) : (a : CC (F.obj (op U₁))) → CC (F.stalk x)) s :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₂ x hxU₂) : (a : CC (F.obj (op U₂))) → CC (F.stalk x)) t :
      CC (F.stalk x))
have h₁ : CC (obj (stalkFunctor C x) F) = CC (F.stalk x) →
  Eq (α := CC (obj (stalkFunctor C x) F))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₁ x hxU₁) : (a : CC (F.obj (op U₁))) → CC (F.stalk x)) s :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ W x hxW) : (a : CC (F.obj (op W))) → CC (F.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map iWU₁.op) : (a : CC (F.obj (op U₁))) → CC (F.obj (op W))) s :
          CC (F.obj (op W))) :
      CC (F.stalk x))
Tactic: convert! congr_arg (F.germ _ x hxW) heq using 1
-- 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 : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
B : Set (Opens (↑X : Type v))
hB : Opens.IsBasis B
F : Presheaf C X
x : (↑X : Type v)
t : ToType (FC := FC) (CC := CC) (F.stalk x)
⊢ ∃ (U : Opens (↑X : Type v)) (m : x ∈ U) (_ : U ∈ B) (s : ToType (FC := FC) (CC := CC) (F.obj (op U))),
    (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x m) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s = t
have obtain : ∀ (U : Opens (↑X : Type v)) (hxU : x ∈ U) (s : ToType (FC := FC) (CC := CC) (F.obj (op U))),
  ∃ (U_1 : Opens (↑X : Type v)) (m : x ∈ U_1) (_ : U_1 ∈ B) (s_1 : ToType (FC := FC) (CC := CC) (F.obj (op U_1))),
    Eq (α := CC (F.stalk x))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U_1 x m) : (a : CC (F.obj (op U_1))) → CC (F.stalk x)) s_1 :
        CC (F.stalk x))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
        CC (F.stalk x))
Tactic: obtain ⟨U, hxU, s, rfl⟩ := F.exists_germ_eq t
Cleared hypotheses: t
-- 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 : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
B : Set (Opens (↑X : Type v))
hB : Opens.IsBasis B
F : Presheaf C X
x : (↑X : Type v)
U : Opens (↑X : Type v)
hxU : x ∈ U
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
⊢ ∃ (U_1 : Opens (↑X : Type v)) (m : x ∈ U_1) (_ : U_1 ∈ B) (s_1 : ToType (FC := FC) (CC := CC) (F.obj (op U_1))),
    Eq (α := CC (F.stalk x))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U_1 x m) : (a : CC (F.obj (op U_1))) → CC (F.stalk x)) s_1 :
        CC (F.stalk x))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
        CC (F.stalk x))
have obtain : ∀ (V : Opens (↑X : Type v)) (hV : V ∈ B) (hxV : x ∈ (↑V : Set (↑X : Type v)))
  (hVU : (↑V : Set (↑X : Type v)) ⊆ (↑U : Set (↑X : Type v))),
  ∃ (U_1 : Opens (↑X : Type v)) (m : x ∈ U_1) (_ : U_1 ∈ B) (s_1 : ToType (FC := FC) (CC := CC) (F.obj (op U_1))),
    Eq (α := CC (F.stalk x))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U_1 x m) : (a : CC (F.obj (op U_1))) → CC (F.stalk x)) s_1 :
        CC (F.stalk x))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
        CC (F.stalk x))
Tactic: obtain ⟨_, ⟨V, hV, rfl⟩, hxV, hVU⟩ := hB.exists_subset_of_mem_open hxU U.2
-- 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 : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
B : Set (Opens (↑X : Type v))
hB : Opens.IsBasis B
F : Presheaf C X
x : (↑X : Type v)
U : Opens (↑X : Type v)
hxU : x ∈ U
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
V : Opens (↑X : Type v)
hV : V ∈ B
hxV : x ∈ (↑V : Set (↑X : Type v))
hVU : (↑V : Set (↑X : Type v)) ⊆ (↑U : Set (↑X : Type v))
⊢ Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V x hxV) : (a : CC (F.obj (op V))) → CC (F.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC)
                (F.map
                  ((homOfLE (x := (↑V : Set (↑X : Type v))) (y := (↑U : Set (↑X : Type v))) hVU).op (X :=
                    (↑V : Set (↑X : Type v))) (Y := (↑U : Set (↑X : Type v))))) :
              (a : CC (F.obj (op U))) → CC (F.obj (op V)))
            s :
          CC (F.obj (op V))) :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))
have rw : Eq (α := CC (F.stalk x))
  ((ConcreteCategory.hom (FC := FC) (CC := CC)
          (F.map
