awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X_1 X Y : LocallyRingedSpace
f : X.toSheafedSpace ≅ Y.toSheafedSpace
⊢ f.inv.hom ≫ f.hom.hom = 𝟙 Y.toPresheafedSpace
have h : (f.inv ≫ f.hom).hom = 𝟙 Y.toPresheafedSpace
Tactic: rw [← InducedCategory.comp_hom, f.inv_hom_id, SheafedSpace.id_hom]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X_1 X Y : LocallyRingedSpace
f : X.toSheafedSpace ≅ Y.toSheafedSpace
h : (f.inv ≫ f.hom).hom = 𝟙 Y.toPresheafedSpace
⊢ f.inv.hom ≫ f.hom.hom = 𝟙 Y.toPresheafedSpace
have h₁ : InducedCategory.Hom.hom (𝟙 Y.toSheafedSpace) = 𝟙 Y.toPresheafedSpace
Tactic: rw [← InducedCategory.comp_hom, f.inv_hom_id, SheafedSpace.id_hom]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X_1 X Y : LocallyRingedSpace
f : X.toSheafedSpace ≅ Y.toSheafedSpace
h : (f.inv ≫ f.hom).hom = 𝟙 Y.toPresheafedSpace
h₁ : InducedCategory.Hom.hom (𝟙 Y.toSheafedSpace) = 𝟙 Y.toPresheafedSpace
⊢ f.inv.hom ≫ f.hom.hom = 𝟙 Y.toPresheafedSpace
have h₂ : 𝟙 Y.toPresheafedSpace = 𝟙 Y.toPresheafedSpace
Tactic: rw [← InducedCategory.comp_hom, f.inv_hom_id, SheafedSpace.id_hom]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X_1 : LocallyRingedSpace
U : TopCat
X : LocallyRingedSpace
f : U ⟶ X.toTopCat
h :
  IsOpenEmbedding (X := (↑U : Type u)) (Y := (↑X.toTopCat : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) f) :
      (a : (↑U : Type u)) → (↑X.toTopCat : Type u))
x : (↑(↑(X.restrict h).toPresheafedSpace : TopCat) : Type u)
⊢ IsLocalRing.{u} (↑((X.restrict h).presheaf.stalk x) : Type u)
have e : (↑(X.presheaf.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) f :
            (a : (↑U : Type u)) → (↑X.toTopCat : Type u))
          x)) :
    Type u) ≃+*
  (↑((X.restrict h).presheaf.stalk x) : Type u)
Tactic: apply @RingEquiv.isLocalRing _ _ _ (X.isLocalRing (f x))
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X_1 X : LocallyRingedSpace
U : (Opens (↑(↑X.toPresheafedSpace : TopCat) : Type u))ᵒᵖ
⊢ (↑(X.presheaf.1 U) : Type u) →+* PUnit.{u + 1}
have refine_1 : Eq.{u + 1} (α := PUnit.{u + 1}) (0 1) 1
Tactic: refine ⟨⟨⟨0, ?_⟩, ?_⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X_1 X : LocallyRingedSpace
U : (Opens (↑(↑X.toPresheafedSpace : TopCat) : Type u))ᵒᵖ
refine_1 : Eq.{u + 1} (α := PUnit.{u + 1}) (0 1) 1
⊢ (↑(X.presheaf.1 U) : Type u) →+* PUnit.{u + 1}
have refine_2 : ∀ (x y : (↑(X.presheaf.1 U) : Type u)),
  OneHom.toFun { toFun := 0, map_one' := sorry } (x * y) =
    HMul.hMul (α := PUnit.{u + 1})
      (OneHom.toFun { toFun := 0, map_one' := sorry : OneHom (↑(X.presheaf.1 U) : Type u) PUnit.{u + 1} } x)
      (OneHom.toFun { toFun := 0, map_one' := sorry : OneHom (↑(X.presheaf.1 U) : Type u) PUnit.{u + 1} } y)
Tactic: refine ⟨⟨⟨0, ?_⟩, ?_⟩, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X_1 X : LocallyRingedSpace
f : ∅ ⟶ X
⊢ f = X.emptyTo
have h : ∀ (carrier : Set (↑(↑X.toPresheafedSpace : TopCat) : Type u)) (is_open' : IsOpen carrier)
  (x : (↑(X.presheaf.obj (op { carrier := carrier, is_open' := is_open' })) : Type u)),
