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

Goal state before:
pp: X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
i : 𝒰.I₀
j : (Precoverage.ZeroHypercover.pullback₁ (pullback.fst (p1 𝒰 f g) (𝒰.f i)) (gluing 𝒰 f g).openCover).I₀
⊢ pullbackFstιToV 𝒰 f g i j ≫ fV 𝒰 f g j i ≫ (gluing 𝒰 f g).ι j =
    pullback.fst (pullback.fst (p1 𝒰 f g) (𝒰.f i)) ((gluing 𝒰 f g).openCover.f j) ≫ pullback.fst (p1 𝒰 f g) (𝒰.f i)

Tactic:
·

Goal state after:
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
i : 𝒰.I₀
j : (Precoverage.ZeroHypercover.pullback₁ (pullback.fst (p1 𝒰 f g) (𝒰.f i)) (gluing 𝒰 f g).openCover).I₀
⊢ pullbackFstιToV 𝒰 f g i j ≫ fV 𝒰 f g j i ≫ (gluing 𝒰 f g).ι j =
    pullback.fst (pullback.fst (p1 𝒰 f g) (𝒰.f i)) ((gluing 𝒰 f g).openCover.f j) ≫ pullback.fst (p1 𝒰 f g) (𝒰.f i)
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: case inl
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
x : ↥(Spec (CommRingCat.of R))
⊢ ((Spec.map (CommRingCat.ofHom (RingHom.fst R S))) x).asIdeal =
    ((PrimeSpectrum.primeSpectrumProd R S).symm (Sum.inl x)).asIdeal

pp: case inr
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
x : ↥(Spec (CommRingCat.of S))
⊢ ((Spec.map (CommRingCat.ofHom (RingHom.snd R S))) x).asIdeal =
    ((PrimeSpectrum.primeSpectrumProd R S).symm (Sum.inr x)).asIdeal

Tactic:
·

Goal state after:
case inl
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
x : ↥(Spec (CommRingCat.of R))
⊢ ((Spec.map (CommRingCat.ofHom (RingHom.fst R S))) x).asIdeal =
    ((PrimeSpectrum.primeSpectrumProd R S).symm (Sum.inl x)).asIdeal
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: case inr
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
x : ↥(Spec (CommRingCat.of S))
⊢ ((Spec.map (CommRingCat.ofHom (RingHom.snd R S))) x).asIdeal =
    ((PrimeSpectrum.primeSpectrumProd R S).symm (Sum.inr x)).asIdeal

Tactic:
·

Goal state after:
case inr
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
x : ↥(Spec (CommRingCat.of S))
⊢ ((Spec.map (CommRingCat.ofHom (RingHom.snd R S))) x).asIdeal =
    ((PrimeSpectrum.primeSpectrumProd R S).symm (Sum.inr x)).asIdeal
Module: Mathlib.AlgebraicGeometry.Pullbacks

Goal state before:
pp: case hom
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ pullback (p1 𝒰 f g) (𝒰.f i) ⟶ pullback (𝒰.f i ≫ f) g

pp: case inv
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ pullback (𝒰.f i ≫ f) g ⟶ pullback (p1 𝒰 f g) (𝒰.f i)

pp: case hom_inv_id
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ autoParam (?hom ≫ ?inv = 𝟙 (pullback (p1 𝒰 f g) (𝒰.f i))) Iso.hom_inv_id._autoParam

pp: case inv_hom_id
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ autoParam (?inv ≫ ?hom = 𝟙 (pullback (𝒰.f i ≫ f) g)) Iso.inv_hom_id._autoParam

Tactic:
·

Goal state after:
case hom
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ pullback (p1 𝒰 f g) (𝒰.f i) ⟶ pullback (𝒰.f i ≫ f) g
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: case inl
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
x : ↥(Spec (CommRingCat.of R))
⊢ IsIso
    (Scheme.Hom.stalkMap (coprodSpec R S) ((coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))) (Sum.inl x)))

pp: case inr
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
x : ↥(Spec (CommRingCat.of S))
⊢ IsIso
    (Scheme.Hom.stalkMap (coprodSpec R S) ((coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))) (Sum.inr x)))

