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

Goal state before:
pp: I : Type u
inst✝ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
c : Cone D
hc : IsLimit c
i : I
U : (D.obj i).Opens
j : Over i
⊢ IsOpenImmersion ((opensDiagramι D i U).app j)

Tactic:
delta opensDiagramι; infer_instance

Goal state after:
Module: Mathlib.AlgebraicGeometry.Birational.Composition

Goal state before:
pp: X Y : Scheme
inst✝² : PreirreducibleSpace ↥X
inst✝¹ : Nonempty ↥Y
f : X ⤏ Y
inst✝ : f.IsDominant
⊢ f.comp (id Y) = f

Tactic:
simp [RationalMap.comp_def]

Goal state after:
Module: Mathlib.AlgebraicGeometry.Birational.Composition

Goal state before:
pp: X₁ X₂ X₃ Y : Scheme
inst✝⁴ : PreirreducibleSpace ↥X₁
inst✝³ : IrreducibleSpace ↥X₂
inst✝² : Nonempty ↥X₃
f₁ : X₁ ⤏ X₂
inst✝¹ : f₁.IsDominant
f₂ : X₂ ⤏ X₃
inst✝ : f₂.IsDominant
f₃ : X₃ ⤏ Y
⊢ (f₁.comp f₂).comp f₃.representative.toRationalMap = f₁.comp (f₂.comp f₃.representative.toRationalMap)

Tactic:
simp_rw [comp_def, ← PartialMap.comp_assoc, PartialMap.toRationalMap_eq_iff]

Goal state after:
X₁ X₂ X₃ Y : Scheme
inst✝⁴ : PreirreducibleSpace ↥X₁
inst✝³ : IrreducibleSpace ↥X₂
inst✝² : Nonempty ↥X₃
f₁ : X₁ ⤏ X₂
inst✝¹ : f₁.IsDominant
f₂ : X₂ ⤏ X₃
inst✝ : f₂.IsDominant
f₃ : X₃ ⤏ Y
⊢ ((f₁.comp f₂).representative.comp f₃.representative).equiv
    ((f₁.representative.comp f₂.representative).comp f₃.representative)
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝² : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U V : (D.obj i).Opens
hU : IsCompact ↑U
hV : IsCompact ↑V
H : c.π.app i ⁻¹ᵁ U = c.π.app i ⁻¹ᵁ V
⊢ ∃ j fji, D.map fji ⁻¹ᵁ U = D.map fji ⁻¹ᵁ V

Tactic:
obtain ⟨j₁, fj₁i, e₁⟩ := exists_map_preimage_le_map_preimage D c hc hU H.le

Goal state after:
I : Type u
inst✝² : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
U V : (D.obj i).Opens
hU : IsCompact ↑U
hV : IsCompact ↑V
H : c.π.app i ⁻¹ᵁ U = c.π.app i ⁻¹ᵁ V
j₁ : I
fj₁i : j₁ ⟶ i
e₁ : D.map fj₁i ⁻¹ᵁ U ≤ D.map fj₁i ⁻¹ᵁ V
⊢ ∃ j fji, D.map fji ⁻¹ᵁ U = D.map fji ⁻¹ᵁ V
Module: Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
tfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)
tfae_3_to_2 :
  (Injective ⇑f ∧ ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable) →
    ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f
tfae_2_to_4 : (∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f) → Surjective (pullback.diagonal f)
hf : Function.Surjective ⇑(pullback.diagonal f)
this : UniversallyInjective f
x : ↥X
algInst✝ : Algebra ↑(Y.residueField (f x)) ↑(X.residueField x) := ⋯
σ₁ σ₂ : Field.Emb ↑(Y.residueField (f x)) ↑(X.residueField x)
⊢ ↑σ₁ = ↑σ₂

Tactic:
let g₁ := (X.SpecToEquivOfField _).symm ⟨_, CommRingCat.ofHom σ₁.toRingHom⟩

