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

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 2
⊢ -(W.a₁ ^ 2) ^ 2 * W.b₈ - 8 * (W.a₁ * W.a₃) ^ 3 - 27 * (W.a₃ ^ 2) ^ 2 + 9 * W.a₁ ^ 2 * (W.a₁ * W.a₃) * W.a₃ ^ 2 =
    W.a₁ ^ 4 * W.b₈ + W.a₃ ^ 4 + W.a₁ ^ 3 * W.a₃ ^ 3

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 2
⊢ -(W.a₁ ^ 2) ^ 2 * W.b₈ - 8 * (W.a₁ * W.a₃) ^ 3 - 27 * (W.a₃ ^ 2) ^ 2 + 9 * W.a₁ ^ 2 * (W.a₁ * W.a₃) * W.a₃ ^ 2 =
    W.a₁ ^ 4 * W.b₈ + W.a₃ ^ 4 + W.a₁ ^ 3 * W.a₃ ^ 3
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case inr
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 b : D.obj i ⟶ X
ha : t.app i = a ≫ f
hb : t.app i = b ≫ f
hab : c.π.app i ≫ a = c.π.app i ≫ b
this✝ : ∀ (i : I), IsAffineHom (c.π.app i)
A : ExistsHomHomCompEqCompAux D t f :=
  { c := c, hc := hc, i := i, a := a, ha := ha, b := b, hb := hb, hab := hab, 𝒰S := S.affineCover, h𝒰S := ⋯,
    𝒰X := fun i ↦ ((Precoverage.ZeroHypercover.pullback₁ f S.affineCover).X i).affineCover, h𝒰X := ⋯ }
𝒰 : (pullback.diagonalObj f).OpenCover := Pullback.diagonalCover f A.𝒰S A.𝒰X
W : (pullback.diagonalObj f).Opens := Pullback.diagonalCoverDiagonalRange f A.𝒰S A.𝒰X
k : A.𝒰D.I₀ → I
hki' : (j : A.𝒰D.I₀) → k j ⟶ A.i'
𝒰Df : (D.obj A.i').OpenCover := A.𝒰D.finiteSubcover
h : Nonempty ↥(D.obj A.i')
O : Finset I := {A.i'} ∪ Finset.image (fun i ↦ k (Cover.idx A.𝒰D ↑i)) Finset.univ
o : 𝒰Df.I₀ := ⋯.some
ho : k (Cover.idx A.𝒰D ↑o) ∈ O
l : I
hl1 : {X : I} → X ∈ O → (l ⟶ X)
hl2 :
  ∀ {X_1 Y : I} (mX : X_1 ∈ O) (mY : Y ∈ O) {f_1 : X_1 ⟶ Y},
    ⟨X_1, ⟨Y, ⟨mX, ⟨mY, f_1⟩⟩⟩⟩ ∈
        Finset.image (fun i ↦ ⟨k (Cover.idx A.𝒰D ↑i), ⟨A.i', ⟨⋯, ⟨⋯, hki' (Cover.idx A.𝒰D ↑i)⟩⟩⟩⟩) Finset.univ →
      hl1 mX ≫ f_1 = hl1 mY
this : ∀ (w v : 𝒰Df.I₀), hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑w) = hl1 ⋯ ≫ hki' (Cover.idx A.𝒰D ↑v)
u : (Precoverage.ZeroHypercover.pullback₁ (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) 𝒰Df).I₀
F : pullback (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) (𝒰Df.f u) ⟶
  pullback (D.map (hki' (Cover.idx A.𝒰D ↑u))) (A.𝒰D.f (Cover.idx A.𝒰D ↑u)) :=
  pullback.map (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) (𝒰Df.f u) (D.map (hki' (Cover.idx A.𝒰D ↑u)))
    (A.𝒰D.f (Cover.idx A.𝒰D ↑u)) (D.map (hl1 ⋯)) (𝟙 (𝒰Df.X u)) (𝟙 (D.obj A.i')) ⋯ ⋯
hF :
  F ≫ pullback.fst (D.map (hki' (Cover.idx A.𝒰D ↑u))) (A.𝒰D.f (Cover.idx A.𝒰D ↑u)) =
    pullback.fst (D.map (hl1 ho ≫ hki' (Cover.idx A.𝒰D ↑o))) (𝒰Df.f u) ≫ D.map (hl1 ⋯)
heq :
  ∀ (j : A.𝒰D.I₀),
    pullback.fst (D.map (hki' j)) (A.𝒰D.f j) ≫ D.map (hki' j) ≫ D.map A.hii' ≫ A.a =
      pullback.fst (D.map (hki' j)) (A.𝒰D.f j) ≫ D.map (hki' j) ≫ D.map A.hii' ≫ A.b
⊢ F ≫
      pullback.fst (D.map (hki' (Cover.idx A.𝒰D ↑u))) (A.𝒰D.f (Cover.idx A.𝒰D ↑u)) ≫
        D.map (hki' (Cover.idx A.𝒰D ↑u)) ≫ D.map A.hii' ≫ A.b =
    F ≫
      pullback.fst (D.map (hki' (Cover.idx A.𝒰D ↑u))) (A.𝒰D.f (Cover.idx A.𝒰D ↑u)) ≫
        D.map (hki' (Cover.idx A.𝒰D ↑u)) ≫ D.map A.hii' ≫ b

