awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
U : (Opens (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1))ᵒᵖ
s' : PullbackCone f.base g.base := PullbackCone.mk s.fst.base s.snd.base ⋯
this :
  CategoryStruct.comp (Y := limit (cospan f.base g.base)) (limit.lift (cospan f.base g.base) s')
      (limit.π (cospan f.base g.base) WalkingCospan.right) =
    s.snd.base
| Set.preimage (β := (↑(↑Y : TopCat) : Type u_1))
    (Function.comp (β := (↑(limit (cospan f.base g.base)) : Type u_1))
      (⇑(TopCat.Hom.hom (limit.π (cospan f.base g.base) WalkingCospan.right)) :
        (a : (↑(limit (cospan f.base g.base)) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1))
      (⇑(TopCat.Hom.hom (limit.lift (cospan f.base g.base) (PullbackCone.mk s.fst.base s.snd.base ⋯)) :
            C((↑(↑s.pt : TopCat) : Type u_1), (↑(limit (cospan f.base g.base)) : Type u_1))) :
        (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(limit (cospan f.base g.base)) : Type u_1)))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
            TopCat.carrier) (X := pullback f.base g.base) (pullback.snd f.base g.base)) :
        (a : (↑(pullback f.base g.base) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ''
      (↑(unop U) : Set (↑(pullback f.base g.base) : Type u_1)))
have rw : Set.preimage (β := (↑(limit (cospan f.base g.base)) : Type u_1))
  (⇑(TopCat.Hom.hom (limit.lift (cospan f.base g.base) (PullbackCone.mk s.fst.base s.snd.base sorry)) :
        C((↑(↑s.pt : TopCat) : Type u_1), (↑(limit (cospan f.base g.base)) : Type u_1))) :
    (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(limit (cospan f.base g.base)) : Type u_1))
  (Set.preimage (β := (↑(↑Y : TopCat) : Type u_1))
    (⇑(TopCat.Hom.hom (limit.π (cospan f.base g.base) WalkingCospan.right)) :
      (a : (↑(limit (cospan f.base g.base)) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
            TopCat.carrier) (X := pullback f.base g.base) (pullback.snd f.base g.base)) :
        (a : (↑(pullback f.base g.base) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ''
      (↑(unop U) : Set (↑(pullback f.base g.base) : Type u_1))))
Tactic: rw [Function.comp_def, ← Set.preimage_preimage]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
U : (Opens (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1))ᵒᵖ
s' : PullbackCone f.base g.base := PullbackCone.mk s.fst.base s.snd.base ⋯
this :
  CategoryStruct.comp (Y := limit (cospan f.base g.base)) (limit.lift (cospan f.base g.base) s')
      (limit.π (cospan f.base g.base) WalkingCospan.right) =
    s.snd.base
| (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := TopCat.carrier)
          s.snd.base) :
      (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ⁻¹'
    (↑(⋯.functor.obj (unop U)) : Set (↑(↑Y : TopCat) : Type u_1))
have rw : Set.preimage (β := (↑(limit (cospan f.base g.base)) : Type u_1))
  (⇑(TopCat.Hom.hom (limit.lift (cospan f.base g.base) (PullbackCone.mk s.fst.base s.snd.base sorry)) :
        C((↑(↑s.pt : TopCat) : Type u_1), (↑(limit (cospan f.base g.base)) : Type u_1))) :
    (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(limit (cospan f.base g.base)) : Type u_1))
  (Set.preimage (β := (↑(↑Y : TopCat) : Type u_1))
    (⇑(TopCat.Hom.hom (limit.π (cospan f.base g.base) WalkingCospan.right)) :
      (a : (↑(limit (cospan f.base g.base)) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
            TopCat.carrier) (X := pullback f.base g.base) (pullback.snd f.base g.base)) :
        (a : (↑(pullback f.base g.base) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ''
      (↑(unop U) : Set (↑(pullback f.base g.base) : Type u_1))))
Tactic: rw [← this]
                  dsimp [s']
                  rw [Function.comp_def, ← Set.preimage_preimage]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
U : (Opens (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1))ᵒᵖ
s' : PullbackCone f.base g.base := PullbackCone.mk s.fst.base s.snd.base ⋯
this :
  CategoryStruct.comp (Y := limit (cospan f.base g.base)) (limit.lift (cospan f.base g.base) s')
