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')
inr :
  ∀ (j : 𝒰.I₀)
    (H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)),
    Nonempty.{u + 1} ↥c.pt
⊢ Nonempty.{u + 1} ↥c.pt

Tactic:
obtain ⟨j, H⟩ :
        ∃ j : 𝒰.I₀, ∀ {i'} (f : i' ⟶ i), Nonempty ((𝒰.pullback₁ (D.map f)).X j) := by
      by_contra! H
      choose i' f hf using H
      let g (j) := IsCofiltered.infTo (insert i (Finset.univ.image i'))
        (Finset.univ.image fun j : 𝒰.I₀ ↦ ⟨_, _, by simp, by simp, f j⟩) (X := j)
      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
      obtain ⟨x, -⟩ :=
        Cover.covers (𝒰.pullback₁ (D.map (g i (by simp)))) (Nonempty.some inferInstance)
      exact (this _).elim x

Premises used: CategoryTheory.PreZeroHypercover.I₀, CategoryTheory.Precoverage.ZeroHypercover.toPreZeroHypercover
-- 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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
⊢ ∀ (i' : Over i), IsAffine.{u} (F.obj i')

Tactic:
have (i' : _) : IsAffine (F.obj i') :=
      have : IsAffineHom (pullback.snd (D.map i'.hom) (𝒰.f j)) :=
        MorphismProperty.pullback_snd _ _ inferInstance
      isAffine_of_isAffineHom (pullback.snd (D.map i'.hom) (𝒰.f j))
-- 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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
⊢ ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')

Tactic:
have (i' : _) : Nonempty (F.obj i') := H i'.hom
-- 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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this_3 : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
this : ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')
e : F ⟶ (F ⋙ Γ.rightOp) ⋙ Scheme.Spec := F.whiskerLeft ΓSpec.adjunction.unit
⊢ ∀ (i_1 : Over i), IsIso (e.app i_1)

Tactic:
have (i : _) : IsIso (e.app i) := IsAffine.affine
-- 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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this_3 : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
this_4 : ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')
e : F ⟶ (F ⋙ Γ.rightOp) ⋙ Scheme.Spec := F.whiskerLeft ΓSpec.adjunction.unit
this : ∀ (i_1 : Over i), IsIso (e.app i_1)
⊢ IsIso e

Tactic:
have : IsIso e := NatIso.isIso_of_isIso_app e
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst_1 : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_2 : IsCofilteredOrEmpty I
inst_3 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_4 : ∀ (i : I), Nonempty.{u + 1} ↥(D.obj i)
inst_5 : ∀ (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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this_3 : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
this_4 : ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')
e : F ⟶ (F ⋙ Γ.rightOp) ⋙ Scheme.Spec := F.whiskerLeft ΓSpec.adjunction.unit
this_5 : ∀ (i_1 : Over i), IsIso (e.app i_1)
this : IsIso e
c' : LimitCone F :=
  { cone := (Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp))),
    isLimit :=
      (Equiv.symm (IsLimit.postcomposeInvEquiv (asIso e) (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) :
          (a : IsLimit (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) →
            IsLimit ((Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))))
        (isLimitOfPreserves Scheme.Spec (limit.isLimit (F ⋙ Γ.rightOp))) }
inst : Nontrivial.{u} (↑(Opposite.unop (limit.cone (F ⋙ Γ.rightOp)).pt) : Type u)
⊢ Nonempty.{u + 1} ↥c'.cone.pt