Tactic:
·

Goal state after:
case inl
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
x : ↥(Spec (CommRingCat.of R))
⊢ IsIso
    (Scheme.Hom.stalkMap (coprodSpec R S) ((coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))) (Sum.inl x)))
Module: Mathlib.AlgebraicGeometry.Pullbacks

Goal state before:
pp: case inv
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ pullback (𝒰.f i ≫ f) g ⟶ pullback (p1 𝒰 f g) (𝒰.f i)

pp: case hom_inv_id
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ autoParam
    (pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ ≫ ?inv =
      𝟙 (pullback (p1 𝒰 f g) (𝒰.f i)))
    Iso.hom_inv_id._autoParam

pp: case inv_hom_id
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ autoParam
    (?inv ≫ pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ =
      𝟙 (pullback (𝒰.f i ≫ f) g))
    Iso.inv_hom_id._autoParam

Tactic:
·

Goal state after:
case inv
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ pullback (𝒰.f i ≫ f) g ⟶ pullback (p1 𝒰 f g) (𝒰.f i)
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: case inr
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
x : ↥(Spec (CommRingCat.of S))
⊢ IsIso
    (Scheme.Hom.stalkMap (coprodSpec R S) ((coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))) (Sum.inr x)))

Tactic:
·

Goal state after:
case inr
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
x : ↥(Spec (CommRingCat.of S))
⊢ IsIso
    (Scheme.Hom.stalkMap (coprodSpec R S) ((coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))) (Sum.inr x)))
Module: Mathlib.AlgebraicGeometry.Pullbacks

Goal state before:
pp: case hom_inv_id
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ autoParam
    (pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ ≫
        pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯ =
      𝟙 (pullback (p1 𝒰 f g) (𝒰.f i)))
    Iso.hom_inv_id._autoParam

pp: case inv_hom_id
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ autoParam
    (pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯ ≫
        pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ =
      𝟙 (pullback (𝒰.f i ≫ f) g))
    Iso.inv_hom_id._autoParam

Tactic:
·

Goal state after:
case hom_inv_id
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ autoParam
    (pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ ≫
        pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯ =
      𝟙 (pullback (p1 𝒰 f g) (𝒰.f i)))
    Iso.hom_inv_id._autoParam
Module: Mathlib.AlgebraicGeometry.Pullbacks

Goal state before:
pp: case hom_inv_id.h₀
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ (pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ ≫
        pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯) ≫
      pullback.fst (p1 𝒰 f g) (𝒰.f i) =
    𝟙 (pullback (p1 𝒰 f g) (𝒰.f i)) ≫ pullback.fst (p1 𝒰 f g) (𝒰.f i)

pp: case h₁
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ (pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ ≫
        pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯) ≫
      pullback.snd (p1 𝒰 f g) (𝒰.f i) =
    𝟙 (pullback (p1 𝒰 f g) (𝒰.f i)) ≫ pullback.snd (p1 𝒰 f g) (𝒰.f i)

Tactic:
·

Goal state after:
case hom_inv_id.h₀
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ (pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ ≫
        pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯) ≫
      pullback.fst (p1 𝒰 f g) (𝒰.f i) =
    𝟙 (pullback (p1 𝒰 f g) (𝒰.f i)) ≫ pullback.fst (p1 𝒰 f g) (𝒰.f i)
Module: Mathlib.AlgebraicGeometry.Pullbacks

Goal state before:
pp: case h₁
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ (pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ ≫
        pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯) ≫
      pullback.snd (p1 𝒰 f g) (𝒰.f i) =
    𝟙 (pullback (p1 𝒰 f g) (𝒰.f i)) ≫ pullback.snd (p1 𝒰 f g) (𝒰.f i)

Tactic:
·

Goal state after:
case h₁
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ (pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ ≫
        pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯) ≫
      pullback.snd (p1 𝒰 f g) (𝒰.f i) =
    𝟙 (pullback (p1 𝒰 f g) (𝒰.f i)) ≫ pullback.snd (p1 𝒰 f g) (𝒰.f i)
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ (coprodSpec R S).base =
    (TopCat.isoOfHomeo
        (PrimeSpectrum.primeSpectrumProdHomeo.trans (coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))))).inv

pp: ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ IsIso
    (TopCat.isoOfHomeo
        (PrimeSpectrum.primeSpectrumProdHomeo.trans (coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))))).inv

