mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: case neg
R : Type u_1
inst✝ : CommRing R
b c d : R
k : ℤ
ih : ∀ (n : ℕ), complEDS b c d k (2 * ↑n) = complEDS b c d k ↑n * complEDS₂ b c d (↑n * k)
n✝ : ℕ
⊢ complEDS b c d k (2 * -↑n✝) = complEDS b c d k (-↑n✝) * complEDS₂ b c d (-↑n✝ * k)

Tactic:
[

Goal state after:
case neg
R : Type u_1
inst✝ : CommRing R
b c d : R
k : ℤ
ih : ∀ (n : ℕ), complEDS b c d k (2 * ↑n) = complEDS b c d k ↑n * complEDS₂ b c d (↑n * k)
n✝ : ℕ
⊢ complEDS b c d k (2 * -↑n✝) = complEDS b c d k (-↑n✝) * complEDS₂ b c d (-↑n✝ * k)
Module: Mathlib.NumberTheory.EllipticDivisibilitySequence

Goal state before:
pp: case neg.succ
R : Type u_1
inst✝ : CommRing R
b c d : R
k : ℤ
ih :
  ∀ (n : ℕ),
    complEDS b c d k (2 * ↑n + 1) =
      complEDS b c d k ↑n ^ 2 * normEDS b c d ((↑n + 1) * k + 1) * normEDS b c d ((↑n + 1) * k - 1) -
        complEDS b c d k (↑n + 1) ^ 2 * normEDS b c d (↑n * k + 1) * normEDS b c d (↑n * k - 1)
m : ℕ
⊢ complEDS b c d k (2 * -↑(m + 1) + 1) =
    complEDS b c d k (-↑(m + 1)) ^ 2 * normEDS b c d ((-↑(m + 1) + 1) * k + 1) *
        normEDS b c d ((-↑(m + 1) + 1) * k - 1) -
      complEDS b c d k (-↑(m + 1) + 1) ^ 2 * normEDS b c d (-↑(m + 1) * k + 1) * normEDS b c d (-↑(m + 1) * k - 1)

Tactic:
[

Goal state after:
case neg.succ
R : Type u_1
inst✝ : CommRing R
b c d : R
k : ℤ
ih :
  ∀ (n : ℕ),
    complEDS b c d k (2 * ↑n + 1) =
      complEDS b c d k ↑n ^ 2 * normEDS b c d ((↑n + 1) * k + 1) * normEDS b c d ((↑n + 1) * k - 1) -
        complEDS b c d k (↑n + 1) ^ 2 * normEDS b c d (↑n * k + 1) * normEDS b c d (↑n * k - 1)
m : ℕ
⊢ complEDS b c d k (2 * -↑(m + 1) + 1) =
    complEDS b c d k (-↑(m + 1)) ^ 2 * normEDS b c d ((-↑(m + 1) + 1) * k + 1) *
        normEDS b c d ((-↑(m + 1) + 1) * k - 1) -
      complEDS b c d k (-↑(m + 1) + 1) ^ 2 * normEDS b c d (-↑(m + 1) * k + 1) * normEDS b c d (-↑(m + 1) * k - 1)
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁵ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁴ : IsCofiltered I
inst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
inst✝ : IsAffine c.pt
this✝¹ : ∀ (i : I), IsAffineHom (c.π.app i)
i : I
hi : (D.obj i).IsQuasiAffine
this✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom ((D ⋙ Γ.rightOp ⋙ Scheme.Spec).map f)
this : ∀ (j : I), CompactSpace ↥((D ⋙ Γ.rightOp ⋙ Scheme.Spec).obj j)
⊢ ∃ i, IsAffine (D.obj i)

Tactic:
obtain ⟨j, fij, hj⟩ := exists_map_eq_top _ _
    (isLimitOfPreserves (Scheme.Γ.rightOp ⋙ Scheme.Spec) hc) (D.obj i).toSpecΓ.opensRange
    ((preimage_opensRange_toSpecΓ (X := c.pt) (c.π.app i)).trans
      (by simp [Hom.opensRange_of_isIso]))

