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 : IsCofilteredOrEmpty I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_3 : ∀ (i : I), Nonempty.{u + 1} ↥(D.obj i)
inst_4 : ∀ (i : I), CompactSpace.{u} ↥(D.obj i)
h : Nonempty I
i : I
this_1 : IsCofiltered I
𝒰 : (D.obj i).OpenCover := OpenCover.finiteSubcover (D.obj i).affineCover
this : ∀ (i' : 𝒰.I₀), IsAffine.{u} (𝒰.X i')
i' : (j : 𝒰.I₀) → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  (m : j ∈ insert i (Finset.image i' Finset.univ)) →
    IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image
          (fun (j : 𝒰.I₀) ↦
            PSigma.mk (β := fun (X : I) ↦
              (Y : I) ×'
                (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
              (i' j)
              (PSigma.mk (β := fun (Y : I) ↦
                (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
                i
                (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                  (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                  (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
          Finset.univ) ⟶
      j :=
  fun (j : I) ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image
        (fun (j : 𝒰.I₀) ↦
          PSigma.mk (β := fun (X : I) ↦
            (Y : I) ×'
              (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
            (i' j)
            (PSigma.mk (β := fun (Y : I) ↦
              (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
              i
              (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
        Finset.univ)
j : 𝒰.I₀
⊢ 𝒰.f j ≫ 𝟙 (D.obj i) = 𝟙 (𝒰.X j) ≫ 𝒰.f j

Tactic:
simp
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofilteredOrEmpty I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_3 : ∀ (i : I), Nonempty.{u + 1} ↥(D.obj i)
inst_4 : ∀ (i : I), CompactSpace.{u} ↥(D.obj i)
h : Nonempty I
i : I
this_1 : IsCofiltered I
𝒰 : (D.obj i).OpenCover := OpenCover.finiteSubcover (D.obj i).affineCover
this : ∀ (i' : 𝒰.I₀), IsAffine.{u} (𝒰.X i')
i' : (j : 𝒰.I₀) → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  (m : j ∈ insert i (Finset.image i' Finset.univ)) →
    IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image
          (fun (j : 𝒰.I₀) ↦
            PSigma.mk (β := fun (X : I) ↦
              (Y : I) ×'
                (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
              (i' j)
              (PSigma.mk (β := fun (Y : I) ↦
                (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
                i
                (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                  (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                  (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
          Finset.univ) ⟶
      j :=
  fun (j : I) ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image
        (fun (j : 𝒰.I₀) ↦
          PSigma.mk (β := fun (X : I) ↦
            (Y : I) ×'
              (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
            (i' j)
            (PSigma.mk (β := fun (Y : I) ↦
              (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
              i
              (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
        Finset.univ)
j : 𝒰.I₀
F : (Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X j ⟶
  (Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j :=
  pullback.map (D.map (g i ⋯)) (𝒰.f j) (D.map (f j)) (𝒰.f j) (D.map (g (i' j) ⋯)) (𝟙 (𝒰.X j)) (𝟙 (D.obj i)) ⋯ ⋯
⊢ IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X j)

Tactic:
Function.isEmpty F

Premises used: Function.isEmpty, CategoryTheory.ConcreteCategory.hom, AlgebraicGeometry.LocallyRingedSpace.Hom.toHom, AlgebraicGeometry.Scheme.Hom.toLRSHom', AlgebraicGeometry.PresheafedSpace.Hom.base, DFunLike.coe
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofilteredOrEmpty I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_3 : ∀ (i : I), Nonempty.{u + 1} ↥(D.obj i)
inst_4 : ∀ (i : I), CompactSpace.{u} ↥(D.obj i)
h : Nonempty I
i : I
this_1 : IsCofiltered I
𝒰 : (D.obj i).OpenCover := OpenCover.finiteSubcover (D.obj i).affineCover
this : ∀ (i' : 𝒰.I₀), IsAffine.{u} (𝒰.X i')
i' : (j : 𝒰.I₀) → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  (m : j ∈ insert i (Finset.image i' Finset.univ)) →
    IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image
          (fun (j : 𝒰.I₀) ↦
            PSigma.mk (β := fun (X : I) ↦
              (Y : I) ×'
                (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
              (i' j)
              (PSigma.mk (β := fun (Y : I) ↦
                (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
                i
                (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                  (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                  (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
          Finset.univ) ⟶
      j :=
  fun (j : I) ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image
        (fun (j : 𝒰.I₀) ↦
          PSigma.mk (β := fun (X : I) ↦
            (Y : I) ×'
              (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
            (i' j)
            (PSigma.mk (β := fun (Y : I) ↦
              (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
              i
              (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
        Finset.univ)
j : 𝒰.I₀
F : (Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X j ⟶
  (Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j :=
  pullback.map (D.map (g i ⋯)) (𝒰.f j) (D.map (f j)) (𝒰.f j) (D.map (g (i' j) ⋯)) (𝟙 (𝒰.X j)) (𝟙 (D.obj i)) ⋯ ⋯
⊢ IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X j)

