awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
dsimp : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ Eq (α := (↑Γ(((Functor.const (Over i)).obj (opensCone D c i U).pt).obj (Over.mk (𝟙 i)), ⊤) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (X := Γ((opensDiagram D i U).obj (Over.mk (𝟙 i)), ⊤))
            (Scheme.Hom.appTop ((opensCone D c i U).π.app (Over.mk (𝟙 i)))) :
          (a : (↑Γ((opensDiagram D i U).obj (Over.mk (𝟙 i)), ⊤) : Type u)) →
            (↑Γ(((Functor.const (Over i)).obj (opensCone D c i U).pt).obj (Over.mk (𝟙 i)), ⊤) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Γ(D.obj i, U)) (Y := Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤))
                ((D.obj i).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u))
            s :
          (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u)) :
      (↑Γ(((Functor.const (Over i)).obj (opensCone D c i U).pt).obj (Over.mk (𝟙 i)), ⊤) : Type u))
    ((CommRingCat.Hom.hom (c.pt.presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
          (a : (↑(c.pt.presheaf.obj (Opposite.op (c.π.app i ⁻¹ᵁ U))) : Type u)) →
            (↑(c.pt.presheaf.obj (Opposite.op ((c.π.app i ⁻¹ᵁ U).ι ''ᵁ ⊤))) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
            s :
          (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u)) :
      (↑(c.pt.presheaf.obj (Opposite.op ((c.π.app i ⁻¹ᵁ U).ι ''ᵁ ⊤))) : Type u))

Tactic:
dsimp [Scheme.Opens.toScheme_presheaf_obj]

Premises used: AlgebraicGeometry.Scheme.Opens.toScheme_presheaf_obj
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
rw : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply]

Premises used: CategoryTheory.ConcreteCategory.comp_apply
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
h : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
congr! 2
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
simp [Scheme.Hom.app_eq_appLE, Scheme.Hom.resLE_appLE]

Premises used: AlgebraicGeometry.Scheme.Hom.app_eq_appLE, AlgebraicGeometry.Scheme.Hom.resLE_appLE
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hf :
  Eq (α := (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier)
            (Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯)) :
          (a : (↑Γ((↑(D.map (𝟙 i) ⁻¹ᵁ U) : Scheme), ⊤) : Type u)) → (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Γ(D.obj i, U)) (Y := Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤))
                ((D.obj i).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u))
            s :
          (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u)) :
      (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    (0 : (↑Γ((opensDiagram D i U).obj j, ⊤) : Type u))
⊢ ∃ (j : I) (f : j ⟶ i),
    Eq (α := (↑Γ(D.obj j, D.map f ⁻¹ᵁ U) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (Scheme.Hom.app (D.map f) U) :
            (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj j, D.map f ⁻¹ᵁ U) : Type u))
          s :
        (↑Γ(D.obj j, D.map f ⁻¹ᵁ U) : Type u))
      (0 : (↑Γ(D.obj j, D.map f ⁻¹ᵁ U) : Type u))

Tactic:
⟨j.left, f.left, ?_⟩

Premises used: CategoryTheory.Over.Hom.left, CategoryTheory.Over.left, Exists.intro
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hf :
  Eq (α := (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier)
            (Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯)) :
          (a : (↑Γ((↑(D.map (𝟙 i) ⁻¹ᵁ U) : Scheme), ⊤) : Type u)) → (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Γ(D.obj i, U)) (Y := Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤))
                ((D.obj i).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u))
            s :
          (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u)) :
      (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    (0 : (↑Γ((opensDiagram D i U).obj j, ⊤) : Type u))
refine :
  Eq (α := (↑Γ(D.obj j.left, D.map (Over.Hom.left f) ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (D.map (Over.Hom.left f)) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj j.left, D.map (Over.Hom.left f) ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(D.obj j.left, D.map (Over.Hom.left f) ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(D.obj j.left, D.map (Over.Hom.left f) ⁻¹ᵁ U) : Type u))
⊢ ∃ (j : I) (f : j ⟶ i),
    Eq (α := (↑Γ(D.obj j, D.map f ⁻¹ᵁ U) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (Scheme.Hom.app (D.map f) U) :
            (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj j, D.map f ⁻¹ᵁ U) : Type u))
          s :
        (↑Γ(D.obj j, D.map f ⁻¹ᵁ U) : Type u))
      (0 : (↑Γ(D.obj j, D.map f ⁻¹ᵁ U) : Type u))

Tactic:
refine ⟨j.left, f.left, ?_⟩
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hf :
  Eq (α := (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier)
            (Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯)) :
          (a : (↑Γ((↑(D.map (𝟙 i) ⁻¹ᵁ U) : Scheme), ⊤) : Type u)) → (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Γ(D.obj i, U)) (Y := Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤))
                ((D.obj i).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u))
            s :
          (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u)) :
      (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    (0 : (↑Γ((opensDiagram D i U).obj j, ⊤) : Type u))
⊢ Over.Hom.left f = j.hom