  Eq (α :=
    (↑(((Opens.map X.emptyTo.base).op ⋙ (toSheafedSpace ∅).presheaf).obj
          (op { carrier := carrier, is_open' := is_open' })) :
      Type u))
    ((CommRingCat.Hom.hom
            (NatTrans.app (f.c ≫ whiskerRight (eqToHom sorry) (toSheafedSpace ∅).presheaf)
              (op { carrier := carrier, is_open' := is_open' })) :
          (a : (↑(X.presheaf.obj (op { carrier := carrier, is_open' := is_open' })) : Type u)) →
            (↑(((Opens.map X.emptyTo.base).op ⋙ ∅.presheaf).obj (op { carrier := carrier, is_open' := is_open' })) :
              Type u))
        x :
      (↑(((Opens.map X.emptyTo.base).op ⋙ (toSheafedSpace ∅).presheaf).obj
            (op { carrier := carrier, is_open' := is_open' })) :
        Type u))
    ((CommRingCat.Hom.hom (X.emptyTo.c.app (op { carrier := carrier, is_open' := is_open' })) :
          (a : (↑(X.presheaf.obj (op { carrier := carrier, is_open' := is_open' })) : Type u)) →
            (↑(((Opens.map X.emptyTo.base).op ⋙ ∅.presheaf).obj (op { carrier := carrier, is_open' := is_open' })) :
              Type u))
        x :
      (↑(((Opens.map X.emptyTo.base).op ⋙ (toSheafedSpace ∅).presheaf).obj
            (op { carrier := carrier, is_open' := is_open' })) :
        Type u))
Tactic: ext ⟨⟩ x
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X : LocallyRingedSpace
U : Opens (↑(↑X.toPresheafedSpace : TopCat) : Type u)
⊢ Eq.{u + 1} (α := Opens (↑(↑X.toRingedSpace.toPresheafedSpace : TopCat) : Type u))
    (X.toRingedSpace.basicOpen (U := U) 0) ⊥
have h : ∀ (x : (↑(↑X.toRingedSpace.toPresheafedSpace : TopCat) : Type u)),
  x ∈ (↑(X.toRingedSpace.basicOpen (U := U) 0) : Set (↑(↑X.toRingedSpace.toPresheafedSpace : TopCat) : Type u)) ↔
    x ∈ (↑⊥ : Set (↑(↑X.toRingedSpace.toPresheafedSpace : TopCat) : Type u))
Tactic: ext x
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X : LocallyRingedSpace
U : Opens (↑(↑X.toPresheafedSpace : TopCat) : Type u)
x : (↑(↑X.toRingedSpace.toPresheafedSpace : TopCat) : Type u)
⊢ x ∈ (↑(X.toRingedSpace.basicOpen (U := U) 0) : Set (↑(↑X.toRingedSpace.toPresheafedSpace : TopCat) : Type u)) ↔
    x ∈ (↑⊥ : Set (↑(↑X.toRingedSpace.toPresheafedSpace : TopCat) : Type u))
have h : ∀ (x_1 : x ∈ U),
  ¬IsUnit (M := (↑(X.toRingedSpace.presheaf.stalk x) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (Y := X.toRingedSpace.presheaf.stalk x) (X.toRingedSpace.presheaf.germ U x sorry) :
            (a : (↑(X.toRingedSpace.presheaf.obj (op U)) : Type u)) → (↑(X.toRingedSpace.presheaf.stalk x) : Type u))
          0 :
        (↑(X.toRingedSpace.presheaf.stalk x) : Type u))
Tactic: simp only [RingedSpace.basicOpen, Opens.coe_mk, Set.mem_setOf_eq,
    Opens.coe_bot, Set.mem_empty_iff_false,
    iff_false, not_exists]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X : LocallyRingedSpace
U : Opens (↑(↑X.toPresheafedSpace : TopCat) : Type u)
x : (↑(↑X.toRingedSpace.toPresheafedSpace : TopCat) : Type u)
hx : x ∈ U