Tactic:
exact Function.isEmpty F

Premises used: Function.isEmpty, CategoryTheory.ConcreteCategory.hom, AlgebraicGeometry.LocallyRingedSpace.Hom.toHom, AlgebraicGeometry.Scheme.Hom.toLRSHom', AlgebraicGeometry.PresheafedSpace.Hom.base, DFunLike.coe
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofilteredOrEmpty I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_3 : ∀ (i : I), Nonempty.{u + 1} ↥(D.obj i)
inst_4 : ∀ (i : I), CompactSpace.{u} ↥(D.obj i)
h : Nonempty I
i : I
this_1 : IsCofiltered I
𝒰 : (D.obj i).OpenCover := OpenCover.finiteSubcover (D.obj i).affineCover
this : ∀ (i' : 𝒰.I₀), IsAffine.{u} (𝒰.X i')
i' : (j : 𝒰.I₀) → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  (m : j ∈ insert i (Finset.image i' Finset.univ)) →
    IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image
          (fun (j : 𝒰.I₀) ↦
            PSigma.mk (β := fun (X : I) ↦
              (Y : I) ×'
                (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
              (i' j)
              (PSigma.mk (β := fun (Y : I) ↦
                (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
                i
                (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                  (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                  (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
          Finset.univ) ⟶
      j :=
  fun (j : I) ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image
        (fun (j : 𝒰.I₀) ↦
          PSigma.mk (β := fun (X : I) ↦
            (Y : I) ×'
              (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
            (i' j)
            (PSigma.mk (β := fun (Y : I) ↦
              (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
              i
              (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
        Finset.univ)
⊢ ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X j)

Tactic:
have (j : 𝒰.I₀) : IsEmpty ((𝒰.pullback₁ (D.map (g i (by simp)))).X j) := by
        let F : (𝒰.pullback₁ (D.map (g i (by simp)))).X j ⟶
            (𝒰.pullback₁ (D.map (f j))).X j :=
          pullback.map _ _ _ _ (D.map (g _ (by simp))) (𝟙 _) (𝟙 _) (by
            rw [← D.map_comp, IsCofiltered.infTo_commutes]
            · simp [g]
            · simp
            · exact Finset.mem_image_of_mem _ (Finset.mem_univ _)) (by simp)
        exact Function.isEmpty 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 : IsCofilteredOrEmpty I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_3 : ∀ (i : I), Nonempty.{u + 1} ↥(D.obj i)
inst_4 : ∀ (i : I), CompactSpace.{u} ↥(D.obj i)
h : Nonempty I
i : I
this_1 : IsCofiltered I
𝒰 : (D.obj i).OpenCover := OpenCover.finiteSubcover (D.obj i).affineCover
this_2 : ∀ (i' : 𝒰.I₀), IsAffine.{u} (𝒰.X i')
i' : (j : 𝒰.I₀) → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  (m : j ∈ insert i (Finset.image i' Finset.univ)) →
    IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image
          (fun (j : 𝒰.I₀) ↦
            PSigma.mk (β := fun (X : I) ↦
              (Y : I) ×'
                (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
              (i' j)
              (PSigma.mk (β := fun (Y : I) ↦
                (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
                i
                (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                  (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                  (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
          Finset.univ) ⟶
      j :=
  fun (j : I) ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image
        (fun (j : 𝒰.I₀) ↦
          PSigma.mk (β := fun (X : I) ↦
            (Y : I) ×'
              (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
            (i' j)
            (PSigma.mk (β := fun (Y : I) ↦
              (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
              i
              (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
        Finset.univ)
this : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X j)
⊢ i ∈ insert i (Finset.image i' Finset.univ)

Tactic:
simp
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofilteredOrEmpty I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_3 : ∀ (i : I), Nonempty.{u + 1} ↥(D.obj i)
inst_4 : ∀ (i : I), CompactSpace.{u} ↥(D.obj i)
h : Nonempty I
i : I
this_1 : IsCofiltered I
𝒰 : (D.obj i).OpenCover := OpenCover.finiteSubcover (D.obj i).affineCover
this_2 : ∀ (i' : 𝒰.I₀), IsAffine.{u} (𝒰.X i')
i' : (j : 𝒰.I₀) → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  (m : j ∈ insert i (Finset.image i' Finset.univ)) →
    IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image
          (fun (j : 𝒰.I₀) ↦
            PSigma.mk (β := fun (X : I) ↦
              (Y : I) ×'
                (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
              (i' j)
              (PSigma.mk (β := fun (Y : I) ↦
                (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
                i
                (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                  (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                  (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
          Finset.univ) ⟶
      j :=
  fun (j : I) ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image
        (fun (j : 𝒰.I₀) ↦
          PSigma.mk (β := fun (X : I) ↦
            (Y : I) ×'
              (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
            (i' j)
            (PSigma.mk (β := fun (Y : I) ↦
              (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
              i
              (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
        Finset.univ)
this : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X j)
obtain :
  ∀
    (x :
      ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X
          (Cover.idx (Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰) (Nonempty.some ⋯)))),
    False
⊢ False