Goal state after:
X Y : Scheme
f : X ⟶ Y
tfae_1_iff_4 : UniversallyInjective f ↔ Surjective (pullback.diagonal f)
tfae_3_to_2 :
  (Injective ⇑f ∧ ∀ (x : ↥X), (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).IsPurelyInseparable) →
    ∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f
tfae_2_to_4 : (∀ (K : Type u) [inst : Field K], Injective fun g ↦ g ≫ f) → Surjective (pullback.diagonal f)
hf : Function.Surjective ⇑(pullback.diagonal f)
this : UniversallyInjective f
x : ↥X
algInst✝ : Algebra ↑(Y.residueField (f x)) ↑(X.residueField x) :=
  (CommRingCat.Hom.hom (Scheme.Hom.residueFieldMap f x)).toAlgebra
σ₁ σ₂ : Field.Emb ↑(Y.residueField (f x)) ↑(X.residueField x)
g₁ : Spec (CommRingCat.of (AlgebraicClosure ↑(X.residueField x))) ⟶ X :=
  (Scheme.SpecToEquivOfField (AlgebraicClosure ↑(X.residueField x)) X).symm ⟨x, CommRingCat.ofHom σ₁.toRingHom⟩
⊢ ↑σ₁ = ↑σ₂
Module: Mathlib.AlgebraicGeometry.RelativeGluing

Goal state before:
pp: case h
S : Scheme
𝒰 : S.OpenCover
inst✝³ : Category.{u_2, u_1} 𝒰.I₀
inst✝² : LocallyDirected 𝒰
d : RelativeGluingData 𝒰
inst✝¹ : Small.{u, u_1} 𝒰.I₀
inst✝ : Quiver.IsThin 𝒰.I₀
i : 𝒰.I₀
s : PullbackCone (𝒰.f i) d.toBase
x : ↥s.pt
⊢ (𝒰.f i) (s.fst x) = (s.snd ≫ d.toBase) x

Tactic:
← s.condition

Goal state after:
case h
S : Scheme
𝒰 : S.OpenCover
inst✝³ : Category.{u_2, u_1} 𝒰.I₀
inst✝² : LocallyDirected 𝒰
d : RelativeGluingData 𝒰
inst✝¹ : Small.{u, u_1} 𝒰.I₀
inst✝ : Quiver.IsThin 𝒰.I₀
i : 𝒰.I₀
s : PullbackCone (𝒰.f i) d.toBase
x : ↥s.pt
⊢ (𝒰.f i) (s.fst x) = (s.fst ≫ 𝒰.f i) x
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: case refine_1.refine_2
P : MorphismProperty Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝⁶ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝⁵ : Category.{v_2, ?u.117} 𝒰.I₀
inst✝⁴ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝³ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
inst✝² : Quiver.IsThin 𝒰.I₀
inst✝¹ : Small.{u, ?u.117} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : d.glued.left.OpenCover := d.relativeGluingData.cover
s : Cocone D
i j : 𝒱.I₀
hij : i ⟶ j
⊢ Cover.trans 𝒱 hij ≫
      ((d.isColimit j).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f j)).mapCocone s)).left ≫
        pullback.fst s.pt.hom (𝒰.f j) =
    ((d.isColimit i).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).mapCocone s)).left ≫
      pullback.fst s.pt.hom (𝒰.f i)

Tactic:
change (d.transitionMap hij).left ≫ _ ≫ _ = _

Goal state after:
case refine_1.refine_2
P : MorphismProperty Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝⁶ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝⁵ : Category.{v_2, ?u.117} 𝒰.I₀
inst✝⁴ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝³ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
inst✝² : Quiver.IsThin 𝒰.I₀
inst✝¹ : Small.{u, ?u.117} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : d.glued.left.OpenCover := d.relativeGluingData.cover
s : Cocone D
i j : 𝒱.I₀
hij : i ⟶ j
⊢ (d.transitionMap hij).left ≫
      ((d.isColimit j).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f j)).mapCocone s)).left ≫
        pullback.fst s.pt.hom (𝒰.f j) =
    ((d.isColimit i).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).mapCocone s)).left ≫
      pullback.fst s.pt.hom (𝒰.f i)
Module: Mathlib.AlgebraicGeometry.EffectiveEpi

