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

Goal state before:
pp: case mpr
X Y : Scheme
U₁ : X.Opens
hU₁ : Dense ↑U₁
V₂ : Y.Opens
hU₂✝ : Dense ↑V₂
hV₁ : Dense ↑U₁
g : ↑U₁ ≅ ↑V₂
hU₂ : Dense ↑V₂
f : ↑U₁ ≅ ↑V₂
e : f = X.isoOfEq ⋯ ≪≫ g ≪≫ Y.isoOfEq ⋯
⊢ { source := U₁, dense_source := hU₁, target := V₂, dense_target := hU₂, iso := f } =
    { source := U₁, dense_source := hV₁, target := V₂, dense_target := hU₂✝, iso := g }

Tactic:
simpa using e

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

Goal state before:
pp: X Y : Scheme
f g : X.PartialMap Y
W : X.Opens
hW : Dense ↑W
hWl : W ≤ f.domain
hWr : W ≤ g.domain
h : X.homOfLE hWl ≫ f.hom = X.homOfLE hWr ≫ g.hom
e₁ : IsDominant f.hom ↔ IsDominant (X.homOfLE hWl ≫ f.hom)
e₂ : IsDominant g.hom ↔ IsDominant (X.homOfLE hWr ≫ g.hom)
⊢ IsDominant (X.homOfLE hWl ≫ f.hom) ↔ IsDominant g.hom

Tactic:
h,

Goal state after:
X Y : Scheme
f g : X.PartialMap Y
W : X.Opens
hW : Dense ↑W
hWl : W ≤ f.domain
hWr : W ≤ g.domain
h : X.homOfLE hWl ≫ f.hom = X.homOfLE hWr ≫ g.hom
e₁ : IsDominant f.hom ↔ IsDominant (X.homOfLE hWl ≫ f.hom)
e₂ : IsDominant g.hom ↔ IsDominant (X.homOfLE hWr ≫ g.hom)
⊢ IsDominant (X.homOfLE hWr ≫ g.hom) ↔ IsDominant g.hom
Module: Mathlib.AlgebraicGeometry.Birational.Composition

Goal state before:
pp: X Y Z : Scheme
inst✝³ : PreirreducibleSpace ↥X
inst✝² : Nonempty ↥Y
f₁ f₂ : X.PartialMap Y
inst✝¹ : IsDominant f₁.hom
inst✝ : IsDominant f₂.hom
g : Y.PartialMap Z
W : X.Opens
hW : Dense ↑W
hW₁ : W ≤ f₁.domain
hW₂ : W ≤ f₂.domain
e : (f₁.restrict W hW hW₁).hom = (f₂.restrict W hW hW₂).hom
⊢ (f₁.restrict W hW hW₁).hom = (X.isoOfEq ⋯).hom ≫ (f₂.restrict W hW hW₂).hom

Tactic:
simpa using e

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

Goal state before:
pp: X Y Z : Scheme
inst✝² : PreirreducibleSpace ↥X
inst✝¹ : Nonempty ↥Y
f : X.PartialMap Y
inst✝ : IsDominant f.hom
g₁ g₂ : Y.PartialMap Z
W : Y.Opens
hW : Dense ↑W
hW₁ : W ≤ g₁.domain
hW₂ : W ≤ g₂.domain
e : (g₁.restrict W hW hW₁).hom = (g₂.restrict W hW hW₂).hom
⊢ (g₁.restrict W hW hW₁).hom = (Y.isoOfEq ⋯).hom ≫ (g₂.restrict W hW hW₂).hom

Tactic:
simpa using e

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)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
i : I
U : (D.obj i).Opens
hU : c.π.app i ⁻¹ᵁ U = ⊤
H : ∀ (j : I) (fji : j ⟶ i), D.map fji ⁻¹ᵁ U ≠ ⊤
j k : I
fkj : k ⟶ j
fji : j ⟶ i
x : ↥(D.obj k)
hx : (D.map (fkj ≫ fji)) x ∉ U
⊢ (D.map fkj) x ∈ (↑(D.map fji ⁻¹ᵁ U))ᶜ