Tactic:
simpa using Over.w f

Premises used: CategoryTheory.Over.w
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hf :
  Eq (α := (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier)
            (Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯)) :
          (a : (↑Γ((↑(D.map (𝟙 i) ⁻¹ᵁ U) : Scheme), ⊤) : Type u)) → (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Γ(D.obj i, U)) (Y := Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤))
                ((D.obj i).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u))
            s :
          (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u)) :
      (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    (0 : (↑Γ((opensDiagram D i U).obj j, ⊤) : Type u))
⊢ Over.Hom.left f = j.hom

Tactic:
have hf' : f.left = j.hom := by simpa using Over.w f
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hf :
  Eq (α := (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier)
            (Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯)) :
          (a : (↑Γ((↑(D.map (𝟙 i) ⁻¹ᵁ U) : Scheme), ⊤) : Type u)) → (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Γ(D.obj i, U)) (Y := Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤))
                ((D.obj i).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u))
            s :
          (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u)) :
      (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    (0 : (↑Γ((opensDiagram D i U).obj j, ⊤) : Type u))
hf' : Over.Hom.left f = j.hom
⊢ D.map (Over.Hom.left f) ⁻¹ᵁ U ≤ (D.map j.hom ⁻¹ᵁ U).ι ''ᵁ ⊤

Tactic:
simp [hf']
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hf :
  Eq (α := (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier)
            (Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯)) :
          (a : (↑Γ((↑(D.map (𝟙 i) ⁻¹ᵁ U) : Scheme), ⊤) : Type u)) → (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Γ(D.obj i, U)) (Y := Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤))
                ((D.obj i).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u))
            s :
          (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u)) :
      (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    (0 : (↑Γ((opensDiagram D i U).obj j, ⊤) : Type u))
hf' : Over.Hom.left f = j.hom
⊢ Scheme.Hom.app (D.map (Over.Hom.left f)) U =
    (D.obj i).presheaf.map (Quiver.Hom.op (homOfLE ⋯)) ≫
      Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯) ≫
        (D.obj j.left).presheaf.map (Quiver.Hom.op (homOfLE ⋯))

Tactic:
simp [Scheme.Hom.app_eq_appLE, Scheme.Hom.resLE_appLE]

Premises used: AlgebraicGeometry.Scheme.Hom.app_eq_appLE, AlgebraicGeometry.Scheme.Hom.resLE_appLE
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofiltered I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U : (D.obj i).Opens
hU : IsCompact (↑U : Set ↥(D.obj i))
s : (↑Γ(D.obj i, U) : Type u)
hs :
  Eq (α := (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Scheme.Hom.app (c.π.app i) U) :
          (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
        s :
      (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(((Functor.const I).obj c.pt).obj i, c.π.app i ⁻¹ᵁ U) : Type u))
this : CompactSpace.{u} ↥((opensDiagram D i U).obj (Over.mk (𝟙 i)))
H : (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤ ≤ U
j : Over i
f : j ⟶ Over.mk (𝟙 i)
hf :
  Eq (α := (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier)
            (Scheme.Hom.appTop (Scheme.Hom.resLE (D.map (Over.Hom.left f)) (D.map (𝟙 i) ⁻¹ᵁ U) (D.map j.hom ⁻¹ᵁ U) ⋯)) :
          (a : (↑Γ((↑(D.map (𝟙 i) ⁻¹ᵁ U) : Scheme), ⊤) : Type u)) → (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (X := Γ(D.obj i, U)) (Y := Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤))
                ((D.obj i).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
              (a : (↑Γ(D.obj i, U) : Type u)) → (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u))
            s :
          (↑Γ(D.obj i, (D.map (𝟙 i) ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u)) :
      (↑Γ((↑(D.map j.hom ⁻¹ᵁ U) : Scheme), ⊤) : Type u))
    (0 : (↑Γ((opensDiagram D i U).obj j, ⊤) : Type u))
hf' : Over.Hom.left f = j.hom
⊢ Eq (α := (↑Γ(D.obj j.left, D.map (Over.Hom.left f) ⁻¹ᵁ U) : Type u))
    (0 : (↑Γ(D.obj j.left, D.map (Over.Hom.left f) ⁻¹ᵁ U) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (X := Γ(D.obj j.left, (D.map j.hom ⁻¹ᵁ U).ι ''ᵁ ⊤)) (Y :=
            Γ(D.obj j.left, D.map (Over.Hom.left f) ⁻¹ᵁ U)) ((D.obj j.left).presheaf.map (Quiver.Hom.op (homOfLE ⋯))) :
          (a : (↑Γ(D.obj j.left, (D.map j.hom ⁻¹ᵁ U).ι ''ᵁ ⊤) : Type u)) →
            (↑Γ(D.obj j.left, D.map (Over.Hom.left f) ⁻¹ᵁ U) : Type u))
        0 :
      (↑Γ(D.obj j.left, D.map (Over.Hom.left f) ⁻¹ᵁ U) : Type u))

Tactic:
simp