Goal state before:
pp: X Y S : Scheme
inst✝³ : IsAffine X
inst✝² : IsAffine Y
π : X ⟶ Y
inst✝¹ : Surjective π
inst✝ : Flat π
f : X ⟶ S
hf : fst π π ≫ f = snd π π ≫ f
b : ↑Y.toPresheafedSpace ⟶ ↑S.toPresheafedSpace
hfac : π.base ≫ b = f.base
𝒰 : Y.OpenCover := Y.openCoverOfIsOpenCover (fun V ↦ (↑V).1) ⋯
i : 𝒰.I₀
this : IsAffine (𝒰.X i)
⊢ Set.range ⇑(ConcreteCategory.hom ((fst π (𝒰.f i)).base ≫ π.base ≫ b)) ⊆ Set.range ⇑(Opens.ι ↑(↑i).2)

Tactic:
← Scheme.Hom.comp_base_assoc,

Goal state after:
X Y S : Scheme
inst✝³ : IsAffine X
inst✝² : IsAffine Y
π : X ⟶ Y
inst✝¹ : Surjective π
inst✝ : Flat π
f : X ⟶ S
hf : fst π π ≫ f = snd π π ≫ f
b : ↑Y.toPresheafedSpace ⟶ ↑S.toPresheafedSpace
hfac : π.base ≫ b = f.base
𝒰 : Y.OpenCover := Y.openCoverOfIsOpenCover (fun V ↦ (↑V).1) ⋯
i : 𝒰.I₀
this : IsAffine (𝒰.X i)
⊢ Set.range ⇑(ConcreteCategory.hom ((fst π (𝒰.f i) ≫ π).base ≫ b)) ⊆ Set.range ⇑(Opens.ι ↑(↑i).2)
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
l : Fin n
m : ℕ
hlj : ¬l = j
⊢ (c.val i).degree ≠ ⊥

Tactic:
ne_eq,

Goal state after:
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
l : Fin n
m : ℕ
hlj : ¬l = j
⊢ ¬(c.val i).degree = ⊥
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁴ : Category.{u, u} I
S X : Scheme
D : I ⥤ Scheme
t : D ⟶ (Functor.const I).obj S
f : X ⟶ S
c : Cone D
hc : IsLimit c
inst✝³ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝² : LocallyOfFiniteType f
inst✝¹ : IsCofiltered I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
a : D.obj i ⟶ X
ha : t.app i = a ≫ f
j : I
b : D.obj j ⟶ X
hb : t.app j = b ≫ f
hab : c.π.app i ≫ a = c.π.app j ≫ b
o : I := IsCofiltered.min i j
⊢ t.app (IsCofiltered.min i j) = (D.map (IsCofiltered.minToRight i j) ≫ b) ≫ f

Tactic:
simp [← hb]

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝³ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝² : IsCofiltered I
inst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
inst✝ : CompactSpace ↥(D.obj i)
s : ↑Γ(D.obj i, ⊤)
hs : (ConcreteCategory.hom (Scheme.Hom.appTop (c.π.app i))) s = 0
U : ↥(D.obj i) → (D.obj i).Opens
hU : ∀ (x : ↥(D.obj i)), IsAffineOpen (U x)
hxU : ∀ (x : ↥(D.obj i)), x ∈ U x
j : ↥(D.obj i) → I
f : (x : ↥(D.obj i)) → j x ⟶ i
H : ∀ (x : ↥(D.obj i)), (ConcreteCategory.hom (Scheme.Hom.app (D.map (f x)) (U x))) ((s |_ U x) ⋯) = 0
t : Finset ↥(D.obj i)
ht : ⋃ x ∈ t, ↑(U x) = ⊤
k : I
fk : {X : I} → X ∈ insert i (Finset.image j t) → (k ⟶ X)
hk :
  ∀ {X Y : I} (mX : X ∈ insert i (Finset.image j t)) (mY : Y ∈ insert i (Finset.image j t)) {f_1 : X ⟶ Y},
    ⟨X, ⟨Y, ⟨mX, ⟨mY, f_1⟩⟩⟩⟩ ∈ Finset.image (fun x ↦ ⟨j ↑x, ⟨i, ⟨⋯, ⟨⋯, f ↑x⟩⟩⟩⟩) t.attach → fk mX ≫ f_1 = fk mY
⊢ (ConcreteCategory.hom (Scheme.Hom.appTop (D.map (fk ⋯)))) s = 0