Tactic:
obtain ⟨x, -⟩ :=
        Cover.covers (𝒰.pullback₁ (D.map (g i (by simp)))) (Nonempty.some inferInstance)

Premises used: AlgebraicGeometry.Scheme.Cover.covers, inferInstance, Nonempty.some, CategoryTheory.Functor.map, CategoryTheory.Precoverage.ZeroHypercover.pullback₁
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofilteredOrEmpty I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_3 : ∀ (i : I), Nonempty.{u + 1} ↥(D.obj i)
inst_4 : ∀ (i : I), CompactSpace.{u} ↥(D.obj i)
h : Nonempty I
i : I
this_1 : IsCofiltered I
𝒰 : (D.obj i).OpenCover := OpenCover.finiteSubcover (D.obj i).affineCover
this_2 : ∀ (i' : 𝒰.I₀), IsAffine.{u} (𝒰.X i')
i' : (j : 𝒰.I₀) → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  (m : j ∈ insert i (Finset.image i' Finset.univ)) →
    IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image
          (fun (j : 𝒰.I₀) ↦
            PSigma.mk (β := fun (X : I) ↦
              (Y : I) ×'
                (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
              (i' j)
              (PSigma.mk (β := fun (Y : I) ↦
                (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
                i
                (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                  (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                  (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
          Finset.univ) ⟶
      j :=
  fun (j : I) ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image
        (fun (j : 𝒰.I₀) ↦
          PSigma.mk (β := fun (X : I) ↦
            (Y : I) ×'
              (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
            (i' j)
            (PSigma.mk (β := fun (Y : I) ↦
              (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
              i
              (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
        Finset.univ)
this : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X j)
x :
  ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X
      (Cover.idx (Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰) (Nonempty.some ⋯)))
⊢ False

Tactic:
(this _).elim x

Premises used: congrArg, true_or, IsEmpty.elim, Finset.mem_image._simp_1, Exists, inferInstance, Nonempty.some, CategoryTheory.Functor.map, funext, And, congr, eq_self, of_eq_true, Finset.mem_univ._simp_1, congrFun', Or, CategoryTheory.Precoverage.ZeroHypercover.pullback₁, Finset.mem_insert._simp_1, AlgebraicGeometry.Scheme.Cover.idx, Eq, true_and, Eq.trans
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_1 : IsCofilteredOrEmpty I
inst_2 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_3 : ∀ (i : I), Nonempty.{u + 1} ↥(D.obj i)
inst_4 : ∀ (i : I), CompactSpace.{u} ↥(D.obj i)
h : Nonempty I
i : I
this_1 : IsCofiltered I
𝒰 : (D.obj i).OpenCover := OpenCover.finiteSubcover (D.obj i).affineCover
this_2 : ∀ (i' : 𝒰.I₀), IsAffine.{u} (𝒰.X i')
i' : (j : 𝒰.I₀) → I
f : (j : 𝒰.I₀) → i' j ⟶ i
hf : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (f j)) 𝒰).X j)
g : (j : I) →
  (m : j ∈ insert i (Finset.image i' Finset.univ)) →
    IsCofiltered.inf (insert i (Finset.image i' Finset.univ))
        (Finset.image
          (fun (j : 𝒰.I₀) ↦
            PSigma.mk (β := fun (X : I) ↦
              (Y : I) ×'
                (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
              (i' j)
              (PSigma.mk (β := fun (Y : I) ↦
                (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                  (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
                i
                (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                  (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                  (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
          Finset.univ) ⟶
      j :=
  fun (j : I) ↦
    IsCofiltered.infTo (insert i (Finset.image i' Finset.univ))
      (Finset.image
        (fun (j : 𝒰.I₀) ↦
          PSigma.mk (β := fun (X : I) ↦
            (Y : I) ×'
              (_ : X ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (X ⟶ Y))
            (i' j)
            (PSigma.mk (β := fun (Y : I) ↦
              (_ : i' j ∈ insert i (Finset.image i' Finset.univ)) ×'
                (_ : Y ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ Y))
              i
              (PSigma.mk (β := fun (x : i' j ∈ insert i (Finset.image i' Finset.univ)) ↦
                (_ : i ∈ insert i (Finset.image i' Finset.univ)) ×' (i' j ⟶ i)) ⋯
                (PSigma.mk (β := fun (x : i ∈ insert i (Finset.image i' Finset.univ)) ↦ i' j ⟶ i) ⋯ (f j)))))
        Finset.univ)
this : ∀ (j : 𝒰.I₀), IsEmpty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X j)
x :
  ↥((Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰).X
      (Cover.idx (Precoverage.ZeroHypercover.pullback₁ (D.map (g i ⋯)) 𝒰) (Nonempty.some ⋯)))
⊢ False

Tactic:
exact (this _).elim x