      (limit.π (cospan f.base g.base) WalkingCospan.right) =
    s.snd.base
⊢ Eq (α := Set (↑(↑s.pt : TopCat) : Type u_1))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
            TopCat.carrier) s.snd.base) :
        (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ⁻¹'
      (↑(⋯.functor.obj (unop U)) : Set (↑(↑Y : TopCat) : Type u_1)))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
            TopCat.carrier) (Y := (↑(pullbackConeOfLeft f g).pt : TopCat)) (pullback.lift s.fst.base s.snd.base ⋯)) :
        (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)) ⁻¹'
      (↑(unop U) : Set (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)))
have e_unop : Eq (α := Set (↑(↑s.pt : TopCat) : Type u_1))
  (Set.preimage (β := (↑(limit (cospan f.base g.base)) : Type u_1))
    (⇑(TopCat.Hom.hom (limit.lift (cospan f.base g.base) (PullbackCone.mk s.fst.base s.snd.base sorry)) :
          C((↑(↑s.pt : TopCat) : Type u_1), (↑(limit (cospan f.base g.base)) : Type u_1))) :
      (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(limit (cospan f.base g.base)) : Type u_1))
    (Set.preimage (β := (↑(↑Y : TopCat) : Type u_1))
      (⇑(TopCat.Hom.hom (limit.π (cospan f.base g.base) WalkingCospan.right)) :
        (a : (↑(limit (cospan f.base g.base)) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
              TopCat.carrier) (X := pullback f.base g.base) (pullback.snd f.base g.base)) :
          (a : (↑(pullback f.base g.base) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ''
        (↑(unop U) : Set (↑(pullback f.base g.base) : Type u_1)))))
  ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := TopCat.carrier)
          (Y := (↑(pullbackConeOfLeft f g).pt : TopCat)) (pullback.lift s.fst.base s.snd.base sorry)) :
      (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)) ⁻¹'
    (↑(unop U) : Set (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)))
Tactic: conv_lhs =>
                  rw [← this]
                  dsimp [s']
                  rw [Function.comp_def, ← Set.preimage_preimage]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
U : (Opens (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1))ᵒᵖ
s' : PullbackCone f.base g.base := PullbackCone.mk s.fst.base s.snd.base ⋯
this :
  CategoryStruct.comp (Y := limit (cospan f.base g.base)) (limit.lift (cospan f.base g.base) s')
      (limit.π (cospan f.base g.base) WalkingCospan.right) =
    s.snd.base
⊢ Eq (α := Set (↑(↑s.pt : TopCat) : Type u_1))
    (Set.preimage (β := (↑(limit (cospan f.base g.base)) : Type u_1))
      (⇑(TopCat.Hom.hom (limit.lift (cospan f.base g.base) (PullbackCone.mk s.fst.base s.snd.base ⋯)) :
            C((↑(↑s.pt : TopCat) : Type u_1), (↑(limit (cospan f.base g.base)) : Type u_1))) :
        (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(limit (cospan f.base g.base)) : Type u_1))
      (Set.preimage (β := (↑(↑Y : TopCat) : Type u_1))
        (⇑(TopCat.Hom.hom (limit.π (cospan f.base g.base) WalkingCospan.right)) :
          (a : (↑(limit (cospan f.base g.base)) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1))
        ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                TopCat.carrier) (X := pullback f.base g.base) (pullback.snd f.base g.base)) :
            (a : (↑(pullback f.base g.base) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ''
          (↑(unop U) : Set (↑(pullback f.base g.base) : Type u_1)))))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
            TopCat.carrier) (Y := (↑(pullbackConeOfLeft f g).pt : TopCat)) (pullback.lift s.fst.base s.snd.base ⋯)) :
        (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)) ⁻¹'
      (↑(unop U) : Set (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)))
have e_unop : Eq (α := Set (↑(↑s.pt : TopCat) : Type u_1))
  ((⇑(TopCat.Hom.hom (limit.lift (cospan f.base g.base) (PullbackCone.mk s.fst.base s.snd.base sorry)) :
          C((↑(↑s.pt : TopCat) : Type u_1), (↑(limit (cospan f.base g.base)) : Type u_1))) :