Tactic:
apply +allowSynthFailures PrimeSpectrum.instNonemptyOfNontrivial

Premises used: PrimeSpectrum.instNonemptyOfNontrivial
-- Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit
Goal:
I : Type u
inst_1 : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst_2 : IsCofilteredOrEmpty I
inst_3 : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst_4 : ∀ (i : I), Nonempty.{u + 1} ↥(D.obj i)
inst_5 : ∀ (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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this_3 : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
this_4 : ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')
e : F ⟶ (F ⋙ Γ.rightOp) ⋙ Scheme.Spec := F.whiskerLeft ΓSpec.adjunction.unit
this_5 : ∀ (i_1 : Over i), IsIso (e.app i_1)
this : IsIso e
c' : LimitCone F :=
  { cone := (Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp))),
    isLimit :=
      (Equiv.symm (IsLimit.postcomposeInvEquiv (asIso e) (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) :
          (a : IsLimit (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) →
            IsLimit ((Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))))
        (isLimitOfPreserves Scheme.Spec (limit.isLimit (F ⋙ Γ.rightOp))) }
i' : (Over i)ᵒᵖ
inst : Nonempty.{u + 1} ↥(↑⊤ : Scheme)
⊢ Nontrivial.{u} (↑((F ⋙ Γ.rightOp).leftOp.obj i') : Type u)

Tactic:
apply +allowSynthFailures Scheme.component_nontrivial

Premises used: AlgebraicGeometry.Scheme.component_nontrivial
-- 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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this_3 : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
this_4 : ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')
e : F ⟶ (F ⋙ Γ.rightOp) ⋙ Scheme.Spec := F.whiskerLeft ΓSpec.adjunction.unit
this_5 : ∀ (i_1 : Over i), IsIso (e.app i_1)
this : IsIso e
c' : LimitCone F :=
  { cone := (Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp))),
    isLimit :=
      (Equiv.symm (IsLimit.postcomposeInvEquiv (asIso e) (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) :
          (a : IsLimit (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) →
            IsLimit ((Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))))
        (isLimitOfPreserves Scheme.Spec (limit.isLimit (F ⋙ Γ.rightOp))) }
i' : (Over i)ᵒᵖ
⊢ Nonempty.{u + 1} ↥(↑⊤ : Scheme)

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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this_3 : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
this_4 : ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')
e : F ⟶ (F ⋙ Γ.rightOp) ⋙ Scheme.Spec := F.whiskerLeft ΓSpec.adjunction.unit
this_5 : ∀ (i_1 : Over i), IsIso (e.app i_1)
this : IsIso e
c' : LimitCone F :=
  { cone := (Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp))),
    isLimit :=
      (Equiv.symm (IsLimit.postcomposeInvEquiv (asIso e) (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) :
          (a : IsLimit (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) →
            IsLimit ((Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))))
        (isLimitOfPreserves Scheme.Spec (limit.isLimit (F ⋙ Γ.rightOp))) }
⊢ ∀ (i' : (Over i)ᵒᵖ), Nontrivial.{u} (↑((F ⋙ Γ.rightOp).leftOp.obj i') : Type u)

Tactic:
have (i' : _) : Nontrivial ((F ⋙ Scheme.Γ.rightOp).leftOp.obj i') := by
        apply +allowSynthFailures Scheme.component_nontrivial
        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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this_3 : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
this_4 : ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')
e : F ⟶ (F ⋙ Γ.rightOp) ⋙ Scheme.Spec := F.whiskerLeft ΓSpec.adjunction.unit
this_5 : ∀ (i_1 : Over i), IsIso (e.app i_1)
this_6 : IsIso e
c' : LimitCone F :=
  { cone := (Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp))),
    isLimit :=
      (Equiv.symm (IsLimit.postcomposeInvEquiv (asIso e) (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) :
          (a : IsLimit (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) →
            IsLimit ((Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))))
        (isLimitOfPreserves Scheme.Spec (limit.isLimit (F ⋙ Γ.rightOp))) }
this : ∀ (i' : (Over i)ᵒᵖ), Nontrivial.{u} (↑((F ⋙ Γ.rightOp).leftOp.obj i') : Type u)
⊢ Nontrivial.{u} (↑(Opposite.unop (limit.cone (F ⋙ Γ.rightOp)).pt) : Type u)

