mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Basic

Goal state before:
pp: X : Scheme
I : X.IdealSheafData
Z : Closeds ↥X
U : ↑X.affineOpens
f : ↑Γ(X, ↑U)
F : Γ(X, ↑U) ⟶ Γ(X, X.basicOpen f) := X.presheaf.map (homOfLE ⋯).op
x : ↑Γ(X, ↑U)
hx : x ∈ PrimeSpectrum.vanishingIdeal (⇑(IsAffineOpen.fromSpec ⋯) ⁻¹' ↑Z)
this :
  ∀ (p : ↥(Spec Γ(X, ↑(X.affineBasicOpen f)))), (IsAffineOpen.fromSpec ⋯) p ∈ Z → (CommRingCat.Hom.hom F) x ∈ p.asIdeal
⊢ x ∈
    Ideal.comap (CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op))
      (PrimeSpectrum.vanishingIdeal (⇑(IsAffineOpen.fromSpec ⋯) ⁻¹' ↑Z))

Tactic:
by
          simpa [PrimeSpectrum.mem_vanishingIdeal] using! this

Goal state after:
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Basic

Goal state before:
pp: case refine_1.a
X Y : Scheme
f : X.Hom Y
inst✝ : QuasiCompact f
U✝ U : ↑Y.affineOpens
s : ↑Γ(Y, ↑U)
⊢ Ideal.map (CommRingCat.Hom.hom (Y.presheaf.map (homOfLE ⋯).op)) (RingHom.ker (CommRingCat.Hom.hom (f.app ↑U))) ≤
    RingHom.ker (CommRingCat.Hom.hom (f.app ↑(Y.affineBasicOpen s)))

pp: case a
X Y : Scheme
f : X.Hom Y
inst✝ : QuasiCompact f
U✝ U : ↑Y.affineOpens
s : ↑Γ(Y, ↑U)
⊢ RingHom.ker (CommRingCat.Hom.hom (f.app ↑(Y.affineBasicOpen s))) ≤
    Ideal.map (CommRingCat.Hom.hom (Y.presheaf.map (homOfLE ⋯).op)) (RingHom.ker (CommRingCat.Hom.hom (f.app ↑U)))

Tactic:
· refine Ideal.map_le_iff_le_comap.mpr fun x hx ↦ ?_
    simp_rw [RingHom.comap_ker, ← CommRingCat.hom_comp, Scheme.affineBasicOpen_coe, f.naturality,
      CommRingCat.hom_comp, ← RingHom.comap_ker]
    exact Ideal.ker_le_comap _ hx

Goal state after:
case a
X Y : Scheme
f : X.Hom Y
inst✝ : QuasiCompact f
U✝ U : ↑Y.affineOpens
s : ↑Γ(Y, ↑U)
⊢ RingHom.ker (CommRingCat.Hom.hom (f.app ↑(Y.affineBasicOpen s))) ≤
    Ideal.map (CommRingCat.Hom.hom (Y.presheaf.map (homOfLE ⋯).op)) (RingHom.ker (CommRingCat.Hom.hom (f.app ↑U)))
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: case e'_2.e'_6
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
x : ↑Γ(X, ↑V)
hx : x ∈ I.ideal V
p : ↥(I.glueDataObj U)
hp : p ∈ Opposite.unop (Opposite.op (I.glueDataObjι U ⁻¹ᵁ (↑U).ι ⁻¹ᵁ ↑V))
f : ↑Γ(X, ↑U)
g : ↑Γ(X, ↑V)
hfg : X.basicOpen f = X.basicOpen g
hf : ↑((I.glueDataObjι U) p) ∈ X.basicOpen f
this : IsIso (Hom.app (I.glueDataObjMap ⋯) (I.glueDataObjι U ⁻¹ᵁ (↑U).ι ⁻¹ᵁ X.basicOpen f))
H : (X.homOfLE ⋯ ≫ (↑U).ι) ⁻¹ᵁ ↑V = ⊤
⊢ x ∈
    Ideal.comap
      (CommRingCat.Hom.hom
        (X.presheaf.map
            ((homOfLE ⋯).op ≫ (homOfLE ⋯).op ≫ ((Hom.opensFunctor (X.basicOpen f).ι).map (eqToHom ⋯).op.unop).op) ≫
          (↑(X.affineBasicOpen f)).topIso.hom))
      (I.ideal (X.affineBasicOpen f))