      (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(limit (cospan f.base g.base)) : Type u_1)) ⁻¹'
    (↑(unop U) : Set (↑(pullback f.base g.base) : Type u_1)))
  ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := TopCat.carrier)
          (Y := (↑(pullbackConeOfLeft f g).pt : TopCat)) (pullback.lift s.fst.base s.snd.base sorry)) :
      (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)) ⁻¹'
    (↑(unop U) : Set (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)))
Tactic: rw [Set.preimage_image_eq _
                    (TopCat.snd_isOpenEmbedding_of_left hf.base_open g.base).injective]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
U V : (Opens (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1))ᵒᵖ
i : U ⟶ V
⊢ (pullbackConeOfLeft f g).pt.presheaf.map i ≫ s.snd.c.app (⋯.functor.op.obj V) ≫ s.pt.presheaf.map (eqToHom ⋯) =
    (s.snd.c.app (⋯.functor.op.obj U) ≫ s.pt.presheaf.map (eqToHom ⋯)) ≫
      CategoryTheory.Functor.map
        (Functor.obj (TopCat.Presheaf.pushforward C (pullback.lift s.fst.base s.snd.base ⋯)) s.pt.presheaf) i
have erw : s.snd.c.app (sorry.functor.op.obj U) ≫
    (Functor.obj (TopCat.Presheaf.pushforward C s.snd.base) s.pt.presheaf).map (sorry.functor.op.map i) ≫
      s.pt.presheaf.map (eqToHom sorry) =
  (s.snd.c.app (sorry.functor.op.obj U) ≫ s.pt.presheaf.map (eqToHom sorry)) ≫
    CategoryTheory.Functor.map
      (Functor.obj (TopCat.Presheaf.pushforward C (pullback.lift s.fst.base s.snd.base sorry)) s.pt.presheaf) i
Tactic: erw [s.snd.c.naturality_assoc]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
U V : (Opens (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1))ᵒᵖ
i : U ⟶ V
⊢ s.snd.c.app (⋯.functor.op.obj U) ≫
      (Functor.obj (TopCat.Presheaf.pushforward C s.snd.base) s.pt.presheaf).map (⋯.functor.op.map i) ≫
        s.pt.presheaf.map (eqToHom ⋯) =
    (s.snd.c.app (⋯.functor.op.obj U) ≫ s.pt.presheaf.map (eqToHom ⋯)) ≫
      CategoryTheory.Functor.map
        (Functor.obj (TopCat.Presheaf.pushforward C (pullback.lift s.fst.base s.snd.base ⋯)) s.pt.presheaf) i
have rw : s.snd.c.app (sorry.functor.op.obj U) ≫
    (Functor.obj (TopCat.Presheaf.pushforward C s.snd.base) s.pt.presheaf).map (sorry.functor.op.map i) ≫
      s.pt.presheaf.map (eqToHom sorry) =
  s.snd.c.app (sorry.functor.op.obj U) ≫
    s.pt.presheaf.map (eqToHom sorry) ≫
      CategoryTheory.Functor.map
        (Functor.obj (TopCat.Presheaf.pushforward C (pullback.lift s.fst.base s.snd.base sorry)) s.pt.presheaf) i
Tactic: rw [Category.assoc]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
U V : (Opens (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1))ᵒᵖ
i : U ⟶ V
⊢ s.snd.c.app (⋯.functor.op.obj U) ≫
      (Functor.obj (TopCat.Presheaf.pushforward C s.snd.base) s.pt.presheaf).map (⋯.functor.op.map i) ≫
        s.pt.presheaf.map (eqToHom ⋯) =
    s.snd.c.app (⋯.functor.op.obj U) ≫
      s.pt.presheaf.map (eqToHom ⋯) ≫
        CategoryTheory.Functor.map
          (Functor.obj (TopCat.Presheaf.pushforward C (pullback.lift s.fst.base s.snd.base ⋯)) s.pt.presheaf) i
have erw : s.snd.c.app (sorry.functor.op.obj U) ≫ s.pt.presheaf.map ((Opens.map s.snd.base).op.map (sorry.functor.op.map i) ≫ eqToHom sorry) =
  s.snd.c.app (sorry.functor.op.obj U) ≫
    s.pt.presheaf.map
      (eqToHom sorry ≫ CategoryTheory.Functor.map (Functor.op (Opens.map (pullback.lift s.fst.base s.snd.base sorry))) i)
Tactic: erw [← s.pt.presheaf.map_comp, ← s.pt.presheaf.map_comp]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
⊢ pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst = s.fst
have refine_1 : (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).base = s.fst.base
Tactic: refine PresheafedSpace.Hom.ext _ _ ?_ <| NatTrans.ext <| funext fun x => ?_
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
refine_1 : (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).base = s.fst.base
⊢ pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst = s.fst
have refine_2 : ∀ (x : (Opens (↑(↑X : TopCat) : Type u_1))ᵒᵖ),
  NatTrans.app