⊢ ¬IsUnit (M := (↑(X.toRingedSpace.presheaf.stalk x) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (Y := X.toRingedSpace.presheaf.stalk x) (X.toRingedSpace.presheaf.germ U x ⋯) :
            (a : (↑(X.toRingedSpace.presheaf.obj (op U)) : Type u)) → (↑(X.toRingedSpace.presheaf.stalk x) : Type u))
          0 :
        (↑(X.toRingedSpace.presheaf.stalk x) : Type u))
have h : ¬Eq.{u + 1} (α := (↑(X.toRingedSpace.presheaf.stalk x) : Type u)) 0 1
Tactic: rw [map_zero, isUnit_zero_iff]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X : LocallyRingedSpace
U : Opens (↑(↑X.toPresheafedSpace : TopCat) : Type u)
x : (↑(↑X.toRingedSpace.toPresheafedSpace : TopCat) : Type u)
hx : x ∈ U
⊢ ¬Eq.{u + 1} (α := (↑(X.toRingedSpace.presheaf.stalk x) : Type u)) 0 1
have h : Ne.{u + 1} (α := (↑(X.presheaf.stalk x) : Type u)) 0 1
Tactic: change (0 : X.presheaf.stalk x) ≠ (1 : X.presheaf.stalk x)
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X : LocallyRingedSpace
U : Opens (↑(↑X.toPresheafedSpace : TopCat) : Type u)
f : (↑(X.presheaf.obj (op U)) : Type u)
hf : IsNilpotent f
⊢ X.toRingedSpace.basicOpen f = ⊥
have obtain : ∀ (n : ℕ) (hn : f ^ n = 0), X.toRingedSpace.basicOpen f = ⊥
Tactic: obtain ⟨n, hn⟩ := hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X : LocallyRingedSpace
U : Opens (↑(↑X.toPresheafedSpace : TopCat) : Type u)
f : (↑(X.presheaf.obj (op U)) : Type u)
n : ℕ
hn : f ^ n = 0
h : n > 0
⊢ X.toRingedSpace.basicOpen f = ⊥
have inr : Eq.{u + 1} (α := Opens (↑(↑X.toRingedSpace.toPresheafedSpace : TopCat) : Type u)) (X.toRingedSpace.basicOpen (U := U) 0)
  ⊥
Tactic: rw [← X.toRingedSpace.basicOpen_pow f n h, hn]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (hypothesis)
Goal:
X : LocallyRingedSpace
U : Opens (↑(↑X.toPresheafedSpace : TopCat) : Type u)
f : (↑(X.presheaf.obj (op U)) : Type u)
n : ℕ
hn : f ^ n = 0
h : n = 0
⊢ X.toRingedSpace.basicOpen f = ⊥
have hn₁ : Eq.{u + 1} (α := (↑(X.presheaf.obj (op U)) : Type u)) 1 0
Tactic: rw [h, pow_zero] at hn
Cleared hypotheses: hn, h
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (hypothesis)
Goal:
X : LocallyRingedSpace
U : Opens (↑(↑X.toPresheafedSpace : TopCat) : Type u)
f : (↑(X.presheaf.obj (op U)) : Type u)
n : ℕ
hn : f ^ n = 0
⊢ X.toRingedSpace.basicOpen f = ⊥
have x : n = 0 ∨ n > 0
Tactic: cases n.eq_zero_or_pos with
  | inr h =>
    rw [← X.toRingedSpace.basicOpen_pow f n h, hn]
    simp [basicOpen_zero]
  | inl h =>
    rw [h, pow_zero] at hn
    simp [eq_zero_of_zero_eq_one hn.symm f, basicOpen_zero]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X Y : LocallyRingedSpace
e : X ≅ Y
x : (↑X.toTopCat : Type u)
⊢ (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) e.inv.base :
        (a : (↑(↑Y.toPresheafedSpace : TopCat) : Type u)) → (↑(↑X.toPresheafedSpace : TopCat) : Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
              e.hom.base :
            (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
          x :
        (↑(↑Y.toPresheafedSpace : TopCat) : Type u)) =
    x
have change : Eq (α := (↑(↑X.toPresheafedSpace : TopCat) : Type u))