Tactic:
Functor.map_comp

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)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
i : I
U : (D.obj i).Opens
hU : c.π.app i ⁻¹ᵁ U = ⊤
H : ∀ (j : I) (fji : j ⟶ i), D.map fji ⁻¹ᵁ U ≠ ⊤
j k : I
fkj : k ⟶ j
fji : j ⟶ i
x : ↥(D.obj k)
hx : (D.map fkj ≫ D.map fji) x ∉ U
⊢ (D.map fkj) x ∈ (↑(D.map fji ⁻¹ᵁ U))ᶜ
Module: Mathlib.AlgebraicGeometry.Cover.Directed

Goal state before:
pp: P : MorphismProperty Scheme
X : Scheme
𝒰 : Cover (precoverage P) X
inst✝¹ : Category.{v_1, u_1} 𝒰.I₀
inst✝ : 𝒰.LocallyDirected
i j k : 𝒰.I₀
fi : i ⟶ k
fj : j ⟶ k
xi : ↥(𝒰.X i)
xj : ↥(𝒰.X j)
h : (𝒰.trans fi) xi = (𝒰.trans fj) xj
⊢ (𝒰.f k) ((𝒰.trans fi) xi) = (ConcreteCategory.hom ((𝒰.trans fj).base ≫ (𝒰.f k).base)) xj

Tactic:
h,

Goal state after:
P : MorphismProperty Scheme
X : Scheme
𝒰 : Cover (precoverage P) X
inst✝¹ : Category.{v_1, u_1} 𝒰.I₀
inst✝ : 𝒰.LocallyDirected
i j k : 𝒰.I₀
fi : i ⟶ k
fj : j ⟶ k
xi : ↥(𝒰.X i)
xj : ↥(𝒰.X j)
h : (𝒰.trans fi) xi = (𝒰.trans fj) xj
⊢ (𝒰.f k) ((𝒰.trans fj) xj) = (ConcreteCategory.hom ((𝒰.trans fj).base ≫ (𝒰.f k).base)) xj
Module: Mathlib.AlgebraicGeometry.Cover.Directed

Goal state before:
pp: P : MorphismProperty Scheme
X : Scheme
𝒰✝ : X.OpenCover
inst✝³ : Category.{v_1, ?u.14} 𝒰✝.I₀
inst✝² : Cover.LocallyDirected 𝒰✝
𝒰 : X.OpenCover
inst✝¹ : Category.{v_2, ?u.64} 𝒰.I₀
inst✝ : Cover.LocallyDirected 𝒰
Y : Scheme
g : (i : 𝒰.I₀) → 𝒰.X i ⟶ Y
h : ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), Cover.trans 𝒰 hij ≫ g j = g i
i j : 𝒰.I₀
⊢ ∀ (x : (Cover.intersectionOfLocallyDirected 𝒰 i j).I₀),
    (Cover.intersectionOfLocallyDirected 𝒰 i j).f x ≫ pullback.fst (𝒰.f i) (𝒰.f j) ≫ g i =
      (Cover.intersectionOfLocallyDirected 𝒰 i j).f x ≫ pullback.snd (𝒰.f i) (𝒰.f j) ≫ g j