Tactic:
]

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

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ W.a₁ ^ 2 + 4 * W.a₂ = W.a₁ ^ 2 + W.a₂

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ W.a₁ ^ 2 + 4 * W.a₂ = W.a₁ ^ 2 + W.a₂
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ 2 * W.a₄ + W.a₁ * W.a₃ = -W.a₄ + W.a₁ * W.a₃

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ 2 * W.a₄ + W.a₁ * W.a₃ = -W.a₄ + W.a₁ * W.a₃
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ W.a₃ ^ 2 + 4 * W.a₆ = W.a₃ ^ 2 + W.a₆

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ W.a₃ ^ 2 + 4 * W.a₆ = W.a₃ ^ 2 + W.a₆
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ W.a₁ ^ 2 * W.a₆ + 4 * W.a₂ * W.a₆ - W.a₁ * W.a₃ * W.a₄ + W.a₂ * W.a₃ ^ 2 - W.a₄ ^ 2 =
    W.a₁ ^ 2 * W.a₆ + W.a₂ * W.a₆ - W.a₁ * W.a₃ * W.a₄ + W.a₂ * W.a₃ ^ 2 - W.a₄ ^ 2

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ W.a₁ ^ 2 * W.a₆ + 4 * W.a₂ * W.a₆ - W.a₁ * W.a₃ * W.a₄ + W.a₂ * W.a₃ ^ 2 - W.a₄ ^ 2 =
    W.a₁ ^ 2 * W.a₆ + W.a₂ * W.a₆ - W.a₁ * W.a₃ * W.a₄ + W.a₂ * W.a₃ ^ 2 - W.a₄ ^ 2
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: case refine_3
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.499} 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.499} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : d.glued.left.OpenCover := d.relativeGluingData.cover
s : Cocone D
m : d.gluedCocone.pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone.ι.app j ≫ m = s.ι.app j
i : 𝒱.I₀
this :
  (d.pullbackGluedIso i).inv ≫ (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).map m =
    (d.isColimit i).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).mapCocone s)
⊢ 𝒱.f i ≫ m.left =
    ((d.isColimit i).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).mapCocone s)).left ≫
      pullback.fst s.pt.hom (𝒰.f i)

Tactic:
]

Goal state after:
case refine_3
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.499} 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.499} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : d.glued.left.OpenCover := d.relativeGluingData.cover
s : Cocone D
m : d.gluedCocone.pt ⟶ s.pt
hm : ∀ (j : J), d.gluedCocone.ι.app j ≫ m = s.ι.app j
i : 𝒱.I₀
this :
  (d.pullbackGluedIso i).inv ≫ (MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).map m =
    (d.isColimit i).desc ((MorphismProperty.Over.pullback P ⊤ (𝒰.f i)).mapCocone s)
⊢ 𝒱.f i ≫ m.left =
    ((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.EllipticCurve.Weierstrass

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ W.b₂ ^ 2 - 24 * W.b₄ = W.b₂ ^ 2