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
          (e.hom.base ≫ e.inv.base) :
        (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑X.toPresheafedSpace : TopCat) : Type u))
      x :
    (↑(↑X.toPresheafedSpace : TopCat) : Type u))
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
          (PresheafedSpace.Hom.base (𝟙 X.toPresheafedSpace)) :
        (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑X.toPresheafedSpace : TopCat) : Type u))
      x :
    (↑(↑X.toPresheafedSpace : TopCat) : Type u))
Tactic: change (e.hom.base ≫ e.inv.base) x = 𝟙 X.toPresheafedSpace x
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X Y : LocallyRingedSpace
e : X ≅ Y
y : (↑Y.toTopCat : Type u)
⊢ (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) e.hom.base :
        (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
              e.inv.base :
            (a : (↑(↑Y.toPresheafedSpace : TopCat) : Type u)) → (↑(↑X.toPresheafedSpace : TopCat) : Type u))
          y :
        (↑(↑X.toPresheafedSpace : TopCat) : Type u)) =
    y
have change : Eq (α := (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
          (e.inv.base ≫ e.hom.base) :
        (a : (↑(↑Y.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
      y :
    (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
          (PresheafedSpace.Hom.base (𝟙 Y.toPresheafedSpace)) :
        (a : (↑(↑Y.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
      y :
    (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
Tactic: change (e.inv.base ≫ e.hom.base) y = 𝟙 Y.toPresheafedSpace y
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X Y : LocallyRingedSpace
f g : X ⟶ Y
hfg : f = g
x x' : (↑X.toTopCat : Type u)
hxx' : x = x'
⊢ Hom.stalkMap f x ≫ X.presheaf.stalkSpecializes ⋯ = Y.presheaf.stalkSpecializes ⋯ ≫ Hom.stalkMap g x'
have subst : Hom.stalkMap f x ≫ X.presheaf.stalkSpecializes sorry = Y.presheaf.stalkSpecializes sorry ≫ Hom.stalkMap f x'
Tactic: subst hfg
Cleared hypotheses: g, hfg
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
x x' : (↑X.toTopCat : Type u)
hxx' : x = x'
⊢ Hom.stalkMap f x ≫ X.presheaf.stalkSpecializes ⋯ = Y.presheaf.stalkSpecializes ⋯ ≫ Hom.stalkMap f x'
have subst : Hom.stalkMap f x ≫ X.presheaf.stalkSpecializes sorry = Y.presheaf.stalkSpecializes sorry ≫ Hom.stalkMap f x
Tactic: subst hxx'
Cleared hypotheses: x', hxx'
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X Y : LocallyRingedSpace
f g : X ⟶ Y
hfg : f = g
x : (↑X.toTopCat : Type u)
⊢ Hom.stalkMap f x = Y.presheaf.stalkSpecializes ⋯ ≫ Hom.stalkMap g x
have subst : Hom.stalkMap f x = Y.presheaf.stalkSpecializes sorry ≫ Hom.stalkMap f x
Tactic: subst hfg
Cleared hypotheses: g, hfg
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X Y : LocallyRingedSpace
e : X ≅ Y
y : (↑Y.toTopCat : Type u)
⊢ Hom.stalkMap e.hom
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
              e.inv.base :
            (a : (↑(↑Y.toPresheafedSpace : TopCat) : Type u)) → (↑(↑X.toPresheafedSpace : TopCat) : Type u))