Tactic:
intro k

Goal state after:
P : MorphismProperty Scheme
X : Scheme
𝒰✝ : X.OpenCover
inst✝³ : Category.{v_1, ?u.14} 𝒰✝.I₀
inst✝² : Cover.LocallyDirected 𝒰✝
𝒰 : X.OpenCover
inst✝¹ : Category.{v_2, ?u.64} 𝒰.I₀
inst✝ : Cover.LocallyDirected 𝒰
Y : Scheme
g : (i : 𝒰.I₀) → 𝒰.X i ⟶ Y
h : ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), Cover.trans 𝒰 hij ≫ g j = g i
i j : 𝒰.I₀
k : (Cover.intersectionOfLocallyDirected 𝒰 i j).I₀
⊢ (Cover.intersectionOfLocallyDirected 𝒰 i j).f k ≫ pullback.fst (𝒰.f i) (𝒰.f j) ≫ g i =
    (Cover.intersectionOfLocallyDirected 𝒰 i j).f k ≫ pullback.snd (𝒰.f i) (𝒰.f j) ≫ g j
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
hij : coeffSubmodule R₀ c' ≤ coeffSubmodule R₀ c ^ (c.val j).degree.succ
hi' : ¬c.val i = 1
⊢ ∀ i_1 ∈ Finset.univ, (Function.update c.val j (c.val j %ₘ c.val i) i_1).degree.succ ≤ (c.val i_1).degree.succ

Tactic:
intro k

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
hij : coeffSubmodule R₀ c' ≤ coeffSubmodule R₀ c ^ (c.val j).degree.succ
hi' : ¬c.val i = 1
k : Fin n
⊢ k ∈ Finset.univ → (Function.update c.val j (c.val j %ₘ c.val i) k).degree.succ ≤ (c.val k).degree.succ
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: case refine_1.refine_2.e_a
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.129} 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.129} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : (a : J) → (D.obj a).left.OpenCover := fun a ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).hom 𝒰
a : J
i j : (𝒱 a).I₀
hij : i ⟶ j
⊢ Cover.trans (𝒱 a) hij ≫ ((d.cocone j).ι.app a).left =
    ((d.cocone i).ι.app a).left ≫ d.relativeGluingData.functor.map hij

Tactic:
exact congr($(d.cocone_ι_transitionMap hij a).left).symm

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case pos
R : Type u_6
inst✝ : CommRing R
M : Submodule ℤ R
hM : 1 ∈ M
S : ConstructibleSetData R[X]
hS : ∀ C ∈ S, ∀ (j : Fin C.n) (k : ℕ), (C.g j).coeff k ∈ M
f : BasicConstructibleSetData R[X] → ConstructibleSetData R
hf₁ :
  ∀ (C : BasicConstructibleSetData R[X]),
    comap Polynomial.C '' (zeroLocus (Set.range { val := C.g }.val) \ zeroLocus {C.f}) = (f C).toSet
hf₂ : ∀ (C : BasicConstructibleSetData R[X]), ∀ C_1 ∈ f C, C_1.n ≤ { val := C.g }.degBound
hf₃ :
  ∀ (C : BasicConstructibleSetData R[X]),
    ∀ C_1 ∈ f C, ∀ (i : Fin C_1.n), C_1.g i ∈ coeffSubmodule ℤ { val := C.g } ^ { val := C.g }.powBound
x : BasicConstructibleSetData R
y : BasicConstructibleSetData R[X]
hy : y ∈ S
hx : x ∈ f y
H : { val := y.g }.degBound ≤ S.degBound
i : Fin x.n
h : S.degBound = 0
this : { val := y.g }.degBound = 0
⊢ { val := y.g }.degBound ^ { val := y.g }.degBound ≤ S.degBound ^ S.degBound

Tactic:
h,

Goal state after:
case pos
R : Type u_6
inst✝ : CommRing R
M : Submodule ℤ R
hM : 1 ∈ M
S : ConstructibleSetData R[X]
hS : ∀ C ∈ S, ∀ (j : Fin C.n) (k : ℕ), (C.g j).coeff k ∈ M
f : BasicConstructibleSetData R[X] → ConstructibleSetData R
hf₁ :
  ∀ (C : BasicConstructibleSetData R[X]),
    comap Polynomial.C '' (zeroLocus (Set.range { val := C.g }.val) \ zeroLocus {C.f}) = (f C).toSet
hf₂ : ∀ (C : BasicConstructibleSetData R[X]), ∀ C_1 ∈ f C, C_1.n ≤ { val := C.g }.degBound
hf₃ :
  ∀ (C : BasicConstructibleSetData R[X]),
    ∀ C_1 ∈ f C, ∀ (i : Fin C_1.n), C_1.g i ∈ coeffSubmodule ℤ { val := C.g } ^ { val := C.g }.powBound
x : BasicConstructibleSetData R
y : BasicConstructibleSetData R[X]
hy : y ∈ S
hx : x ∈ f y
H : { val := y.g }.degBound ≤ S.degBound
i : Fin x.n
h : S.degBound = 0
this : { val := y.g }.degBound = 0
⊢ { val := y.g }.degBound ^ { val := y.g }.degBound ≤ 0 ^ 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass

Goal state before:
pp: case c
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ 2 * W.b₄ = -W.b₄

Tactic:
· linear_combination W.b₄ * CharP.cast_eq_zero R 3

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

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsElliptic
h : W.c₄ = 0
⊢ W.c₄ ^ 3 = 0

Tactic:
h,

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsElliptic
h : W.c₄ = 0
⊢ 0 ^ 3 = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass

Goal state before:
pp: R : Type u
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsElliptic
inst✝ : CharP R 2
h : W.a₁ = 0
⊢ W.a₁ ^ 12 = 0

Tactic:
h,

Goal state after:
R : Type u
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsElliptic
inst✝ : CharP R 2
h : W.a₁ = 0
⊢ 0 ^ 12 = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass

Goal state before:
pp: R : Type u
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsElliptic
inst✝ : CharP R 3
h : W.b₂ = 0
⊢ W.b₂ ^ 6 = 0

Tactic:
h,

Goal state after:
R : Type u
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsElliptic
inst✝ : CharP R 3
h : W.b₂ = 0
⊢ 0 ^ 6 = 0
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
n : ℕ
M : Submodule ℤ R
hM : 1 ∈ M
k : ℕ
d : Multiset (Fin n)
S : ConstructibleSetData (MvPolynomial (Fin n) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) d)
⊢ ∃ T,
    comap MvPolynomial.C '' S.toSet = T.toSet ∧
      ∀ C ∈ T,
        C.n ≤ numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n ∧
          ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n

Tactic:
induction n generalizing k M with

Goal state after:
case succ
R : Type u_2
inst✝ : CommRing R
n✝ n : ℕ
IH :
  ∀ {M : Submodule ℤ R},
    1 ∈ M →
      ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),
        (∀ C ∈ S, C.n ≤ k) →
          (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) d)) →
            ∃ T,
              comap MvPolynomial.C '' S.toSet = T.toSet ∧
                ∀ C ∈ T,
                  C.n ≤ numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n ∧
                    ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n
M : Submodule ℤ R
hM : 1 ∈ M
k : ℕ
d : Multiset (Fin (n + 1))
S : ConstructibleSetData (MvPolynomial (Fin (n + 1)) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS :
  ∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin (n + 1)) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin (n + 1)) d)
⊢ ∃ T,
    comap MvPolynomial.C '' S.toSet = T.toSet ∧
      ∀ C ∈ T,
        C.n ≤ numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) (n + 1) ∧
          ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) (n + 1)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : Affine R
⊢ evalEval 0 0 W.polynomial = -W.a₆

Tactic:
by
  simp only [evalEval_polynomial, zero_add, zero_sub, mul_zero, zero_pow <| Nat.succ_ne_zero _]

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
x : ↥(D.obj k)
l : ↥(D.obj i)
hl : l ∈ t
hlU : (D.map (fk ⋯)) x ∈ ↑(U l)
h₁ : fk ⋯ = fk ⋯ ≫ f l
⊢ D.map (fk ⋯) ⁻¹ᵁ U l ≤ D.map (fk ⋯) ⁻¹ᵁ D.map (f l) ⁻¹ᵁ U l

Tactic:
rw [← Scheme.Hom.comp_preimage, ← D.map_comp, h₁]

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

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W : Affine R
S : Type s
inst✝ : CommRing S
f : R →+* S
x y : R
h : W.Equation x y
⊢ f (evalEval x y W.polynomial) = 0