Tactic:
·

Goal state after:
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ (coprodSpec R S).base =
    (TopCat.isoOfHomeo
        (PrimeSpectrum.primeSpectrumProdHomeo.trans (coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))))).inv
Module: Mathlib.AlgebraicGeometry.Pullbacks

Goal state before:
pp: case inv_hom_id
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ autoParam
    (pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯ ≫
        pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ =
      𝟙 (pullback (𝒰.f i ≫ f) g))
    Iso.inv_hom_id._autoParam

Tactic:
·

Goal state after:
case inv_hom_id
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ autoParam
    (pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯ ≫
        pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ =
      𝟙 (pullback (𝒰.f i ≫ f) g))
    Iso.inv_hom_id._autoParam
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ IsIso
    (TopCat.isoOfHomeo
        (PrimeSpectrum.primeSpectrumProdHomeo.trans (coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))))).inv

Tactic:
·

Goal state after:
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ IsIso
    (TopCat.isoOfHomeo
        (PrimeSpectrum.primeSpectrumProdHomeo.trans (coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))))).inv
Module: Mathlib.AlgebraicGeometry.Pullbacks

Goal state before:
pp: case inv_hom_id.h₀
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ (pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯ ≫
        pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯) ≫
      pullback.fst (𝒰.f i ≫ f) g =
    𝟙 (pullback (𝒰.f i ≫ f) g) ≫ pullback.fst (𝒰.f i ≫ f) g

pp: case h₁
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ (pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯ ≫
        pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯) ≫
      pullback.snd (𝒰.f i ≫ f) g =
    𝟙 (pullback (𝒰.f i ≫ f) g) ≫ pullback.snd (𝒰.f i ≫ f) g

Tactic:
·

Goal state after:
case inv_hom_id.h₀
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ (pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯ ≫
        pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯) ≫
      pullback.fst (𝒰.f i ≫ f) g =
    𝟙 (pullback (𝒰.f i ≫ f) g) ≫ pullback.fst (𝒰.f i ≫ f) g
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: case h₁
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S✝ : Type u
inst✝¹ : CommRing R✝
inst✝ : CommRing S✝
R S : CommRingCat
⊢ coprod.inl ≫
      coprodComparison Scheme.Spec (op R) (op S) ≫
        Spec.map
          ((limit.isoLimitCone { cone := R.prodFan S, isLimit := R.prodFanIsLimit S }).inv ≫
            (opProdIsoCoprod R S).unop.inv) =
    coprod.inl ≫ coprodSpec ↑R ↑S

pp: case h₂
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S✝ : Type u
inst✝¹ : CommRing R✝
inst✝ : CommRing S✝
R S : CommRingCat
⊢ coprod.inr ≫
      coprodComparison Scheme.Spec (op R) (op S) ≫
        Spec.map
          ((limit.isoLimitCone { cone := R.prodFan S, isLimit := R.prodFanIsLimit S }).inv ≫
            (opProdIsoCoprod R S).unop.inv) =
    coprod.inr ≫ coprodSpec ↑R ↑S

Tactic:
·

Goal state after:
case h₁
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S✝ : Type u
inst✝¹ : CommRing R✝
inst✝ : CommRing S✝
R S : CommRingCat
⊢ coprod.inl ≫
      coprodComparison Scheme.Spec (op R) (op S) ≫
        Spec.map
          ((limit.isoLimitCone { cone := R.prodFan S, isLimit := R.prodFanIsLimit S }).inv ≫
            (opProdIsoCoprod R S).unop.inv) =
    coprod.inl ≫ coprodSpec ↑R ↑S
Module: Mathlib.AlgebraicGeometry.Pullbacks

Goal state before:
pp: case h₁
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ (pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯ ≫
        pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯) ≫
      pullback.snd (𝒰.f i ≫ f) g =
    𝟙 (pullback (𝒰.f i ≫ f) g) ≫ pullback.snd (𝒰.f i ≫ f) g