          y) ≫
      Hom.stalkMap e.inv y =
    Y.presheaf.stalkSpecializes ⋯
have rw : Y.presheaf.stalkSpecializes sorry ≫ Hom.stalkMap (𝟙 Y) y = Y.presheaf.stalkSpecializes sorry
Tactic: rw [← stalkMap_comp, LocallyRingedSpace.stalkMap_congr_hom (e.inv ≫ e.hom) (𝟙 _) (by simp)]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X Y : LocallyRingedSpace
e : X ≅ Y
x : (↑X.toTopCat : Type u)
⊢ Hom.stalkMap e.inv
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
              e.hom.base :
            (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
          x) ≫
      Hom.stalkMap e.hom x =
    X.presheaf.stalkSpecializes ⋯
have rw : X.presheaf.stalkSpecializes sorry ≫ Hom.stalkMap (𝟙 X) x = X.presheaf.stalkSpecializes sorry
Tactic: rw [← stalkMap_comp, LocallyRingedSpace.stalkMap_congr_hom (e.hom ≫ e.inv) (𝟙 _) (by simp)]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
U : Opens (↑Y.toTopCat : Type u)
s : (↑(Y.presheaf.obj (op U)) : Type u)
⊢ (Opens.map f.base).obj (Y.toRingedSpace.basicOpen s) =
    X.toRingedSpace.basicOpen (U := (Opens.map f.base).obj U)
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (X := Y.presheaf.obj (op U)) (f.c.app (op U)) :
          (a : (↑(Y.presheaf.obj (op U)) : Type u)) →
            (↑(((Presheaf.pushforward CommRingCat f.base).obj X.presheaf).obj (op U)) : Type u))
        s)
have h : ∀ (x : (↑(↑X.toPresheafedSpace : TopCat) : Type u)),
  x ∈ (↑((Opens.map f.base).obj (Y.toRingedSpace.basicOpen s)) : Set (↑(↑X.toPresheafedSpace : TopCat) : Type u)) ↔
    x ∈
      (↑(X.toRingedSpace.basicOpen (U := (Opens.map f.base).obj U)
            ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                  CommRingCat.carrier) (X := Y.presheaf.obj (op U)) (f.c.app (op U)) :
                (a : (↑(Y.presheaf.obj (op U)) : Type u)) →
                  (↑(((Presheaf.pushforward CommRingCat f.base).obj X.presheaf).obj (op U)) : Type u))
              s)) :
        Set (↑(↑X.toPresheafedSpace : TopCat) : Type u))
Tactic: ext x
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
U : Opens (↑Y.toTopCat : Type u)
s : (↑(Y.presheaf.obj (op U)) : Type u)
x : (↑(↑X.toPresheafedSpace : TopCat) : Type u)
⊢ x ∈ (↑((Opens.map f.base).obj (Y.toRingedSpace.basicOpen s)) : Set (↑(↑X.toPresheafedSpace : TopCat) : Type u)) ↔
    x ∈
      (↑(X.toRingedSpace.basicOpen (U := (Opens.map f.base).obj U)
            ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                  CommRingCat.carrier) (X := Y.presheaf.obj (op U)) (f.c.app (op U)) :
                (a : (↑(Y.presheaf.obj (op U)) : Type u)) →
                  (↑(((Presheaf.pushforward CommRingCat f.base).obj X.presheaf).obj (op U)) : Type u))
              s)) :
        Set (↑(↑X.toPresheafedSpace : TopCat) : Type u))
have h : x ∈ (↑((Opens.map f.base).obj (Y.toRingedSpace.basicOpen s)) : Set (↑(↑X.toPresheafedSpace : TopCat) : Type u)) →
  x ∈
    (↑(X.toRingedSpace.basicOpen (U := (Opens.map f.base).obj U)
          ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Y.presheaf.obj (op U)) (f.c.app (op U)) :
              (a : (↑(Y.presheaf.obj (op U)) : Type u)) →
                (↑(((Presheaf.pushforward CommRingCat f.base).obj X.presheaf).obj (op U)) : Type u))
            s)) :
      Set (↑(↑X.toPresheafedSpace : TopCat) : Type u))
Tactic: constructor
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
U : Opens (↑Y.toTopCat : Type u)
s : (↑(Y.presheaf.obj (op U)) : Type u)
x : (↑(↑X.toPresheafedSpace : TopCat) : Type u)
h :
  x ∈ (↑((Opens.map f.base).obj (Y.toRingedSpace.basicOpen s)) : Set (↑(↑X.toPresheafedSpace : TopCat) : Type u)) →
    x ∈
      (↑(X.toRingedSpace.basicOpen (U := (Opens.map f.base).obj U)
            ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                  CommRingCat.carrier) (X := Y.presheaf.obj (op U)) (f.c.app (op U)) :
                (a : (↑(Y.presheaf.obj (op U)) : Type u)) →
                  (↑(((Presheaf.pushforward CommRingCat f.base).obj X.presheaf).obj (op U)) : Type u))
              s)) :
        Set (↑(↑X.toPresheafedSpace : TopCat) : Type u))
⊢ x ∈ (↑((Opens.map f.base).obj (Y.toRingedSpace.basicOpen s)) : Set (↑(↑X.toPresheafedSpace : TopCat) : Type u)) ↔
    x ∈
      (↑(X.toRingedSpace.basicOpen (U := (Opens.map f.base).obj U)
            ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                  CommRingCat.carrier) (X := Y.presheaf.obj (op U)) (f.c.app (op U)) :
                (a : (↑(Y.presheaf.obj (op U)) : Type u)) →
                  (↑(((Presheaf.pushforward CommRingCat f.base).obj X.presheaf).obj (op U)) : Type u))
              s)) :
        Set (↑(↑X.toPresheafedSpace : TopCat) : Type u))
have h₁ : x ∈
    (↑(X.toRingedSpace.basicOpen (U := (Opens.map f.base).obj U)
          ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Y.presheaf.obj (op U)) (f.c.app (op U)) :
              (a : (↑(Y.presheaf.obj (op U)) : Type u)) →
                (↑(((Presheaf.pushforward CommRingCat f.base).obj X.presheaf).obj (op U)) : Type u))
            s)) :
      Set (↑(↑X.toPresheafedSpace : TopCat) : Type u)) →
  x ∈ (↑((Opens.map f.base).obj (Y.toRingedSpace.basicOpen s)) : Set (↑(↑X.toPresheafedSpace : TopCat) : Type u))
Tactic: constructor
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.LocallyRingedSpace (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
U : Opens (↑Y.toTopCat : Type u)
s : (↑(Y.presheaf.obj (op U)) : Type u)
x : (↑(↑X.toPresheafedSpace : TopCat) : Type u)
hxU : x ∈ (Opens.map f.base).obj U
hx :
  IsUnit (M := (↑(X.toRingedSpace.presheaf.stalk x) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Y := X.toRingedSpace.presheaf.stalk x)
            (X.toRingedSpace.presheaf.germ ((Opens.map f.base).obj U) x hxU) :
          (a : (↑(X.toRingedSpace.presheaf.obj (op ((Opens.map f.base).obj U))) : Type u)) →
            (↑(X.toRingedSpace.presheaf.stalk x) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Y.presheaf.obj (op U)) (f.c.app (op U)) :