Tactic:
h,

Goal state after:
R : Type r
inst✝¹ : CommRing R
W : Affine R
S : Type s
inst✝ : CommRing S
f : R →+* S
x y : R
h : W.Equation x y
⊢ f 0 = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
x✝² x✝¹ c x✝ : Ideal W.CoordinateRing
⊢ x✝² ⊔ (x✝¹ ⊔ (c ⊔ x✝)) = x✝² ⊔ x✝ ⊔ x✝¹ ⊔ c

Tactic:
← sup_assoc,

Goal state after:
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
x✝² x✝¹ c x✝ : Ideal W.CoordinateRing
⊢ x✝² ⊔ x✝¹ ⊔ (c ⊔ x✝) = x✝² ⊔ x✝ ⊔ x✝¹ ⊔ c
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
W : ℤ → R
a b c d : ℤ
⊢ atomRel W (2 * a) (2 * b) (2 * c) (2 * d) = rel W (a - d) (b - d) (c - d) (2 * d)

Tactic:
simp_rw [rel_eq, mul_sub, sub_add_cancel]

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

Goal state before:
pp: case neg
F : Type u
inst✝¹ : Field F
W : Affine F
inst✝ : DecidableEq F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c
hx : ¬x₁ = x₂
⊢ ∃ x,
    (∃ a a_1 a_2, a_2 * (Y - W.negPolynomial) = x + a * C (X - C x₁) + a_1 * C (X - C x₂)) ∧
      (AdjoinRoot.mk W.polynomial) x = 1

Tactic:
· replace hx := sub_ne_zero_of_ne hx
    refine ⟨_, ⟨⟨C <| C (x₁ - x₂)⁻¹, C <| C <| (x₁ - x₂)⁻¹ * -1, 0, ?_⟩, map_one _⟩⟩
    rw [← mul_right_inj' <| C_ne_zero.mpr <| C_ne_zero.mpr hx]
    simp only [← mul_assoc, mul_add, ← C_mul, mul_inv_cancel₀ hx]
    C_simp
    ring1

Goal state after:
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case 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)
⊢ (j : 𝒰.I₀) → 𝒰.X j ⟶ Spec ((fun k ↦ Γ(D.obj i, V k)) j)

Tactic:
intro k

Goal state after:
case 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
⊢ 𝒰.X k ⟶ Spec ((fun k ↦ Γ(D.obj i, V k)) k)
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case refine_3
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)
⊢ ∀ (j : 𝒰.I₀),
    IsPullback ((fun k ↦ IsOpenImmersion.lift (V k).ι (𝒰.f k ≫ c.π.app i) ⋯ ≫ ⋯.isoSpec.hom) j) (𝒰.f j)
      ((fun k ↦ ⋯.isoSpec.inv ≫ (V k).ι) j) (c.π.app i)

Tactic:
intro k

Goal state after:
case refine_3
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
⊢ IsPullback ((fun k ↦ IsOpenImmersion.lift (V k).ι (𝒰.f k ≫ c.π.app i) ⋯ ≫ ⋯.isoSpec.hom) k) (𝒰.f k)
    ((fun k ↦ ⋯.isoSpec.inv ≫ (V k).ι) k) (c.π.app i)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (X * W.preΨ₄ ^ 2 * W.Ψ₂Sq -
      ((W.preΨ₄ * 1 ^ 3 * if Even 0 then W.Ψ₂Sq ^ 2 else 1) -
            W.preΨ' (0 + 1) * W.preΨ' (0 + 3) ^ 3 * if Even 0 then 1 else W.Ψ₂Sq ^ 2) *
          W.preΨ' 3 *
        if Even 3 then W.Ψ₂Sq else 1) =
    X * W.preΨ₄ ^ 2 * W.Ψ₂Sq - W.Ψ₃ * (W.preΨ₄ * W.Ψ₂Sq ^ 2 - W.Ψ₃ ^ 3)