Tactic:
simp only [homOfLE_leOfHom, Scheme.Hom.comp_base,
    TopologicalSpace.Opens.map_comp_obj, eqToHom_op, eqToHom_unop, ← Functor.map_comp,
    Scheme.Opens.topIso_hom, Category.assoc]

Goal state after:
case e'_2.e'_6
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
x : ↑Γ(X, ↑V)
hx : x ∈ I.ideal V
p : ↥(I.glueDataObj U)
hp : p ∈ Opposite.unop (Opposite.op (I.glueDataObjι U ⁻¹ᵁ (↑U).ι ⁻¹ᵁ ↑V))
f : ↑Γ(X, ↑U)
g : ↑Γ(X, ↑V)
hfg : X.basicOpen f = X.basicOpen g
hf : ↑((I.glueDataObjι U) p) ∈ X.basicOpen f
this : IsIso (Hom.app (I.glueDataObjMap ⋯) (I.glueDataObjι U ⁻¹ᵁ (↑U).ι ⁻¹ᵁ X.basicOpen f))
H : (X.homOfLE ⋯ ≫ (↑U).ι) ⁻¹ᵁ ↑V = ⊤
⊢ x ∈
    Ideal.comap
      (CommRingCat.Hom.hom
        (X.presheaf.map
          ((homOfLE ⋯).op ≫ (homOfLE ⋯).op ≫ ((Hom.opensFunctor (X.basicOpen f).ι).map (eqToHom ⋯)).op ≫ eqToHom ⋯)))
      (I.ideal (X.affineBasicOpen f))
Module: Mathlib.AlgebraicGeometry.PullbackCarrier

Goal state before:
pp: X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
t : ↥(pullback f g)
⊢ Spec.map (ofPointTensor t) ≫ (Triplet.ofPoint t).SpecTensorTo = (pullback f g).fromSpecResidueField t

Tactic:
apply pullback.hom_ext

Goal state after:
case h₀
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
t : ↥(pullback f g)
⊢ (Spec.map (ofPointTensor t) ≫ (Triplet.ofPoint t).SpecTensorTo) ≫ pullback.fst f g =
    (pullback f g).fromSpecResidueField t ≫ pullback.fst f g

case h₁
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
t : ↥(pullback f g)
⊢ (Spec.map (ofPointTensor t) ≫ (Triplet.ofPoint t).SpecTensorTo) ≫ pullback.snd f g =
    (pullback f g).fromSpecResidueField t ≫ pullback.snd f g
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: X : Scheme
I : X.IdealSheafData
i : ↑X.affineOpens
⊢ I.glueDataT i i = 𝟙 (I.glueDataObjPullback (i, i).1 (i, i).2)

Tactic:
apply pullback.hom_ext

Goal state after:
case h₀
X : Scheme
I : X.IdealSheafData
i : ↑X.affineOpens
⊢ I.glueDataT i i ≫ pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯) =
    𝟙 (I.glueDataObjPullback (i, i).1 (i, i).2) ≫ pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)

case h₁
X : Scheme
I : X.IdealSheafData
i : ↑X.affineOpens
⊢ I.glueDataT i i ≫ pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯) =
    𝟙 (I.glueDataObjPullback (i, i).1 (i, i).2) ≫ pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯)
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ pullback.lift (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, k).2 ⋯)
        (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, i).2 ⋯) ⋯ ≫
      pullback.snd (pullback.fst (I.glueDataObjι (j, k).1) (X.homOfLE ⋯))
        (pullback.fst (I.glueDataObjι (j, i).1) (X.homOfLE ⋯)) =
    pullback.fst (pullback.fst (I.glueDataObjι (i, j).1) (X.homOfLE ⋯))
        (pullback.fst (I.glueDataObjι (i, k).1) (X.homOfLE ⋯)) ≫
      I.glueDataT i j