              (a : (↑(Y.presheaf.obj (op U)) : Type u)) →
                (↑(((Presheaf.pushforward CommRingCat f.base).obj X.presheaf).obj (op U)) : Type u))
            s :
          (↑((obj (Presheaf.pushforward CommRingCat f.base) X.presheaf).obj (op U)) : Type u)) :
      (↑(X.toRingedSpace.presheaf.stalk x) : Type u))
⊢ x ∈ (↑((Opens.map f.base).obj (Y.toRingedSpace.basicOpen s)) : Set (↑(↑X.toPresheafedSpace : TopCat) : Type u))
have h : Membership.mem (α := (↑(↑Y.toPresheafedSpace : TopCat) : Type u)) (RingedSpace.basicOpen Y.toSheafedSpace s)
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) f.base :
        (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
      x :
    (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
Tactic: simp only [Opens.map_coe, Set.mem_preimage, SetLike.mem_coe, toRingedSpace] at hx ⊢
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Y
H : IsOpenImmersion f
⊢ Functor.obj (TopCat.Presheaf.pushforward C (Iso.hom (Iso.refl (↑X : TopCat)))) X.presheaf ≅
    (Y.restrict (U := (↑X : TopCat)) (f := f.base) ⋯).presheaf
have symm : (Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry).presheaf ≅
  Functor.obj (TopCat.Presheaf.pushforward C (Iso.hom (Iso.refl (↑X : TopCat)))) X.presheaf
Tactic: symm
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Y
H : IsOpenImmersion f
⊢ (Y.restrict (U := (↑X : TopCat)) (f := f.base) ⋯).presheaf ≅
    Functor.obj (TopCat.Presheaf.pushforward C (Iso.hom (Iso.refl (↑X : TopCat)))) X.presheaf
have app : (X_1 : (Opens (↑(↑(Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry) : TopCat) : Type u_1))ᵒᵖ) →
  (Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry).presheaf.obj X_1 ≅
    (Functor.obj (TopCat.Presheaf.pushforward C (Iso.hom (Iso.refl (↑X : TopCat)))) X.presheaf).obj X_1
Tactic: fapply NatIso.ofComponents
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Y
H : IsOpenImmersion f
app :
  (X_1 : (Opens (↑(↑(Y.restrict (U := (↑X : TopCat)) (f := f.base) ⋯) : TopCat) : Type u_1))ᵒᵖ) →
    (Y.restrict (U := (↑X : TopCat)) (f := f.base) ⋯).presheaf.obj X_1 ≅
      (Functor.obj (TopCat.Presheaf.pushforward C (Iso.hom (Iso.refl (↑X : TopCat)))) X.presheaf).obj X_1
⊢ (Y.restrict (U := (↑X : TopCat)) (f := f.base) ⋯).presheaf ≅
    Functor.obj (TopCat.Presheaf.pushforward C (Iso.hom (Iso.refl (↑X : TopCat)))) X.presheaf
have naturality : autoParam
  (∀ {X_1 Y_1 : (Opens (↑(↑(Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry) : TopCat) : Type u_1))ᵒᵖ}
    (f_1 : X_1 ⟶ Y_1),
    (Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry).presheaf.map f_1 ≫
        Iso.hom (asIso (f.c.app (op ((opensFunctor f).obj (unop Y_1)))) ≪≫ Functor.mapIso X.presheaf (eqToIso sorry)) =
      Iso.hom (asIso (f.c.app (op ((opensFunctor f).obj (unop X_1)))) ≪≫ Functor.mapIso X.presheaf (eqToIso sorry)) ≫
        (Functor.obj (TopCat.Presheaf.pushforward C (Iso.hom (Iso.refl (↑X : TopCat)))) X.presheaf).map f_1)
  NatIso.ofComponents._auto_1
Tactic: fapply NatIso.ofComponents