      ((pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).c ≫ Functor.whiskerRight (eqToHom sorry) s.pt.presheaf)
      x =
    s.fst.c.app x
Tactic: refine PresheafedSpace.Hom.ext _ _ ?_ <| NatTrans.ext <| funext fun x => ?_
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : (Opens (↑(↑X : TopCat) : Type u_1))ᵒᵖ
⊢ NatTrans.app
      ((pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).c ≫ Functor.whiskerRight (eqToHom ⋯) s.pt.presheaf)
      x =
    s.fst.c.app x
have refine_2 : ∀ (x : Opens (↑(↑X : TopCat) : Type u_1)),
  Eq (α := X.presheaf.obj (op x) ⟶ ((Opens.map s.fst.base).op ⋙ s.pt.presheaf).obj (op x))
    (NatTrans.app
      ((pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).c ≫ Functor.whiskerRight (eqToHom sorry) s.pt.presheaf)
      (op x))
    (s.fst.c.app (op x))
Tactic: induction x with | op x => ?_
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens (↑(↑X : TopCat) : Type u_1)
⊢ Eq (α := X.presheaf.obj (op x) ⟶ ((Opens.map s.fst.base).op ⋙ s.pt.presheaf).obj (op x))
    (NatTrans.app
      ((pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).c ≫ Functor.whiskerRight (eqToHom ⋯) s.pt.presheaf)
      (op x))
    (s.fst.c.app (op x))
have refine_2 : Eq (α := X.presheaf.obj (op x) ⟶ ((Opens.map s.fst.base).op ⋙ s.pt.presheaf).obj (op x))
  (CategoryStruct.comp (Y :=
    (Functor.obj (TopCat.Presheaf.pushforward C (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).base)
          s.pt.presheaf).obj
      (op x))
    (CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj (unop (op x))))) (invApp f (unop (op x)))
          (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) sorry) (unop (op x)))) ≫
            Y.presheaf.map (eqToHom sorry)) ≫
        s.snd.c.app (sorry.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).fst.base).op.obj (op x))) ≫
          s.pt.presheaf.map (eqToHom sorry) :
      X.presheaf.obj (op x) ⟶
        (Functor.obj (TopCat.Presheaf.pushforward C (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).base)
              s.pt.presheaf).obj
          (op x))
    (NatTrans.app (Functor.whiskerRight (eqToHom sorry) s.pt.presheaf) (op x) :
      (Functor.obj (TopCat.Presheaf.pushforward C (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).base)
              s.pt.presheaf).obj
          (op x) ⟶
        ((Opens.map s.fst.base).op ⋙ s.pt.presheaf).obj (op x)))
  (s.fst.c.app (op x))
Tactic: change ((_ ≫ _) ≫ _ ≫ _) ≫ _ = _
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens (↑(↑X : TopCat) : Type u_1)
⊢ Eq (α := X.presheaf.obj (op x) ⟶ ((Opens.map s.fst.base).op ⋙ s.pt.presheaf).obj (op x))
    (CategoryStruct.comp (Y :=
      (Functor.obj (TopCat.Presheaf.pushforward C (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).base)
            s.pt.presheaf).obj
        (op x))
      (CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj (unop (op x))))) (invApp f (unop (op x)))
            (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) ⋯) (unop (op x)))) ≫
              Y.presheaf.map (eqToHom ⋯)) ≫
          s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).fst.base).op.obj (op x))) ≫
            s.pt.presheaf.map (eqToHom ⋯) :
        X.presheaf.obj (op x) ⟶
          (Functor.obj
                (TopCat.Presheaf.pushforward C (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).base)
                s.pt.presheaf).obj
            (op x))
      (NatTrans.app (Functor.whiskerRight (eqToHom ⋯) s.pt.presheaf) (op x) :
        (Functor.obj (TopCat.Presheaf.pushforward C (pullbackConeOfLeftLift f g s ≫ (pullbackConeOfLeft f g).fst).base)
                s.pt.presheaf).obj
            (op x) ⟶
          ((Opens.map s.fst.base).op ⋙ s.pt.presheaf).obj (op x)))
    (s.fst.c.app (op x))
have refine_2 : CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj x))) (invApp f x)
    (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) sorry) x)) ≫
      CategoryStruct.comp (Y :=
        (Functor.obj (TopCat.Presheaf.pushforward C (pullbackConeOfLeft f g).fst.base)
              (pullbackConeOfLeft f g).pt.presheaf).obj
          (op x))