Tactic:
apply pullback.hom_ext

Goal state after:
case h₀
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ (pullback.lift (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, k).2 ⋯)
          (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, i).2 ⋯) ⋯ ≫
        pullback.snd (pullback.fst (I.glueDataObjι (j, k).1) (X.homOfLE ⋯))
          (pullback.fst (I.glueDataObjι (j, i).1) (X.homOfLE ⋯))) ≫
      pullback.fst (I.glueDataObjι (j, i).1) (X.homOfLE ⋯) =
    (pullback.fst (pullback.fst (I.glueDataObjι (i, j).1) (X.homOfLE ⋯))
          (pullback.fst (I.glueDataObjι (i, k).1) (X.homOfLE ⋯)) ≫
        I.glueDataT i j) ≫
      pullback.fst (I.glueDataObjι (j, i).1) (X.homOfLE ⋯)

case h₁
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ (pullback.lift (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, k).2 ⋯)
          (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, i).2 ⋯) ⋯ ≫
        pullback.snd (pullback.fst (I.glueDataObjι (j, k).1) (X.homOfLE ⋯))
          (pullback.fst (I.glueDataObjι (j, i).1) (X.homOfLE ⋯))) ≫
      pullback.snd (I.glueDataObjι (j, i).1) (X.homOfLE ⋯) =
    (pullback.fst (pullback.fst (I.glueDataObjι (i, j).1) (X.homOfLE ⋯))
          (pullback.fst (I.glueDataObjι (i, k).1) (X.homOfLE ⋯)) ≫
        I.glueDataT i j) ≫
      pullback.snd (I.glueDataObjι (j, i).1) (X.homOfLE ⋯)
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
      pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
        pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯ =
    𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)))

Tactic:
apply pullback.hom_ext

Goal state after:
case h₀
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ (pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
        pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
          pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
      pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) =
    𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))

case h₁
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ (pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
        pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
          pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
      pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) =
    𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: case h₀
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ (pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
        pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
          pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
      pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) =
    𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))

Tactic:
apply pullback.hom_ext

Goal state after:
case h₀.h₀
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ ((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
          pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
            pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
        pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯) =
    (𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
        pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)

case h₁
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ ((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
          pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
            pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
        pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯) =
    (𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
        pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯)
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: case h₁
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ (pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
        pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
          pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
      pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) =
    𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))

Tactic:
apply pullback.hom_ext

Goal state after:
case h₁.h₀
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ ((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
          pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
            pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
        pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯) =
    (𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
        pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)

case h₁
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ ((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫
          pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫
            pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ⋯) ⋯) ≫
        pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯) =
    (𝟙 (pullback (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
        pullback.snd (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯))) ≫
      pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯)
Module: Mathlib.RingTheory.Finiteness.FinitePresentationLocal

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
s✝ : Set S
hs✝ : Ideal.span s✝ = ⊤
s : Finset S
h₁ : ↑s ⊆ s✝
hs : Ideal.span ↑s = ⊤
h : ∀ (i : ↥s), FinitePresentation R (Localization.Away ↑i)
hfintype : FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : ↥s → MvPolynomial (Fin n) R
hg' : ∀ (g : ↥s), f (g' g) = ↑g
h' : ↥s → MvPolynomial (Fin n) R
hh' : ∀ (g : ↥s), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
q : MvPolynomial (Fin n) R
⊢ l ∈ Finsupp.supported S S ↑s.attach

Tactic:
rintro a -

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
s✝ : Set S
hs✝ : Ideal.span s✝ = ⊤
s : Finset S
h₁ : ↑s ⊆ s✝
hs : Ideal.span ↑s = ⊤
h : ∀ (i : ↥s), FinitePresentation R (Localization.Away ↑i)
hfintype : FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : ↥s → MvPolynomial (Fin n) R
hg' : ∀ (g : ↥s), f (g' g) = ↑g
h' : ↥s → MvPolynomial (Fin n) R
hh' : ∀ (g : ↥s), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
q : MvPolynomial (Fin n) R
a : ↑↑s
⊢ a ∈ ↑s.attach
Module: Mathlib.AlgebraicGeometry.Morphisms.Separated