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ W.b₂ ^ 2 - 24 * W.b₄ = W.b₂ ^ 2
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ -W.b₂ ^ 3 + 36 * W.b₂ * W.b₄ - 216 * W.b₆ = -W.b₂ ^ 3

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ -W.b₂ ^ 3 + 36 * W.b₂ * W.b₄ - 216 * W.b₆ = -W.b₂ ^ 3
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ -W.b₂ ^ 2 * W.b₈ - 8 * W.b₄ ^ 3 - 27 * W.b₆ ^ 2 + 9 * W.b₂ * W.b₄ * W.b₆ = -W.b₂ ^ 2 * W.b₈ - 8 * W.b₄ ^ 3

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ -W.b₂ ^ 2 * W.b₈ - 8 * W.b₄ ^ 3 - 27 * W.b₆ ^ 2 + 9 * W.b₂ * W.b₄ * W.b₆ = -W.b₂ ^ 2 * W.b₈ - 8 * W.b₄ ^ 3
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 2
⊢ { a := 4, b := W.b₂, c := 2 * W.b₄, d := W.b₆ } = { a := 0, b := W.b₂, c := 0, d := W.b₆ }

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 2
⊢ { a := 4, b := W.b₂, c := 2 * W.b₄, d := W.b₆ } = { a := 0, b := W.b₂, c := 0, d := W.b₆ }
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
⊢ 0 ^ 0 ≤ 0 ^ 0

Tactic:
]

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

Goal state before:
pp: R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ { a := 4, b := W.b₂, c := 2 * W.b₄, d := W.b₆ } = { a := 1, b := W.b₂, c := -W.b₄, d := W.b₆ }

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ { a := 4, b := W.b₂, c := 2 * W.b₄, d := W.b₆ } = { a := 1, b := W.b₂, c := -W.b₄, d := W.b₆ }
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✝ : CommRing R
W : WeierstrassCurve R
hu : IsUnit 2
⊢ IsUnit (2 ^ 4) ∧ IsUnit W.Δ ↔ IsUnit W.Δ

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
hu : IsUnit 2
⊢ IsUnit (2 ^ 4) ∧ IsUnit W.Δ ↔ IsUnit W.Δ
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case ht
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
⊢ S.degBound ≠ 0

Tactic:
]

Goal state after:
case ht
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
⊢ S.degBound ≠ 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass

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

Tactic:
]

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
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: R : Type u
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsElliptic
inst✝ : IsReduced R
⊢ W.c₄ = 0 ↔ W.c₄ = 0

Tactic:
]

Goal state after:
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
⊢ ↑W.Δ'⁻¹ * W.a₁ ^ (4 * 3) = ↑W.Δ'⁻¹ * W.a₁ ^ 12

Tactic:
]

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
C✝ C' C : VariableChange R
⊢ { u := { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.u * C.u,
      r :=
        { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.r * ↑C.u ^ 2 +
          C.r,
      s :=
        ↑C.u * { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.s + C.s,
      t :=
        { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.t * ↑C.u ^ 3 +
            { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.r * C.s *
              ↑C.u ^ 2 +
          C.t } =
    { u := 1, r := 0, s := 0, t := 0 }

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
C✝ C' C : VariableChange R
⊢ { u := { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.u * C.u,
      r :=
        { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.r * ↑C.u ^ 2 +
          C.r,
      s :=
        ↑C.u * { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.s + C.s,
      t :=
        { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.t * ↑C.u ^ 3 +
            { u := C.u⁻¹, r := -C.r * ↑C.u⁻¹ ^ 2, s := -C.s * ↑C.u⁻¹, t := (C.r * C.s - C.t) * ↑C.u⁻¹ ^ 3 }.r * C.s *
              ↑C.u ^ 2 +
          C.t } =
    { u := 1, r := 0, s := 0, t := 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
⊢ W.a₁ ^ 12 = 0 ↔ W.a₁ ^ 12 = 0

Tactic:
]

Goal state after:
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
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: k : ℕ
D : ℕ → ℕ
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: R : Type u
inst✝³ : CommRing R
W : WeierstrassCurve R
inst✝² : W.IsElliptic
inst✝¹ : CharP R 2
inst✝ : IsReduced R
⊢ W.a₁ = 0 ↔ W.a₁ = 0