              ((homOfLE (x := (↑V : Set (↑X : Type v))) (y := (↑U : Set (↑X : Type v))) hVU).op (X :=
                (↑V : Set (↑X : Type v))) (Y := (↑U : Set (↑X : Type v)))) ≫
            F.germ V x hxV) :
        (a : CC (F.obj (op U))) → CC (F.stalk x))
      s :
    CC (F.stalk x))
  ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
    CC (F.stalk x))
Tactic: rw [← ConcreteCategory.comp_apply, F.germ_res']
-- 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 : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
B : Set (Opens (↑X : Type v))
hB : Opens.IsBasis B
F : Presheaf C X
x : (↑X : Type v)
U : Opens (↑X : Type v)
hxU : x ∈ U
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
V : Opens (↑X : Type v)
hV : V ∈ B
hxV : x ∈ (↑V : Set (↑X : Type v))
hVU : (↑V : Set (↑X : Type v)) ⊆ (↑U : Set (↑X : Type v))
rw :
  Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC)
            (F.map
                ((homOfLE (x := (↑V : Set (↑X : Type v))) (y := (↑U : Set (↑X : Type v))) hVU).op (X :=
                  (↑V : Set (↑X : Type v))) (Y := (↑U : Set (↑X : Type v)))) ≫
              F.germ V x hxV) :
          (a : CC (F.obj (op U))) → CC (F.stalk x))
        s :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))
⊢ Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V x hxV) : (a : CC (F.obj (op V))) → CC (F.stalk x))
        ((ConcreteCategory.hom (FC := FC) (CC := CC)
                (F.map
                  ((homOfLE (x := (↑V : Set (↑X : Type v))) (y := (↑U : Set (↑X : Type v))) hVU).op (X :=
                    (↑V : Set (↑X : Type v))) (Y := (↑U : Set (↑X : Type v))))) :
              (a : CC (F.obj (op U))) → CC (F.obj (op V)))
            s :
          CC (F.obj (op V))) :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))
have rw₁ : Eq (α := CC (F.stalk x))
  ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x sorry) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
    CC (F.stalk x))
  ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x hxU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
    CC (F.stalk x))
Tactic: rw [← ConcreteCategory.comp_apply, F.germ_res']
-- 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 : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
B : Set (Opens (↑X : Type v))
hB : Opens.IsBasis B
F : Presheaf C X
U V : Opens (↑X : Type v)
x : (↑X : Type v)
mU : x ∈ U
mV : x ∈ V
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
t : ToType (FC := FC) (CC := CC) (F.obj (op V))
h :
  Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x mU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V x mV) : (a : CC (F.obj (op V))) → CC (F.stalk x)) t :
      CC (F.stalk x))
⊢ ∃ (W : Opens (↑X : Type v)) (_ : x ∈ W) (_ : W ∈ B) (hWU : W ≤ U) (hWV : W ≤ V),
    Eq (α := CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE hWU).op) :
            (a : CC (F.obj (op U))) → CC (F.obj (op W)))
          s :
        CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE hWV).op) :
            (a : CC (F.obj (op V))) → CC (F.obj (op W)))
          t :
        CC (F.obj (op W)))
have obtain : ∀ (W : Opens (↑X : Type v)) (hxW : x ∈ W) (hWU : W ⟶ U) (hWV : W ⟶ V)
  (e :
    Eq (α := CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map hWU.op) : (a : CC (F.obj (op U))) → CC (F.obj (op W))) s :
        CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map hWV.op) : (a : CC (F.obj (op V))) → CC (F.obj (op W))) t :
        CC (F.obj (op W)))),
  ∃ (W : Opens (↑X : Type v)) (_ : x ∈ W) (_ : W ∈ B) (hWU : W ≤ U) (hWV : W ≤ V),
    Eq (α := CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE hWU).op) :
            (a : CC (F.obj (op U))) → CC (F.obj (op W)))
          s :
        CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE hWV).op) :
            (a : CC (F.obj (op V))) → CC (F.obj (op W)))
          t :
        CC (F.obj (op W)))
Tactic: obtain ⟨W, hxW, hWU, hWV, e⟩ := F.germ_eq x mU mV _ _ h
-- 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 : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
B : Set (Opens (↑X : Type v))