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.I₀) → (pullback f (𝒰.f i)).OpenCover
x : ↥X
i : 𝒰.I₀ := Cover.idx 𝒰 (f x)
y : ↥(𝒰.X i)
hy✝ : (𝒰.f i) y = f x
z : ↥(pullback f (𝒰.f i))
hz₁ : (pullback.fst f (𝒰.f i)) z = x
hz₂ : (pullback.snd f (𝒰.f i)) z = y
j : (𝒱 i).I₀ := Cover.idx (𝒱 i) z
w : ↥((𝒱 i).X j)
hy : ((𝒱 i).f j) w = z
⊢ (pullback.diagonal ((𝒱 i).f j ≫ pullback.snd f (𝒰.f i)) ≫
        pullback.map ((𝒱 i).f j ≫ pullback.snd f (𝒰.f i)) ((𝒱 i).f j ≫ pullback.snd f (𝒰.f i)) (pullback.snd f (𝒰.f i))
          (pullback.snd f (𝒰.f i)) ((𝒱 i).f j) ((𝒱 i).f j) (𝟙 (𝒰.X i)) ⋯ ⋯) ≫
      pullback.map (pullback.snd f (𝒰.f i)) (pullback.snd f (𝒰.f i)) f f (pullback.fst f (𝒰.f i))
        (pullback.fst f (𝒰.f i)) (𝒰.f i) ⋯ ⋯ =
    ((𝒱 i).f j ≫ pullback.fst f (𝒰.f i)) ≫ pullback.diagonal f

Tactic:
apply pullback.hom_ext

Goal state after:
case h₀
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.I₀) → (pullback f (𝒰.f i)).OpenCover
x : ↥X
i : 𝒰.I₀ := ⋯
y : ↥(𝒰.X i)
hy✝ : (𝒰.f i) y = f x
z : ↥(pullback f (𝒰.f i))
hz₁ : (pullback.fst f (𝒰.f i)) z = x
hz₂ : (pullback.snd f (𝒰.f i)) z = y
j : (𝒱 i).I₀ := ⋯
w : ↥((𝒱 i).X j)
hy : ((𝒱 i).f j) w = z
⊢ ((pullback.diagonal ((𝒱 i).f j ≫ pullback.snd f (𝒰.f i)) ≫
          pullback.map ((𝒱 i).f j ≫ pullback.snd f (𝒰.f i)) ((𝒱 i).f j ≫ pullback.snd f (𝒰.f i))
            (pullback.snd f (𝒰.f i)) (pullback.snd f (𝒰.f i)) ((𝒱 i).f j) ((𝒱 i).f j) (𝟙 (𝒰.X i)) ⋯ ⋯) ≫
        pullback.map (pullback.snd f (𝒰.f i)) (pullback.snd f (𝒰.f i)) f f (pullback.fst f (𝒰.f i))
          (pullback.fst f (𝒰.f i)) (𝒰.f i) ⋯ ⋯) ≫
      pullback.fst f f =
    (((𝒱 i).f j ≫ pullback.fst f (𝒰.f i)) ≫ pullback.diagonal f) ≫ pullback.fst f f

case h₁
X Y : Scheme
f : X ⟶ Y
𝒰 : Y.OpenCover
𝒱 : (i : 𝒰.I₀) → (pullback f (𝒰.f i)).OpenCover
x : ↥X
i : 𝒰.I₀ := ⋯
y : ↥(𝒰.X i)
hy✝ : (𝒰.f i) y = f x
z : ↥(pullback f (𝒰.f i))
hz₁ : (pullback.fst f (𝒰.f i)) z = x
hz₂ : (pullback.snd f (𝒰.f i)) z = y
j : (𝒱 i).I₀ := ⋯
w : ↥((𝒱 i).X j)
hy : ((𝒱 i).f j) w = z
⊢ ((pullback.diagonal ((𝒱 i).f j ≫ pullback.snd f (𝒰.f i)) ≫
          pullback.map ((𝒱 i).f j ≫ pullback.snd f (𝒰.f i)) ((𝒱 i).f j ≫ pullback.snd f (𝒰.f i))
            (pullback.snd f (𝒰.f i)) (pullback.snd f (𝒰.f i)) ((𝒱 i).f j) ((𝒱 i).f j) (𝟙 (𝒰.X i)) ⋯ ⋯) ≫
        pullback.map (pullback.snd f (𝒰.f i)) (pullback.snd f (𝒰.f i)) f f (pullback.fst f (𝒰.f i))
          (pullback.fst f (𝒰.f i)) (𝒰.f i) ⋯ ⋯) ≫
      pullback.snd f f =
    (((𝒱 i).f j ≫ pullback.fst f (𝒰.f i)) ≫ pullback.diagonal f) ≫ pullback.snd f f
Module: Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion

Goal state before:
pp: X Y : Scheme
inst✝¹ : IsAffine Y
f : X ⟶ Y
inst✝ : CompactSpace ↥X
hfopen : IsOpenMap ⇑f
hfinj₁ : Function.Injective ⇑f
hfinj₂ : Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop f))
x : ↥X
φ : Γ(Y, ⊤) ⟶ Γ(X, ⊤) := Scheme.Hom.appTop f
𝒰 : X.OpenCover := X.affineCover.finiteSubcover
res : (i : 𝒰.I₀) → Γ(X, ⊤) ⟶ Γ(𝒰.X i, ⊤) := fun i ↦ Scheme.Hom.appTop (𝒰.f i)
g : ↑Γ(Y, ⊤)
h : (ConcreteCategory.hom (X.presheaf.germ (f ⁻¹ᵁ ⊤) x True.intro)) ((ConcreteCategory.hom (Scheme.Hom.app f ⊤)) g) = 0
U : Opens ↑↑X.toRingedSpace.toPresheafedSpace
w : U ⟶ ⊤
hx : x ∈ U
hg : (ConcreteCategory.hom (X.toRingedSpace.presheaf.map w.op)) ((ConcreteCategory.hom φ) g) = 0
s : ↑Γ(Y, ⊤)
hyv : f x ∈ Y.basicOpen s
bsle : Y.basicOpen s ≤ { carrier := ⇑f '' U.carrier, is_open' := ⋯ }
W : (i : 𝒰.I₀) → Opens ↥(𝒰.X i) :=
  fun i ↦ (𝒰.X i).basicOpen ((ConcreteCategory.hom (res i)) ((ConcreteCategory.hom φ) s))
hwle : ∀ (i : 𝒰.I₀), W i ≤ 𝒰.f i ⁻¹ᵁ U
h0 : ∀ (i : 𝒰.I₀), (ConcreteCategory.hom (Scheme.Hom.appLE (𝒰.f i) ⊤ (W i) ⋯)) ((ConcreteCategory.hom φ) g) = 0
h1 : ∀ (i : 𝒰.I₀), ∃ n, (ConcreteCategory.hom (res i)) ((ConcreteCategory.hom φ) (s ^ n * g)) = 0
⊢ (ConcreteCategory.hom (Y.presheaf.Γgerm (f x))) g = 0

Tactic:
have h2 : ∃ n, ∀ i, (res i) (φ (s ^ n * g)) = 0 := by
    choose fn hfn using h1
    refine ⟨Finset.sup Finset.univ fn, fun i ↦ ?_⟩
    rw [map_mul, map_pow, map_mul, map_pow]
    simp only [map_mul, map_pow, map_mul, map_pow] at hfn
    apply pow_mul_eq_zero_of_le (Finset.le_sup (Finset.mem_univ i)) (hfn i)