Tactic:
CommRingCat.FilteredColimits.nontrivial
        (isColimitCoconeLeftOpOfCone _ (limit.isLimit (F ⋙ Scheme.Γ.rightOp)))

Premises used: CategoryTheory.Limits.isColimitCoconeLeftOpOfCone, AlgebraicGeometry.Scheme.Γ, CategoryTheory.Functor.comp, CommRingCat.FilteredColimits.nontrivial, CategoryTheory.Limits.limit.isLimit, CategoryTheory.Functor.rightOp
-- 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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this_3 : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
this_4 : ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')
e : F ⟶ (F ⋙ Γ.rightOp) ⋙ Scheme.Spec := F.whiskerLeft ΓSpec.adjunction.unit
this_5 : ∀ (i_1 : Over i), IsIso (e.app i_1)
this_6 : IsIso e
c' : LimitCone F :=
  { cone := (Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp))),
    isLimit :=
      (Equiv.symm (IsLimit.postcomposeInvEquiv (asIso e) (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) :
          (a : IsLimit (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) →
            IsLimit ((Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))))
        (isLimitOfPreserves Scheme.Spec (limit.isLimit (F ⋙ Γ.rightOp))) }
this : ∀ (i' : (Over i)ᵒᵖ), Nontrivial.{u} (↑((F ⋙ Γ.rightOp).leftOp.obj i') : Type u)
⊢ Nontrivial.{u} (↑(Opposite.unop (limit.cone (F ⋙ Γ.rightOp)).pt) : Type u)

Tactic:
exact CommRingCat.FilteredColimits.nontrivial
        (isColimitCoconeLeftOpOfCone _ (limit.isLimit (F ⋙ Scheme.Γ.rightOp)))

Premises used: CategoryTheory.Limits.isColimitCoconeLeftOpOfCone, AlgebraicGeometry.Scheme.Γ, CategoryTheory.Functor.comp, CommRingCat.FilteredColimits.nontrivial, CategoryTheory.Limits.limit.isLimit, CategoryTheory.Functor.rightOp
-- 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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this_3 : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
this_4 : ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')
e : F ⟶ (F ⋙ Γ.rightOp) ⋙ Scheme.Spec := F.whiskerLeft ΓSpec.adjunction.unit
this_5 : ∀ (i_1 : Over i), IsIso (e.app i_1)
this : IsIso e
c' : LimitCone F :=
  { cone := (Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp))),
    isLimit :=
      (Equiv.symm (IsLimit.postcomposeInvEquiv (asIso e) (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) :
          (a : IsLimit (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) →
            IsLimit ((Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))))
        (isLimitOfPreserves Scheme.Spec (limit.isLimit (F ⋙ Γ.rightOp))) }
⊢ Nonempty.{u + 1} ↥c'.cone.pt

Tactic:
have : Nonempty c'.1.pt := by
      apply +allowSynthFailures PrimeSpectrum.instNonemptyOfNontrivial
      have (i' : _) : Nontrivial ((F ⋙ Scheme.Γ.rightOp).leftOp.obj i') := by
        apply +allowSynthFailures Scheme.component_nontrivial
        simp
      exact CommRingCat.FilteredColimits.nontrivial
        (isColimitCoconeLeftOpOfCone _ (limit.isLimit (F ⋙ Scheme.Γ.rightOp)))
-- 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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this_3 : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
this_4 : ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')
e : F ⟶ (F ⋙ Γ.rightOp) ⋙ Scheme.Spec := F.whiskerLeft ΓSpec.adjunction.unit
this_5 : ∀ (i_1 : Over i), IsIso (e.app i_1)
this_6 : IsIso e
c' : LimitCone F :=
  { cone := (Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp))),
    isLimit :=
      (Equiv.symm (IsLimit.postcomposeInvEquiv (asIso e) (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) :
          (a : IsLimit (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) →
            IsLimit ((Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))))
        (isLimitOfPreserves Scheme.Spec (limit.isLimit (F ⋙ Γ.rightOp))) }
this : Nonempty.{u + 1} ↥c'.cone.pt
α : F ⟶ Over.forget i ⋙ D :=
  Functor.whiskerRight (Functor.whiskerLeft (Over.post D) (Adjunction.counit (Over.mapPullbackAdj (𝒰.f j))))
    (Over.forget (D.obj i))
⊢ Nonempty.{u + 1} ↥c.pt