Tactic:
]

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

Goal state before:
pp: k : ℕ
D : ℕ → ℕ
⊢ k = k

Tactic:
]

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

Goal state before:
pp: k : ℕ
D : ℕ → ℕ
n : ℕ
⊢ numBound k D (n + 1) ^ numBound k D (n + 1) * degBound k D n =
    numBound k D (n + 1) ^ numBound k D (n + 1) * degBound k D n

Tactic:
]

Goal state after:
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
⊢ ↑W.Δ'⁻¹ * W.b₂ ^ (2 * 3) = ↑W.Δ'⁻¹ * W.b₂ ^ 6

Tactic:
]

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

Goal state before:
pp: k : ℕ
D : ℕ → ℕ
n : ℕ
⊢ numBound k D n * degBound k D n * D n = numBound k D n * degBound k D n * D n

Tactic:
]

Goal state after:
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
⊢ W.b₂ ^ 6 = 0 ↔ W.b₂ ^ 6 = 0

Tactic:
]

Goal state after:
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
⊢ 0 = 0

Tactic:
]

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

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

Tactic:
]

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

Goal state before:
pp: R : Type u
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsElliptic
A : Type v
inst✝ : CommRing A
f : R →+* A
⊢ f W.Δ = f W.Δ

Tactic:
]

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

Goal state before:
pp: k₀ k : ℕ
D : ℕ → ℕ
n : ℕ
⊢ (numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
          ((k₀ * k) ^ (k₀ * k) • D) n) ^
        (numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
          ((k₀ * k) ^ (k₀ * k) • D) n) *
      ((k₀ * k) ^ (k₀ * k) * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n) =
    (k₀ * k) ^ (k₀ * k) *
      ((numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
            ((k₀ * k) ^ (k₀ * k) • D) n) ^
          (numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
            ((k₀ * k) ^ (k₀ * k) • D) n) *
        degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n)

Tactic:
]

Goal state after:
k₀ k : ℕ
D : ℕ → ℕ
n : ℕ
⊢ (numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
          ((k₀ * k) ^ (k₀ * k) • D) n) ^
        (numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
          ((k₀ * k) ^ (k₀ * k) • D) n) *
      ((k₀ * k) ^ (k₀ * k) * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n) =
    (k₀ * k) ^ (k₀ * k) *
      ((numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
            ((k₀ * k) ^ (k₀ * k) • D) n) ^
          (numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
            ((k₀ * k) ^ (k₀ * k) • D) n) *
        degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass

Goal state before:
pp: R : Type u
inst✝² : CommRing R
W : WeierstrassCurve R
inst✝¹ : W.IsElliptic
A : Type v
inst✝ : CommRing A
f : R →+* A
⊢ f ↑W.Δ'⁻¹ * f W.c₄ ^ 3 = f ↑W.Δ'⁻¹ * f W.c₄ ^ 3

Tactic:
]

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

Goal state before:
pp: k₀ k : ℕ
D : ℕ → ℕ
n : ℕ
⊢ numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
        ((k₀ * k) ^ (k₀ * k) * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n) *
      Nat.casesOn (n + 1) k D =
    numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
      ((k₀ * k) ^ (k₀ * k) • D) n

Tactic:
]

Goal state after:
k₀ k : ℕ
D : ℕ → ℕ
n : ℕ
⊢ numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
        ((k₀ * k) ^ (k₀ * k) * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n) *
      Nat.casesOn (n + 1) k D =
    numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n *
      ((k₀ * k) ^ (k₀ * k) • D) n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W : Affine R
inst✝ : Nontrivial R
⊢ { a := 0, b := 1, c := { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly,
        d := { a := -1, b := -W.a₂, c := -W.a₄, d := -W.a₆ }.toPoly }.toPoly ≠
    0

Tactic:
]