Goal state after:
X Y : Scheme
inst✝¹ : IsAffine Y
f : X ⟶ Y
inst✝ : CompactSpace ↥X
hfopen : IsOpenMap ⇑f
hfinj₁ : Function.Injective ⇑f
hfinj₂ : Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop f))
x : ↥X
φ : Γ(Y, ⊤) ⟶ Γ(X, ⊤) := Scheme.Hom.appTop f
𝒰 : X.OpenCover := X.affineCover.finiteSubcover
res : (i : 𝒰.I₀) → Γ(X, ⊤) ⟶ Γ(𝒰.X i, ⊤) := fun i ↦ Scheme.Hom.appTop (𝒰.f i)
g : ↑Γ(Y, ⊤)
h : (ConcreteCategory.hom (X.presheaf.germ (f ⁻¹ᵁ ⊤) x True.intro)) ((ConcreteCategory.hom (Scheme.Hom.app f ⊤)) g) = 0
U : Opens ↑↑X.toRingedSpace.toPresheafedSpace
w : U ⟶ ⊤
hx : x ∈ U
hg : (ConcreteCategory.hom (X.toRingedSpace.presheaf.map w.op)) ((ConcreteCategory.hom φ) g) = 0
s : ↑Γ(Y, ⊤)
hyv : f x ∈ Y.basicOpen s
bsle : Y.basicOpen s ≤ { carrier := ⇑f '' U.carrier, is_open' := ⋯ }
W : (i : 𝒰.I₀) → Opens ↥(𝒰.X i) :=
  fun i ↦ (𝒰.X i).basicOpen ((ConcreteCategory.hom (res i)) ((ConcreteCategory.hom φ) s))
hwle : ∀ (i : 𝒰.I₀), W i ≤ 𝒰.f i ⁻¹ᵁ U
h0 : ∀ (i : 𝒰.I₀), (ConcreteCategory.hom (Scheme.Hom.appLE (𝒰.f i) ⊤ (W i) ⋯)) ((ConcreteCategory.hom φ) g) = 0
h1 : ∀ (i : 𝒰.I₀), ∃ n, (ConcreteCategory.hom (res i)) ((ConcreteCategory.hom φ) (s ^ n * g)) = 0
h2 : ∃ n, ∀ (i : 𝒰.I₀), (ConcreteCategory.hom (res i)) ((ConcreteCategory.hom φ) (s ^ n * g)) = 0
⊢ (ConcreteCategory.hom (Y.presheaf.Γgerm (f x))) g = 0
Module: Mathlib.AlgebraicGeometry.Noetherian

Goal state before:
pp: case basicOpen
X : Scheme
ι : Sort u_1
S : ι → ↑X.affineOpens
hS : ⨆ i, ↑(S i) = ⊤
hS' : ∀ (i : ι), IsNoetherianRing ↑Γ(X, ↑(S i))
U : ↑X.affineOpens
f : ↑Γ(X, ↑U)
hN : IsNoetherianRing ↑Γ(X, ↑U)
⊢ IsNoetherianRing ↑Γ(X, ↑(X.affineBasicOpen f))

Tactic:
have := U.prop.isLocalization_basicOpen f

Goal state after:
case basicOpen
X : Scheme
ι : Sort u_1
S : ι → ↑X.affineOpens
hS : ⨆ i, ↑(S i) = ⊤
hS' : ∀ (i : ι), IsNoetherianRing ↑Γ(X, ↑(S i))
U : ↑X.affineOpens
f : ↑Γ(X, ↑U)
hN : IsNoetherianRing ↑Γ(X, ↑U)
this : IsLocalization.Away f ↑Γ(X, X.basicOpen f)
⊢ IsNoetherianRing ↑Γ(X, ↑(X.affineBasicOpen f))
Module: Mathlib.AlgebraicGeometry.Noetherian

Goal state before:
pp: case openCover
X : Scheme
ι : Sort u_1
S : ι → ↑X.affineOpens
hS : ⨆ i, ↑(S i) = ⊤
hS' : ∀ (i : ι), IsNoetherianRing ↑Γ(X, ↑(S i))
U : ↑X.affineOpens
s : Finset ↑Γ(X, ↑U)
x✝ : Ideal.span ↑s = ⊤
hN : ∀ (f : ↥s), IsNoetherianRing ↑Γ(X, ↑(X.affineBasicOpen ↑f))
f : ↑Γ(X, ↑U)
hf : f ∈ s
this : IsNoetherianRing ↑Γ(X, X.basicOpen f)
⊢ IsNoetherianRing (Localization.Away ↑⟨f, hf⟩)