Tactic:
this.map (((Functor.Initial.isLimitWhiskerEquiv (Over.forget i) c).symm hc).lift
        ((Cone.postcompose α).obj c'.1))

Premises used: CategoryTheory.Functor.Initial.isLimitWhiskerEquiv, CategoryTheory.ConcreteCategory.hom, CategoryTheory.Over.forget, AlgebraicGeometry.LocallyRingedSpace.Hom.toHom, AlgebraicGeometry.Scheme.Hom.toLRSHom', AlgebraicGeometry.PresheafedSpace.Hom.base, Nonempty.map, Equiv.symm, DFunLike.coe, CategoryTheory.Functor.obj, CategoryTheory.Limits.IsLimit.lift, CategoryTheory.Limits.LimitCone.cone, CategoryTheory.Limits.Cone.postcompose
-- 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')
j : 𝒰.I₀
H : ∀ {i' : I} (f : i' ⟶ i), Nonempty.{u + 1} ↥((Precoverage.ZeroHypercover.pullback₁ (D.map f) 𝒰).X j)
F : Over i ⥤ Scheme := Over.post D ⋙ Over.pullback (𝒰.f j) ⋙ Over.forget (𝒰.X j)
this_3 : ∀ (i' : Over i), IsAffine.{u} (F.obj i')
this_4 : ∀ (i' : Over i), Nonempty.{u + 1} ↥(F.obj i')
e : F ⟶ (F ⋙ Γ.rightOp) ⋙ Scheme.Spec := F.whiskerLeft ΓSpec.adjunction.unit
this_5 : ∀ (i_1 : Over i), IsIso (e.app i_1)
this_6 : IsIso e
c' : LimitCone F :=
  { cone := (Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp))),
    isLimit :=
      (Equiv.symm (IsLimit.postcomposeInvEquiv (asIso e) (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) :
          (a : IsLimit (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))) →
            IsLimit ((Cone.postcompose (asIso e).inv).obj (Scheme.Spec.mapCone (limit.cone (F ⋙ Γ.rightOp)))))
        (isLimitOfPreserves Scheme.Spec (limit.isLimit (F ⋙ Γ.rightOp))) }
this : Nonempty.{u + 1} ↥c'.cone.pt
α : F ⟶ Over.forget i ⋙ D :=
  Functor.whiskerRight (Functor.whiskerLeft (Over.post D) (Adjunction.counit (Over.mapPullbackAdj (𝒰.f j))))
    (Over.forget (D.obj i))
⊢ Nonempty.{u + 1} ↥c.pt

Tactic:
exact this.map (((Functor.Initial.isLimitWhiskerEquiv (Over.forget i) c).symm hc).lift
        ((Cone.postcompose α).obj c'.1))

Premises used: CategoryTheory.Functor.Initial.isLimitWhiskerEquiv, CategoryTheory.ConcreteCategory.hom, CategoryTheory.Over.forget, AlgebraicGeometry.LocallyRingedSpace.Hom.toHom, AlgebraicGeometry.Scheme.Hom.toLRSHom', AlgebraicGeometry.PresheafedSpace.Hom.base, Nonempty.map, Equiv.symm, DFunLike.coe, CategoryTheory.Functor.obj, CategoryTheory.Limits.IsLimit.lift, CategoryTheory.Limits.LimitCone.cone, CategoryTheory.Limits.Cone.postcompose