Tactic:
·

Goal state after:
case h₁
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g
s : PullbackCone f g
i : 𝒰.I₀
⊢ (pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯ ≫
        pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯) ≫
      pullback.snd (𝒰.f i ≫ f) g =
    𝟙 (pullback (𝒰.f i ≫ f) g) ≫ pullback.snd (𝒰.f i ≫ f) g
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: case h₂
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S✝ : Type u
inst✝¹ : CommRing R✝
inst✝ : CommRing S✝
R S : CommRingCat
⊢ coprod.inr ≫
      coprodComparison Scheme.Spec (op R) (op S) ≫
        Spec.map
          ((limit.isoLimitCone { cone := R.prodFan S, isLimit := R.prodFanIsLimit S }).inv ≫
            (opProdIsoCoprod R S).unop.inv) =
    coprod.inr ≫ coprodSpec ↑R ↑S

Tactic:
·

Goal state after:
case h₂
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S✝ : Type u
inst✝¹ : CommRing R✝
inst✝ : CommRing S✝
R S : CommRingCat
⊢ coprod.inr ≫
      coprodComparison Scheme.Spec (op R) (op S) ≫
        Spec.map
          ((limit.isoLimitCone { cone := R.prodFan S, isLimit := R.prodFanIsLimit S }).inv ≫
            (opProdIsoCoprod R S).unop.inv) =
    coprod.inr ≫ coprodSpec ↑R ↑S
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: case he
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
⊢ IsIdempotentElem (Function.update 0 i 1)

pp: case H
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
⊢ ∀ (x y : (a : ι) → R a),
    (algebraMap ((a : ι) → R a) (R i)) x = (algebraMap ((a : ι) → R a) (R i)) y ↔
      Function.update 0 i 1 * x = Function.update 0 i 1 * y

pp: case H'
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
⊢ Function.Surjective ⇑(algebraMap ((a : ι) → R a) (R i))

Tactic:
·

Goal state after:
case he
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
⊢ IsIdempotentElem (Function.update 0 i 1)
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: case H
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
⊢ ∀ (x y : (a : ι) → R a),
    (algebraMap ((a : ι) → R a) (R i)) x = (algebraMap ((a : ι) → R a) (R i)) y ↔
      Function.update 0 i 1 * x = Function.update 0 i 1 * y

pp: case H'
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
⊢ Function.Surjective ⇑(algebraMap ((a : ι) → R a) (R i))

Tactic:
·

Goal state after:
case H
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
⊢ ∀ (x y : (a : ι) → R a),
    (algebraMap ((a : ι) → R a) (R i)) x = (algebraMap ((a : ι) → R a) (R i)) y ↔
      Function.update 0 i 1 * x = Function.update 0 i 1 * y
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: case H.mp
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
x y : (a : ι) → R a
⊢ (algebraMap ((a : ι) → R a) (R i)) x = (algebraMap ((a : ι) → R a) (R i)) y →
    Function.update 0 i 1 * x = Function.update 0 i 1 * y

pp: case H.mpr
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
x y : (a : ι) → R a
⊢ Function.update 0 i 1 * x = Function.update 0 i 1 * y →
    (algebraMap ((a : ι) → R a) (R i)) x = (algebraMap ((a : ι) → R a) (R i)) y

Tactic:
·

Goal state after:
case H.mp
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
x y : (a : ι) → R a
⊢ (algebraMap ((a : ι) → R a) (R i)) x = (algebraMap ((a : ι) → R a) (R i)) y →
    Function.update 0 i 1 * x = Function.update 0 i 1 * y
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: case H.mpr
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
x y : (a : ι) → R a
⊢ Function.update 0 i 1 * x = Function.update 0 i 1 * y →
    (algebraMap ((a : ι) → R a) (R i)) x = (algebraMap ((a : ι) → R a) (R i)) y

Tactic:
·

Goal state after:
case H.mpr
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
x y : (a : ι) → R a
⊢ Function.update 0 i 1 * x = Function.update 0 i 1 * y →
    (algebraMap ((a : ι) → R a) (R i)) x = (algebraMap ((a : ι) → R a) (R i)) y
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: case H'
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
⊢ Function.Surjective ⇑(algebraMap ((a : ι) → R a) (R i))