Tactic:
have := U.prop.isLocalization_basicOpen f

Goal state after:
case openCover
X : Scheme
ι : Sort u_1
S : ι → ↑X.affineOpens
hS : ⨆ i, ↑(S i) = ⊤
hS' : ∀ (i : ι), IsNoetherianRing ↑Γ(X, ↑(S i))
U : ↑X.affineOpens
s : Finset ↑Γ(X, ↑U)
x✝ : Ideal.span ↑s = ⊤
hN : ∀ (f : ↥s), IsNoetherianRing ↑Γ(X, ↑(X.affineBasicOpen ↑f))
f : ↑Γ(X, ↑U)
hf : f ∈ s
this✝ : IsNoetherianRing ↑Γ(X, X.basicOpen f)
this : IsLocalization.Away f ↑Γ(X, X.basicOpen f)
⊢ IsNoetherianRing (Localization.Away ↑⟨f, hf⟩)
Module: Mathlib.AlgebraicGeometry.Geometrically.Integral

Goal state before:
pp: X S : Scheme
f : X ⟶ S
inst✝⁴ : GeometricallyIntegral f
inst✝³ : Flat f
inst✝² : UniversallyOpen f
inst✝¹ : IsIntegral S
inst✝ : IsLocallyNoetherian S
⊢ IsIntegral X

Tactic:
rw [isIntegral_iff_irreducibleSpace_and_isReduced]

Goal state after:
X S : Scheme
f : X ⟶ S
inst✝⁴ : GeometricallyIntegral f
inst✝³ : Flat f
inst✝² : UniversallyOpen f
inst✝¹ : IsIntegral S
inst✝ : IsLocallyNoetherian S
⊢ IrreducibleSpace ↥X ∧ IsReduced X
Module: Mathlib.AlgebraicGeometry.Geometrically.Integral

Goal state before:
pp: X S : Scheme
f : X ⟶ S
inst✝² : GeometricallyIntegral f
inst✝¹ : Subsingleton ↥S
inst✝ : IsIntegral S
⊢ IsIntegral X

Tactic:
rw [isIntegral_iff_irreducibleSpace_and_isReduced]

Goal state after:
X S : Scheme
f : X ⟶ S
inst✝² : GeometricallyIntegral f
inst✝¹ : Subsingleton ↥S
inst✝ : IsIntegral S
⊢ IrreducibleSpace ↥X ∧ IsReduced X
Module: Mathlib.RingTheory.RingHom.Integral

Goal state before:
pp: R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
hs : Ideal.span s = ⊤
H : ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] x ↦ x.IsIntegral) (Localization.awayMap f ↑r)
r : S
this✝¹ : Algebra R S := f.toAlgebra
t : R
ht : t ∈ s
this✝ : Algebra (Localization.Away t) (Localization.Away (f t)) := (Localization.awayMap f t).toAlgebra
this : IsScalarTower R (Localization.Away t) (Localization.Away (f t))
⊢ ∃ n, ↑⟨t, ht⟩ ^ n • r ∈ Subalgebra.toSubmodule (integralClosure R S)

Tactic:
have : _root_.IsIntegral (Localization.Away t) (algebraMap S (Localization.Away (f t)) r) :=
    H ⟨t, ht⟩ (algebraMap _ _ r)

Goal state after:
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
hs : Ideal.span s = ⊤
H : ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] x ↦ x.IsIntegral) (Localization.awayMap f ↑r)
r : S
this✝² : Algebra R S := f.toAlgebra
t : R
ht : t ∈ s
this✝¹ : Algebra (Localization.Away t) (Localization.Away (f t)) := (Localization.awayMap f t).toAlgebra
this✝ : IsScalarTower R (Localization.Away t) (Localization.Away (f t))
this : _root_.IsIntegral (Localization.Away t) ((algebraMap S (Localization.Away (f t))) r)
⊢ ∃ n, ↑⟨t, ht⟩ ^ n • r ∈ Subalgebra.toSubmodule (integralClosure R S)