Tactic:
apply (D.obj k).IsSheaf.section_ext

Goal state after:
I : Type u
inst✝³ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝² : IsCofiltered I
inst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
i : I
inst✝ : CompactSpace ↥(D.obj i)
s : ↑Γ(D.obj i, ⊤)
hs : (ConcreteCategory.hom (Scheme.Hom.appTop (c.π.app i))) s = 0
U : ↥(D.obj i) → (D.obj i).Opens
hU : ∀ (x : ↥(D.obj i)), IsAffineOpen (U x)
hxU : ∀ (x : ↥(D.obj i)), x ∈ U x
j : ↥(D.obj i) → I
f : (x : ↥(D.obj i)) → j x ⟶ i
H : ∀ (x : ↥(D.obj i)), (ConcreteCategory.hom (Scheme.Hom.app (D.map (f x)) (U x))) ((s |_ U x) ⋯) = 0
t : Finset ↥(D.obj i)
ht : ⋃ x ∈ t, ↑(U x) = ⊤
k : I
fk : {X : I} → X ∈ insert i (Finset.image j t) → (k ⟶ X)
hk :
  ∀ {X Y : I} (mX : X ∈ insert i (Finset.image j t)) (mY : Y ∈ insert i (Finset.image j t)) {f_1 : X ⟶ Y},
    ⟨X, ⟨Y, ⟨mX, ⟨mY, f_1⟩⟩⟩⟩ ∈ Finset.image (fun x ↦ ⟨j ↑x, ⟨i, ⟨⋯, ⟨⋯, f ↑x⟩⟩⟩⟩) t.attach → fk mX ≫ f_1 = fk mY
⊢ ∀ x ∈ Opposite.unop (Opposite.op ⊤),
    ∃ V,
      ∃ (hV : V ≤ Opposite.unop (Opposite.op ⊤)),
        x ∈ V ∧
          (ConcreteCategory.hom ((D.obj k).presheaf.map (homOfLE hV).op))
              ((ConcreteCategory.hom (Scheme.Hom.appTop (D.map (fk ⋯)))) s) =
            (ConcreteCategory.hom ((D.obj k).presheaf.map (homOfLE hV).op)) 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
p q : R[X]
hpq : p • 1 + q • (mk W') Y = 0
h :
  ![p, q] 0 • (CoordinateRing.basis W') 0 + ∑ i, ![p, q] i.succ • (CoordinateRing.basis W') i.succ = 0 →
    ∀ (i : Fin 2), ![p, q] i = 0
⊢ p = 0 ∧ q = 0

Tactic:
basis_zero,

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Affine R
p q : R[X]
hpq : p • 1 + q • (mk W') Y = 0
h : ![p, q] 0 • 1 + ∑ i, ![p, q] i.succ • (CoordinateRing.basis W') i.succ = 0 → ∀ (i : Fin 2), ![p, q] i = 0
⊢ p = 0 ∧ q = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
x : W'.CoordinateRing
h :
  (CoordinateRing.basis W').equivFun x 0 • (CoordinateRing.basis W') 0 +
      (CoordinateRing.basis W').equivFun x (Fin.succ 0) • (CoordinateRing.basis W') (Fin.succ 0) =
    x
⊢ ∃ p q, p • 1 + q • (mk W') Y = x