Tactic:
·

Goal state after:
case H'
ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R✝ S : Type u
inst✝² : CommRing R✝
inst✝¹ : CommRing S
i : ι
R : ι → Type (max u_1 u)
inst✝ : (i : ι) → CommRing (R i)
this : Algebra ((i : ι) → R i) (R i) := ⋯
⊢ Function.Surjective ⇑(algebraMap ((a : ι) → R a) (R i))
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: ι : Type u
f : ι → Scheme
σ : Type v
g : σ → Scheme
X Y : Scheme
R S : Type u
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Finite σ
inst✝ : ∀ (i : σ), IsAffine (g i)
⊢ IsAffine (∐ g)

Tactic:
obtain ⟨ι, ⟨e⟩⟩ := Small.equiv_small.{u} (α := σ)
  have : Finite ι := e.finite_iff.mp ‹_›
  have (i : _) : IsAffine ((g ∘ e.symm) i) := by dsimp; infer_instance
  exact IsAffine.of_isIso ((Sigma.reindex e.symm g).inv ≫
    (Sigma.mapIso (fun i ↦ Scheme.isoSpec _)).hom ≫ sigmaSpec _)

Goal state after:
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: ι : Type u
X : Scheme
inst✝ : Finite ι
U : ι → X.Opens
hU : ∀ (i : ι), IsAffineOpen (U i)
hU' : _root_.Pairwise (Disjoint on U)
this : IsOpenImmersion (Sigma.desc fun i ↦ (U i).ι)
⊢ iSup U = Scheme.Hom.opensRange (Sigma.desc fun i ↦ (U i).ι)

pp: ι : Type u
X : Scheme
inst✝ : Finite ι
U : ι → X.Opens
hU : ∀ (i : ι), IsAffineOpen (U i)
hU' : _root_.Pairwise (Disjoint on U)
this : IsOpenImmersion (Sigma.desc fun i ↦ (U i).ι)
⊢ IsAffine (∐ fun i ↦ ↑(U i))

Tactic:
·

Goal state after:
ι : Type u
X : Scheme
inst✝ : Finite ι
U : ι → X.Opens
hU : ∀ (i : ι), IsAffineOpen (U i)
hU' : _root_.Pairwise (Disjoint on U)
this : IsOpenImmersion (Sigma.desc fun i ↦ (U i).ι)
⊢ iSup U = Scheme.Hom.opensRange (Sigma.desc fun i ↦ (U i).ι)
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: ι : Type u
X : Scheme
inst✝ : Finite ι
U : ι → X.Opens
hU : ∀ (i : ι), IsAffineOpen (U i)
hU' : _root_.Pairwise (Disjoint on U)
this : IsOpenImmersion (Sigma.desc fun i ↦ (U i).ι)
⊢ IsAffine (∐ fun i ↦ ↑(U i))

Tactic:
·

Goal state after:
ι : Type u
X : Scheme
inst✝ : Finite ι
U : ι → X.Opens
hU : ∀ (i : ι), IsAffineOpen (U i)
hU' : _root_.Pairwise (Disjoint on U)
this : IsOpenImmersion (Sigma.desc fun i ↦ (U i).ι)
⊢ IsAffine (∐ fun i ↦ ↑(U i))
Module: Mathlib.AlgebraicGeometry.Limits

Goal state before:
pp: X : Scheme
U V : X.Opens
hU : IsAffineOpen U
hV : IsAffineOpen V
H : Disjoint U V
⊢ IsAffineOpen (U ⊔ V)

Tactic:
convert!
    iSup_of_disjoint (U := fun i : Unit ⊕ Unit ↦ i.elim (fun _ ↦ U) (fun _ ↦ V)) (by simp_all)
      (by simp_all [_root_.Pairwise, Unique.forall_iff, ← Opens.coe_disjoint, disjoint_comm])

Goal state after:
X : Scheme
U V : X.Opens
hU : IsAffineOpen U
hV : IsAffineOpen V
H : Disjoint U V
⊢ U ⊔ V = ⨆ i, Sum.elim (fun x ↦ U) (fun x ↦ V) i