Goal state after:
R : Type r
inst✝¹ : CommRing R
W : Affine R
inst✝ : Nontrivial R
⊢ { a := 0, b := 1, c := { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly,
        d := { a := -1, b := -W.a₂, c := -W.a₄, d := -W.a₆ }.toPoly }.toPoly ≠
    0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W : Affine R
inst✝ : Nontrivial R
⊢ { a := 0, b := 1, c := { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly,
          d := { a := -1, b := -W.a₂, c := -W.a₄, d := -W.a₆ }.toPoly }.toPoly.degree =
    2

Tactic:
]

Goal state after:
R : Type r
inst✝¹ : CommRing R
W : Affine R
inst✝ : Nontrivial R
⊢ { a := 0, b := 1, c := { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly,
          d := { a := -1, b := -W.a₂, c := -W.a₄, d := -W.a₆ }.toPoly }.toPoly.degree =
    2
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W : Affine R
inst✝ : Nontrivial R
⊢ { a := 0, b := 1, c := { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly,
          d := { a := -1, b := -W.a₂, c := -W.a₄, d := -W.a₆ }.toPoly }.toPoly.degree =
    2

Tactic:
exact Cubic.degree_of_b_ne_zero' one_ne_zero

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

Goal state before:
pp: k₁ k₂ : ℕ
D₁ D₂ : ℕ → ℕ
hk : k₁ ≤ k₂
n : ℕ
hD : ∀ i < n + 1, D₁ i ≤ D₂ i
⊢ numBound k₁ D₁ (n + 1) ^ numBound k₁ D₁ (n + 1) * degBound k₁ D₁ n ≤
    numBound k₂ D₂ (n + 1) ^ numBound k₂ D₂ (n + 1) * degBound k₂ D₂ n

Tactic:
]

Goal state after:
k₁ k₂ : ℕ
D₁ D₂ : ℕ → ℕ
hk : k₁ ≤ k₂
n : ℕ
hD : ∀ i < n + 1, D₁ i ≤ D₂ i
⊢ numBound k₁ D₁ (n + 1) ^ numBound k₁ D₁ (n + 1) * degBound k₁ D₁ n ≤
    numBound k₂ D₂ (n + 1) ^ numBound k₂ D₂ (n + 1) * degBound k₂ D₂ n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W : Affine R
inst✝ : Nontrivial R
⊢ { a := 0, b := 1, c := { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly,
          d := { a := -1, b := -W.a₂, c := -W.a₄, d := -W.a₆ }.toPoly }.toPoly.natDegree =
    2

Tactic:
]

Goal state after:
R : Type r
inst✝¹ : CommRing R
W : Affine R
inst✝ : Nontrivial R
⊢ { a := 0, b := 1, c := { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly,
          d := { a := -1, b := -W.a₂, c := -W.a₄, d := -W.a₆ }.toPoly }.toPoly.natDegree =
    2
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: k₁ k₂ : ℕ
D₁ D₂ : ℕ → ℕ
hk : k₁ ≤ k₂
n : ℕ
hD : ∀ i < n + 1, D₁ i ≤ D₂ i
⊢ numBound k₁ D₁ n * degBound k₁ D₁ n * D₁ n ≤ numBound k₂ D₂ n * degBound k₂ D₂ n * D₂ n

Tactic:
]

Goal state after:
k₁ k₂ : ℕ
D₁ D₂ : ℕ → ℕ
hk : k₁ ≤ k₂
n : ℕ
hD : ∀ i < n + 1, D₁ i ≤ D₂ i
⊢ numBound k₁ D₁ n * degBound k₁ D₁ n * D₁ n ≤ numBound k₂ D₂ n * degBound k₂ D₂ n * D₂ n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W : Affine R
inst✝ : IsDomain R
h : ¬Irreducible W.polynomial
f g : R[X]
h0 : 3 = f.degree + g.degree
h1 : { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly.degree = (f + g).degree
⊢ False

Tactic:
]

Goal state after:
R : Type r
inst✝¹ : CommRing R
W : Affine R
inst✝ : IsDomain R
h : ¬Irreducible W.polynomial
f g : R[X]
h0 : 3 = f.degree + g.degree
h1 : { a := 0, b := 0, c := W.a₁, d := W.a₃ }.toPoly.degree = (f + g).degree
⊢ False