hB : Opens.IsBasis B
F : Presheaf C X
U V : Opens (↑X : Type v)
x : (↑X : Type v)
mU : x ∈ U
mV : x ∈ V
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
t : ToType (FC := FC) (CC := CC) (F.obj (op V))
h :
  Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x mU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V x mV) : (a : CC (F.obj (op V))) → CC (F.stalk x)) t :
      CC (F.stalk x))
W : Opens (↑X : Type v)
hxW : x ∈ W
hWU : W ⟶ U
hWV : W ⟶ V
e :
  Eq (α := CC (F.obj (op W)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map hWU.op) : (a : CC (F.obj (op U))) → CC (F.obj (op W))) s :
      CC (F.obj (op W)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map hWV.op) : (a : CC (F.obj (op V))) → CC (F.obj (op W))) t :
      CC (F.obj (op W)))
⊢ ∃ (W : Opens (↑X : Type v)) (_ : x ∈ W) (_ : W ∈ B) (hWU : W ≤ U) (hWV : W ≤ V),
    Eq (α := CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE hWU).op) :
            (a : CC (F.obj (op U))) → CC (F.obj (op W)))
          s :
        CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE hWV).op) :
            (a : CC (F.obj (op V))) → CC (F.obj (op W)))
          t :
        CC (F.obj (op W)))
have obtain : ∀ (W' : Opens (↑X : Type v)) (hW' : W' ∈ B) (hxW' : x ∈ (↑W' : Set (↑X : Type v)))
  (hW'W : (↑W' : Set (↑X : Type v)) ⊆ (↑W : Set (↑X : Type v))),
  ∃ (W : Opens (↑X : Type v)) (_ : x ∈ W) (_ : W ∈ B) (hWU : W ≤ U) (hWV : W ≤ V),
    Eq (α := CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE hWU).op) :
            (a : CC (F.obj (op U))) → CC (F.obj (op W)))
          s :
        CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE hWV).op) :
            (a : CC (F.obj (op V))) → CC (F.obj (op W)))
          t :
        CC (F.obj (op W)))
Tactic: obtain ⟨_, ⟨W', hW', rfl⟩, hxW', hW'W⟩ := hB.exists_subset_of_mem_open hxW W.2
-- 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 : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
B : Set (Opens (↑X : Type v))
hB : Opens.IsBasis B
F : Presheaf C X
U V : Opens (↑X : Type v)
x : (↑X : Type v)
mU : x ∈ U
mV : x ∈ V
s : ToType (FC := FC) (CC := CC) (F.obj (op U))
t : ToType (FC := FC) (CC := CC) (F.obj (op V))
h :
  Eq (α := CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U x mU) : (a : CC (F.obj (op U))) → CC (F.stalk x)) s :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ V x mV) : (a : CC (F.obj (op V))) → CC (F.stalk x)) t :
      CC (F.stalk x))
W : Opens (↑X : Type v)
hxW : x ∈ W
hWU : W ⟶ U
hWV : W ⟶ V
e :
  Eq (α := CC (F.obj (op W)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map hWU.op) : (a : CC (F.obj (op U))) → CC (F.obj (op W))) s :
      CC (F.obj (op W)))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map hWV.op) : (a : CC (F.obj (op V))) → CC (F.obj (op W))) t :
      CC (F.obj (op W)))
W' : Opens (↑X : Type v)
hW' : W' ∈ B
hxW' : x ∈ (↑W' : Set (↑X : Type v))
hW'W : (↑W' : Set (↑X : Type v)) ⊆ (↑W : Set (↑X : Type v))
⊢ ∃ (W : Opens (↑X : Type v)) (_ : x ∈ W) (_ : W ∈ B) (hWU : W ≤ U) (hWV : W ≤ V),
    Eq (α := CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE hWU).op) :
            (a : CC (F.obj (op U))) → CC (F.obj (op W)))
          s :
        CC (F.obj (op W)))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE hWV).op) :
            (a : CC (F.obj (op V))) → CC (F.obj (op W)))
          t :
        CC (F.obj (op W)))
have refine : Eq (α := CC (F.obj (op W')))
  ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE sorry).op) : (a : CC (F.obj (op U))) → CC (F.obj (op W')))
      s :
    CC (F.obj (op W')))
  ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.map (homOfLE sorry).op) : (a : CC (F.obj (op V))) → CC (F.obj (op W')))
      t :
    CC (F.obj (op W')))
Tactic: refine ⟨W', hxW', hW', hW'W.trans hWU.le, hW'W.trans hWV.le, ?_⟩
-- 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 : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
B : Set (Opens (↑X : Type v))
hB : Opens.IsBasis B
F G : Presheaf C X
α : F ⟶ G
hα :
  ∀ U ∈ B,