Tactic:
if_pos Even.zero,

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (X * W.preΨ₄ ^ 2 * W.Ψ₂Sq -
      (W.preΨ₄ * 1 ^ 3 * W.Ψ₂Sq ^ 2 - W.preΨ' (0 + 1) * W.preΨ' (0 + 3) ^ 3 * if Even 0 then 1 else W.Ψ₂Sq ^ 2) *
          W.preΨ' 3 *
        if Even 3 then W.Ψ₂Sq else 1) =
    X * W.preΨ₄ ^ 2 * W.Ψ₂Sq - W.Ψ₃ * (W.preΨ₄ * W.Ψ₂Sq ^ 2 - W.Ψ₃ ^ 3)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (X * W.preΨ₄ ^ 2 * W.Ψ₂Sq -
      (W.preΨ₄ * 1 ^ 3 * W.Ψ₂Sq ^ 2 - 1 * W.Ψ₃ ^ 3 * if Even 0 then 1 else W.Ψ₂Sq ^ 2) * W.Ψ₃ *
        if Even 3 then W.Ψ₂Sq else 1) =
    X * W.preΨ₄ ^ 2 * W.Ψ₂Sq - W.Ψ₃ * (W.preΨ₄ * W.Ψ₂Sq ^ 2 - W.Ψ₃ ^ 3)

Tactic:
if_pos Even.zero,

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (X * W.preΨ₄ ^ 2 * W.Ψ₂Sq - (W.preΨ₄ * 1 ^ 3 * W.Ψ₂Sq ^ 2 - 1 * W.Ψ₃ ^ 3 * 1) * W.Ψ₃ * if Even 3 then W.Ψ₂Sq else 1) =
    X * W.preΨ₄ ^ 2 * W.Ψ₂Sq - W.Ψ₃ * (W.preΨ₄ * W.Ψ₂Sq ^ 2 - W.Ψ₃ ^ 3)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ C X * (C W.preΨ₄ * W.ψ₂) ^ 2 - (C W.preΨ₄ * W.ψ₂ * W.ψ₂ ^ 3 - 1 * W.ψ 3 ^ 3) * W.ψ 3 =
    C X * C W.preΨ₄ ^ 2 * W.ψ₂ ^ 2 - C W.preΨ₄ * W.ψ₂ ^ 4 * C W.Ψ₃ + C W.Ψ₃ ^ 4

Tactic:
ψ_three

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ C X * (C W.preΨ₄ * W.ψ₂) ^ 2 - (C W.preΨ₄ * W.ψ₂ * W.ψ₂ ^ 3 - 1 * C W.Ψ₃ ^ 3) * C W.Ψ₃ =
    C X * C W.preΨ₄ ^ 2 * W.ψ₂ ^ 2 - C W.preΨ₄ * W.ψ₂ ^ 4 * C W.Ψ₃ + C W.Ψ₃ ^ 4
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case inr
n : ℕ
hn : 2 * n + 1 ≠ 0
⊢ 2 * ↑(expDegree (2 * n + 1)) = ↑(2 * n + 1) ^ 2 - if Even (2 * n + 1) then 4 else 1

Tactic:
push_cast [expDegree, show (2 * n + 1) ^ 2 = 2 * (2 * n * (n + 1)) + 1 by ring1,
      n.not_even_two_mul_add_one, Nat.add_sub_cancel, Nat.mul_div_cancel_left _ two_pos]
    ring1

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
⊢ -48 * W.a₄ = 0

Tactic:
linear_combination -24 * W.a₄ * CharP.cast_eq_zero R 2

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

Goal state before:
pp: F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = E'.a₄ ^ 3 * E.a₆ ^ 2
ha₄ : ¬E.a₄ = 0
ha₆ : ¬E.a₆ = 0
h : E'.a₄ = 0
⊢ False

Tactic:
h,

Goal state after:
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = 0 ^ 3 * E.a₆ ^ 2
ha₄ : ¬E.a₄ = 0
ha₆ : ¬E.a₆ = 0
h : E'.a₄ = 0
⊢ False