Goal state after:
I : Type u
inst✝⁵ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁴ : IsCofiltered I
inst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
inst✝ : IsAffine c.pt
this✝¹ : ∀ (i : I), IsAffineHom (c.π.app i)
i : I
hi : (D.obj i).IsQuasiAffine
this✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom ((D ⋙ Γ.rightOp ⋙ Scheme.Spec).map f)
this : ∀ (j : I), CompactSpace ↥((D ⋙ Γ.rightOp ⋙ Scheme.Spec).obj j)
j : I
fij : j ⟶ i
hj : (D ⋙ Γ.rightOp ⋙ Scheme.Spec).map fij ⁻¹ᵁ Hom.opensRange (D.obj i).toSpecΓ = ⊤
⊢ ∃ i, IsAffine (D.obj i)
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁵ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁴ : IsCofiltered I
inst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
J : Type u_1
inst✝ : Finite J
U : J → c.pt.Opens
hU : IsOpenCover U
hU' : ∀ (i : J), IsAffineOpen (U i)
j : J → I
V : (k : J) → (D.obj (j k)).Opens
hV : ∀ (k : J), IsAffineOpen (V k)
hVU : ∀ (k : J), c.π.app (j k) ⁻¹ᵁ V k = U k
val✝ : Fintype J
i : I
fi : (k : J) → i ⟶ j k
⊢ c.π.app i ⁻¹ᵁ ⨆ k, D.map (fi k) ⁻¹ᵁ V k = ⊤

Tactic:
[

Goal state after:
I : Type u
inst✝⁵ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁴ : IsCofiltered I
inst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
J : Type u_1
inst✝ : Finite J
U : J → c.pt.Opens
hU : IsOpenCover U
hU' : ∀ (i : J), IsAffineOpen (U i)
j : J → I
V : (k : J) → (D.obj (j k)).Opens
hV : ∀ (k : J), IsAffineOpen (V k)
hVU : ∀ (k : J), c.π.app (j k) ⁻¹ᵁ V k = U k
val✝ : Fintype J
i : I
fi : (k : J) → i ⟶ j k
⊢ c.π.app i ⁻¹ᵁ ⨆ k, D.map (fi k) ⁻¹ᵁ V k = ⊤
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: case intro.refine_1
I : Type u
inst✝⁵ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁴ : IsCofiltered I
inst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
J : Type u_1
inst✝ : Finite J
U : J → c.pt.Opens
hU : IsOpenCover U
hU' : ∀ (i : J), IsAffineOpen (U i)
j : J → I
V : (k : J) → (D.obj (j k)).Opens
hV : ∀ (k : J), IsAffineOpen (V k)
hVU : ∀ (k : J), c.π.app (j k) ⁻¹ᵁ V k = U k
val✝ : Fintype J
i : I
fi : (k : J) → i ⟶ j k
k : I
fkj : k ⟶ i
e : D.map fkj ⁻¹ᵁ ⨆ k, D.map (fi k) ⁻¹ᵁ V k = ⊤
⊢ D.map fkj ⁻¹ᵁ ⨆ k, D.map (fi k) ⁻¹ᵁ V k ≤ ⨆ x, D.map (fkj ≫ fi x) ⁻¹ᵁ V x

Tactic:
[

Goal state after:
case intro.refine_1
I : Type u
inst✝⁵ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝⁴ : IsCofiltered I
inst✝³ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝² : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝¹ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
J : Type u_1
inst✝ : Finite J
U : J → c.pt.Opens
hU : IsOpenCover U
hU' : ∀ (i : J), IsAffineOpen (U i)
j : J → I
V : (k : J) → (D.obj (j k)).Opens
hV : ∀ (k : J), IsAffineOpen (V k)
hVU : ∀ (k : J), c.π.app (j k) ⁻¹ᵁ V k = U k
val✝ : Fintype J
i : I
fi : (k : J) → i ⟶ j k
k : I
fkj : k ⟶ i
e : D.map fkj ⁻¹ᵁ ⨆ k, D.map (fi k) ⁻¹ᵁ V k = ⊤
⊢ D.map fkj ⁻¹ᵁ ⨆ k, D.map (fi k) ⁻¹ᵁ V k ≤ ⨆ x, D.map (fkj ≫ fi x) ⁻¹ᵁ V x
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ W.Ψ (-n) = -W.Ψ n