Tactic:
basis_zero,

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Affine R
x : W'.CoordinateRing
h :
  (CoordinateRing.basis W').equivFun x 0 • 1 +
      (CoordinateRing.basis W').equivFun x (Fin.succ 0) • (CoordinateRing.basis W') (Fin.succ 0) =
    x
⊢ ∃ p q, p • 1 + q • (mk W') Y = x
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
⊢ XYIdeal W x (C (W.negY x y)) * XYIdeal W x (C y) = XIdeal W x

Tactic:
by
  have Y_rw : (Y - C (C y)) * (Y - C (C <| W.negY x y)) -
      C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) =
        W.polynomial * 1 := by
    linear_combination (norm := (rw [negY, polynomial]; C_simp; ring1))
      congr_arg C (congr_arg C ((equation_iff ..).mp h.left).symm)
  simp_rw [XYIdeal, XClass, YClass, span_pair_mul_span_pair, mul_comm, ← map_mul,
    AdjoinRoot.mk_eq_mk.mpr ⟨1, Y_rw⟩, map_mul, span_insert, ← span_singleton_mul_span_singleton,
    ← Ideal.mul_sup, ← span_insert]
  convert! mul_top (_ : Ideal W.CoordinateRing) using 2
  on_goal 2 => infer_instance
  simp_rw [← Set.image_singleton (f := mk W), ← Set.image_insert_eq, ← map_span]
  convert! map_top (R := F[X][Y]) (mk W) using 1
  apply congr_arg
  simp_rw [eq_top_iff_one, mem_span_insert', mem_span_singleton']
  rcases ((nonsingular_iff' ..).mp h).right with hx | hy
  · let W_X := W.a₁ * y - (3 * x ^ 2 + 2 * W.a₂ * x + W.a₄)
    refine
      ⟨C <| C W_X⁻¹ * -(X + C (2 * x + W.a₂)), C <| C <| W_X⁻¹ * W.a₁, 0, C <| C <| W_X⁻¹ * -1, ?_⟩
    rw [← mul_right_inj' <| C_ne_zero.mpr <| C_ne_zero.mpr hx]
    simp only [W_X, mul_add, ← mul_assoc, ← C_mul, mul_inv_cancel₀ hx]
    C_simp
    ring1
  · let W_Y := 2 * y + W.a₁ * x + W.a₃
    refine ⟨0, C <| C W_Y⁻¹, C <| C <| W_Y⁻¹ * -1, 0, ?_⟩
    rw [negY, ← mul_right_inj' <| C_ne_zero.mpr <| C_ne_zero.mpr hy]
    simp only [W_Y, mul_add, ← mul_assoc, ← C_mul, mul_inv_cancel₀ hy]
    C_simp
    ring1

Goal state after:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
W : ℤ → R
h : IsEllipticNet W
x : R
p q r s : ℤ
⊢ rel (x • W) p q r s = 0

Tactic:
linear_combination (norm := (simp_rw [rel, Pi.smul_apply, smul_eq_mul]; ring1)) x ^ 4 * h p q r s

Goal state after:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
b c d : R
m : ℕ
⊢ preNormEDS' b c d (2 * (m + 3)) =
    preNormEDS' b c d (m + 2) ^ 2 * preNormEDS' b c d (m + 3) * preNormEDS' b c d (m + 5) -
      preNormEDS' b c d (m + 1) * preNormEDS' b c d (m + 3) * preNormEDS' b c d (m + 4) ^ 2

Tactic:
rw [show 2 * (m + 3) = 2 * m + 1 + 5 by rfl, preNormEDS', dif_neg m.not_even_two_mul_add_one]

Goal state after:
R : Type u_1
inst✝ : CommRing R
b c d : R
m : ℕ
⊢ (have this := ⋯;
    preNormEDS' b c d ((2 * m + 1) / 2 + 2) ^ 2 * preNormEDS' b c d ((2 * m + 1) / 2 + 3) *
        preNormEDS' b c d ((2 * m + 1) / 2 + 5) -
      preNormEDS' b c d ((2 * m + 1) / 2 + 1) * preNormEDS' b c d ((2 * m + 1) / 2 + 3) *
        preNormEDS' b c d ((2 * m + 1) / 2 + 4) ^ 2) =
    preNormEDS' b c d (m + 2) ^ 2 * preNormEDS' b c d (m + 3) * preNormEDS' b c d (m + 5) -
      preNormEDS' b c d (m + 1) * preNormEDS' b c d (m + 3) * preNormEDS' b c d (m + 4) ^ 2
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
⊢ (ClassGroup.mk W.FunctionField) (XYIdeal' ⋯) * (ClassGroup.mk W.FunctionField) (XYIdeal' h) = 1