        (Y.presheaf.map (eqToHom sorry) :
          Y.presheaf.obj ((Opens.map g.base).op.obj (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) sorry) x))) ⟶
            Y.presheaf.obj
              (sorry.functor.op.obj (Functor.obj (Functor.op (Opens.map (pullback.fst f.base g.base))) (op x))))
        (s.snd.c.app (sorry.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).fst.base).op.obj (op x))) ≫
            s.pt.presheaf.map (eqToHom sorry) ≫ NatTrans.app (Functor.whiskerRight (eqToHom sorry) s.pt.presheaf) (op x) :
          (Functor.obj (TopCat.Presheaf.pushforward C (pullbackConeOfLeft f g).fst.base)
                  (pullbackConeOfLeft f g).pt.presheaf).obj
              (op x) ⟶
            ((Opens.map s.fst.base).op ⋙ s.pt.presheaf).obj (op x))) =
  s.fst.c.app (op x)
Tactic: simp_rw [Category.assoc]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens (↑(↑X : TopCat) : Type u_1)
⊢ CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj x))) (invApp f x)
      (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) ⋯) x)) ≫
        CategoryStruct.comp (Y :=
          (Functor.obj (TopCat.Presheaf.pushforward C (pullbackConeOfLeft f g).fst.base)
                (pullbackConeOfLeft f g).pt.presheaf).obj
            (op x))
          (Y.presheaf.map (eqToHom ⋯) :
            Y.presheaf.obj ((Opens.map g.base).op.obj (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) ⋯) x))) ⟶
              Y.presheaf.obj
                (⋯.functor.op.obj (Functor.obj (Functor.op (Opens.map (pullback.fst f.base g.base))) (op x))))
          (s.snd.c.app (⋯.functor.op.obj ((Opens.map (pullbackConeOfLeft f g).fst.base).op.obj (op x))) ≫
              s.pt.presheaf.map (eqToHom ⋯) ≫ NatTrans.app (Functor.whiskerRight (eqToHom ⋯) s.pt.presheaf) (op x) :
            (Functor.obj (TopCat.Presheaf.pushforward C (pullbackConeOfLeft f g).fst.base)
                    (pullbackConeOfLeft f g).pt.presheaf).obj
                (op x) ⟶
              ((Opens.map s.fst.base).op ⋙ s.pt.presheaf).obj (op x))) =
    s.fst.c.app (op x)
have refine_2 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Tactic: erw [← s.pt.presheaf.map_comp]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens (↑(↑X : TopCat) : Type u_1)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have refine_2 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Tactic: erw [s.snd.c.naturality_assoc]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens (↑(↑X : TopCat) : Type u_1)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have this : Eq (α :=
  Z.presheaf.obj (op ((opensFunctor f).obj x)) ⟶
    (Functor.obj (TopCat.Presheaf.pushforward C (s.fst ≫ f).base) s.pt.presheaf).obj (op ((opensFunctor f).obj x)))
  ((s.fst ≫ f).c.app (op ((opensFunctor f).obj x)))
  ((s.snd ≫ g).c.app (op ((opensFunctor f).obj x)) ≫ s.pt.presheaf.map (eqToHom sorry))
Tactic: have := congr_app s.condition (op (opensFunctor f |>.obj x))
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
x : Opens (↑(↑X : TopCat) : Type u_1)
this :
  Eq (α :=
    Z.presheaf.obj (op ((opensFunctor f).obj x)) ⟶
      (Functor.obj (TopCat.Presheaf.pushforward C (s.fst ≫ f).base) s.pt.presheaf).obj (op ((opensFunctor f).obj x)))
    (f.c.app (op ((opensFunctor f).obj x)) ≫ s.fst.c.app (op ((Opens.map f.base).obj ((opensFunctor f).obj x))))
    ((g.c.app (op ((opensFunctor f).obj x)) ≫ s.snd.c.app (op ((Opens.map g.base).obj ((opensFunctor f).obj x)))) ≫
      s.pt.presheaf.map (eqToHom ⋯))
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have this₁ : Eq (α :=
  Z.presheaf.obj (op ((opensFunctor f).obj x)) ⟶
    (Functor.obj (TopCat.Presheaf.pushforward C (s.snd ≫ g).base) s.pt.presheaf).obj (op ((opensFunctor f).obj x)))
  ((f.c.app (op ((opensFunctor f).obj x)) ≫ s.fst.c.app (op ((Opens.map f.base).obj ((opensFunctor f).obj x)))) ≫
    inv (s.pt.presheaf.map (eqToHom sorry)))
  (g.c.app (op ((opensFunctor f).obj x)) ≫ s.snd.c.app (op ((Opens.map g.base).obj ((opensFunctor f).obj x))))
Tactic: rw [← IsIso.comp_inv_eq] at this
Cleared hypotheses: this