    Function.Injective.{v + 1, v + 1} (α := CC (F.obj (op U))) (β := CC (G.obj (op U)))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (α.app (op U))) : (a : CC (F.obj (op U))) → CC (G.obj (op U)))
x : (↑X : Type v)
s t : CC (obj (stalkFunctor C x) F)
hst :
  Eq (α := CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) α) :
          (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
        s :
      CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) α) :
          (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
        t :
      CC (obj (stalkFunctor C x) G))
⊢ s = t
have obtain : ∀ (U₁ : Opens (↑X : Type v)) (hxU₁ : x ∈ U₁) (hU₁ : U₁ ∈ B) (s : ToType (FC := FC) (CC := CC) (F.obj (op U₁)))
  (hst :
    Eq (α := CC (obj (stalkFunctor C x) G))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) α) :
            (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
          ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₁ x hxU₁) : (a : CC (F.obj (op U₁))) → CC (F.stalk x))
              s :
            CC (F.stalk x)) :
        CC (obj (stalkFunctor C x) G))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) α) :
            (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
          t :
        CC (obj (stalkFunctor C x) G))),
  (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₁ x hxU₁) : (a : CC (F.obj (op U₁))) → CC (F.stalk x)) s = t
Tactic: obtain ⟨U₁, hxU₁, hU₁, s, rfl⟩ := exists_mem_germ_eq_of_isBasis hB _ x s
Cleared hypotheses: s, hst
-- 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 : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst_2 : (X Y : C) → FunLike.{u_1 + 1, v + 1, v + 1} (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC (CC := CC)
inst_3 : PreservesFilteredColimits (forget C (FC := FC) (CC := CC))
B : Set (Opens (↑X : Type v))
hB : Opens.IsBasis B
F G : Presheaf C X
α : F ⟶ G
hα :
  ∀ U ∈ B,
    Function.Injective.{v + 1, v + 1} (α := CC (F.obj (op U))) (β := CC (G.obj (op U)))
      (⇑(ConcreteCategory.hom (FC := FC) (CC := CC) (α.app (op U))) : (a : CC (F.obj (op U))) → CC (G.obj (op U)))
x : (↑X : Type v)
t : CC (obj (stalkFunctor C x) F)
U₁ : Opens (↑X : Type v)
hxU₁ : x ∈ U₁
hU₁ : U₁ ∈ B
s : ToType (FC := FC) (CC := CC) (F.obj (op U₁))
hst :
  Eq (α := CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) α) :
          (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₁ x hxU₁) : (a : CC (F.obj (op U₁))) → CC (F.stalk x)) s :
          CC (F.stalk x)) :
      CC (obj (stalkFunctor C x) G))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) α) :
          (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
        t :
      CC (obj (stalkFunctor C x) G))
⊢ (ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₁ x hxU₁) : (a : CC (F.obj (op U₁))) → CC (F.stalk x)) s = t
have obtain : ∀ (U₂ : Opens (↑X : Type v)) (hxU₂ : x ∈ U₂) (hU₂ : U₂ ∈ B) (t : ToType (FC := FC) (CC := CC) (F.obj (op U₂)))
  (hst :
    Eq (α := CC (obj (stalkFunctor C x) G))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) α) :
            (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
          ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₁ x hxU₁) : (a : CC (F.obj (op U₁))) → CC (F.stalk x))
              s :
            CC (F.stalk x)) :
        CC (obj (stalkFunctor C x) G))
      ((ConcreteCategory.hom (FC := FC) (CC := CC) (map (stalkFunctor C x) α) :
            (a : CC ((stalkFunctor C x).obj F)) → CC ((stalkFunctor C x).obj G))
          ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₂ x hxU₂) : (a : CC (F.obj (op U₂))) → CC (F.stalk x))
              t :
            CC (F.stalk x)) :
        CC (obj (stalkFunctor C x) G))),
  Eq (α := CC (obj (stalkFunctor C x) F))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₁ x hxU₁) : (a : CC (F.obj (op U₁))) → CC (F.stalk x)) s :
      CC (F.stalk x))
    ((ConcreteCategory.hom (FC := FC) (CC := CC) (F.germ U₂ x hxU₂) : (a : CC (F.obj (op U₂))) → CC (F.stalk x)) t :
      CC (F.stalk x))
Tactic: obtain ⟨U₂, hxU₂, hU₂, t, rfl⟩ := exists_mem_germ_eq_of_isBasis hB _ x t
Cleared hypotheses: t, hst