Tactic:
rw [← map_mul]
  exact (ClassGroup.mk_eq_one_of_coe_ideal <| (coeIdeal_mul ..).symm.trans <|
    FractionalIdeal.coeIdeal_inj.mpr <| XYIdeal_neg_mul h).mpr ⟨_, XClass_ne_zero x, rfl⟩

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Affine R
p q : R[X]
⊢ ((CoordinateRing.basis W').repr ((p • 1 + q • (mk W') Y) * (CoordinateRing.basis W') 0)) 0 *
        ((CoordinateRing.basis W').repr ((p • 1 + q • (mk W') Y) * (CoordinateRing.basis W') 1)) 1 -
      ((CoordinateRing.basis W').repr ((p • 1 + q • (mk W') Y) * (CoordinateRing.basis W') 1)) 0 *
        ((CoordinateRing.basis W').repr ((p • 1 + q • (mk W') Y) * (CoordinateRing.basis W') 0)) 1 =
    p ^ 2 - p * q * (C W'.a₁ * X + C W'.a₃) - q ^ 2 * (X ^ 3 + C W'.a₂ * X ^ 2 + C W'.a₄ * X + C W'.a₆)

Tactic:
basis_zero,

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Affine R
p q : R[X]
⊢ ((CoordinateRing.basis W').repr ((p • 1 + q • (mk W') Y) * 1)) 0 *
        ((CoordinateRing.basis W').repr ((p • 1 + q • (mk W') Y) * (CoordinateRing.basis W') 1)) 1 -
      ((CoordinateRing.basis W').repr ((p • 1 + q • (mk W') Y) * (CoordinateRing.basis W') 1)) 0 *
        ((CoordinateRing.basis W').repr ((p • 1 + q • (mk W') Y) * 1)) 1 =
    p ^ 2 - p * q * (C W'.a₁ * X + C W'.a₃) - q ^ 2 * (X ^ 3 + C W'.a₂ * X ^ 2 + C W'.a₄ * X + C W'.a₆)
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case refine_3.refine_2
I : Type u
inst✝⁶ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁵ : IsCofiltered I
inst✝⁴ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝³ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝² : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
𝒰 : c.pt.OpenCover
inst✝¹ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
inst✝ : Finite 𝒰.I₀
i : I
V : 𝒰.I₀ → (D.obj i).Opens
hV : TopologicalSpace.IsOpenCover V
hV' : ∀ (j : 𝒰.I₀), IsAffineOpen (V j) ∧ Hom.opensRange (𝒰.f j) = c.π.app i ⁻¹ᵁ V j
hV'' : ∀ (k : 𝒰.I₀), Set.range ⇑(𝒰.f k) = ⇑(c.π.app i) ⁻¹' ↑(V k)
k : 𝒰.I₀
s : PullbackCone ⋯.fromSpec (c.π.app i)
y : ↥s.pt
⊢ (s.snd ≫ c.π.app i) y ∈ V k

Tactic:
← s.condition

Goal state after:
case refine_3.refine_2
I : Type u
inst✝⁶ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁵ : IsCofiltered I
inst✝⁴ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝³ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝² : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
𝒰 : c.pt.OpenCover
inst✝¹ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
inst✝ : Finite 𝒰.I₀
i : I
V : 𝒰.I₀ → (D.obj i).Opens
hV : TopologicalSpace.IsOpenCover V
hV' : ∀ (j : 𝒰.I₀), IsAffineOpen (V j) ∧ Hom.opensRange (𝒰.f j) = c.π.app i ⁻¹ᵁ V j
hV'' : ∀ (k : 𝒰.I₀), Set.range ⇑(𝒰.f k) = ⇑(c.π.app i) ⁻¹' ↑(V k)
k : 𝒰.I₀
s : PullbackCone ⋯.fromSpec (c.π.app i)
y : ↥s.pt
⊢ (s.fst ≫ ⋯.fromSpec) y ∈ V k
Module: Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms

Goal state before:
pp: F : Type u_2
inst✝³ : Field F
W : WeierstrassCurve F
inst✝² : W.IsElliptic
inst✝¹ : W.IsCharTwoJNeZeroNF
inst✝ : CharP F 2
⊢ W.j = 1 / W.a₆

Tactic:
by
  rw [j, Units.val_inv_eq_inv_val, ← div_eq_inv_mul, coe_Δ',
    c₄_of_isCharTwoJNeZeroNF_of_char_two, Δ_of_isCharTwoJNeZeroNF_of_char_two, one_pow]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsCharTwoJEqZeroNF
inst✝ : CharP R 2
⊢ -W.a₄ ^ 2 = W.a₄ ^ 2

Tactic:
linear_combination -W.a₄ ^ 2 * CharP.cast_eq_zero R 2

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic

Goal state before:
pp: case a
F : Type u
inst✝ : Field F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ P x * ![P x / P z ^ 2, P y / P z ^ 3, 1] z ^ 2 + -P x * 1 -
      (![P x / P z ^ 2, P y / P z ^ 3, 1] x * P z ^ 2 + -P x * (P z ^ 2 / P z ^ 2)) =
    0

Tactic:
matrix_simp

Goal state after:
case a
F : Type u
inst✝ : Field F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ P x * 1 ^ 2 + -P x * 1 - (P x / P z ^ 2 * P z ^ 2 + -P x * (P z ^ 2 / P z ^ 2)) = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic

Goal state before:
pp: case a
F : Type u
inst✝ : Field F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ P y * ![P x / P z ^ 2, P y / P z ^ 3, 1] z ^ 3 + -P y * 1 -
      (![P x / P z ^ 2, P y / P z ^ 3, 1] y * P z ^ 3 + -P y * (P z ^ 3 / P z ^ 3)) =
    0

Tactic:
matrix_simp

Goal state after:
case a
F : Type u
inst✝ : Field F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ P y * 1 ^ 3 + -P y * 1 - (P y / P z ^ 3 * P z ^ 3 + -P y * (P z ^ 3 / P z ^ 3)) = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : Nontrivial R
n : ℕ
h : ↑n ≠ 0
⊢ W.preΨ' n ≠ 0

Tactic:
by_cases hn : 2 < n
  · exact ne_zero_of_natDegree_gt <| W.natDegree_preΨ'_pos hn h
  · rcases n with _ | _ | _ <;> aesop

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case neg.succ.inl
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : NoZeroDivisors R
m : ℕ
h : ↑↑(m + 1) ≠ 0
hn : m ≤ 0
hm : (↑(m + 1)).natAbs = m + 1
⊢ W.ΨSq ↑(m + 1) ≠ 0

Tactic:
Nat.le_zero

Goal state after:
case neg.succ.inl
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : NoZeroDivisors R
m : ℕ
h : ↑↑(m + 1) ≠ 0
hn : m = 0
hm : (↑(m + 1)).natAbs = m + 1
⊢ W.ΨSq ↑(m + 1) ≠ 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case neg.succ.inr
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : NoZeroDivisors R
m : ℕ
h : ↑(-↑(m + 1)) ≠ 0
hn : m ≤ 0
hm : (-↑(m + 1)).natAbs = m + 1
⊢ W.ΨSq (-↑(m + 1)) ≠ 0

Tactic:
Nat.le_zero

Goal state after:
case neg.succ.inr
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : NoZeroDivisors R
m : ℕ
h : ↑(-↑(m + 1)) ≠ 0
hn : m = 0
hm : (-↑(m + 1)).natAbs = m + 1
⊢ W.ΨSq (-↑(m + 1)) ≠ 0