Tactic:
[

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ W.Ψ (-n) = -W.Ψ n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
m : ℤ
⊢ W.Ψ (2 * m) * W.ψ₂ = W.Ψ (m - 1) ^ 2 * W.Ψ m * W.Ψ (m + 2) - W.Ψ (m - 2) * W.Ψ m * W.Ψ (m + 1) ^ 2

Tactic:
[

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
m : ℤ
⊢ W.Ψ (2 * m) * W.ψ₂ = W.Ψ (m - 1) ^ 2 * W.Ψ m * W.Ψ (m + 2) - W.Ψ (m - 2) * W.Ψ m * W.Ψ (m + 1) ^ 2
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
m : ℤ
⊢ W.Ψ (2 * m + 1) =
    W.Ψ (m + 2) * W.Ψ m ^ 3 - W.Ψ (m - 1) * W.Ψ (m + 1) ^ 3 +
      W.toAffine.polynomial * (16 * W.toAffine.polynomial - 8 * W.ψ₂ ^ 2) *
        C (if Even m then W.preΨ (m + 2) * W.preΨ m ^ 3 else -W.preΨ (m - 1) * W.preΨ (m + 1) ^ 3)

Tactic:
[

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
m : ℤ
⊢ W.Ψ (2 * m + 1) =
    W.Ψ (m + 2) * W.Ψ m ^ 3 - W.Ψ (m - 1) * W.Ψ (m + 1) ^ 3 +
      W.toAffine.polynomial * (16 * W.toAffine.polynomial - 8 * W.ψ₂ ^ 2) *
        C (if Even m then W.preΨ (m + 2) * W.preΨ m ^ 3 else -W.preΨ (m - 1) * W.preΨ (m + 1) ^ 3)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Affine F
x₁ y₁ x₂ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
H : x₁ = x₂
⊢ some x₁ y₁ h₁ = some x₂ y₂ h₂ ∨ some x₁ y₁ h₁ = -some x₂ y₂ h₂

Tactic:
[

Goal state after:
F : Type u
inst✝ : Field F
W : Affine F
x₁ y₁ x₂ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
H : x₁ = x₂
⊢ some x₁ y₁ h₁ = some x₂ y₂ h₂ ∨ some x₁ y₁ h₁ = -some x₂ y₂ h₂
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ (mk W) (W.Ψ n) ^ 2 = (mk W) (C (W.ΨSq n))

Tactic:
[

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ (mk W) (W.Ψ n) ^ 2 = (mk W) (C (W.ΨSq n))
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

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

Tactic:
[

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

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ W.Φ (-n) = W.Φ n

Tactic:
[

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ W.Φ (-n) = W.Φ n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ (mk W) (W.ψ n) = (mk W) (W.Ψ n)

Tactic:
[

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ (mk W) (W.ψ n) = (mk W) (W.Ψ n)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ W.φ (-n) = W.φ n

Tactic:
[

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ W.φ (-n) = W.φ n
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ (mk W) (W.φ n) = (mk W) (C (W.Φ n))

Tactic:
[

Goal state after:
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ (mk W) (W.φ n) = (mk W) (C (W.Φ n))
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W : WeierstrassCurve R
f : R →+* S
⊢ (W.map f).ψ₂ = Polynomial.map (mapRingHom f) W.ψ₂

Tactic:
[

Goal state after:
R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W : WeierstrassCurve R
f : R →+* S
⊢ (W.map f).ψ₂ = Polynomial.map (mapRingHom f) W.ψ₂
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₁
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ({ u := 1, r := 0, s := s, t := 0 } • E).a₁ = E'.a₁

Tactic:
[

Goal state after:
case h.a₁
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ({ u := 1, r := 0, s := s, t := 0 } • E).a₁ = E'.a₁
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₂
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ({ u := 1, r := 0, s := s, t := 0 } • E).a₂ = E'.a₂

Tactic:
[

Goal state after:
case h.a₂
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ({ u := 1, r := 0, s := s, t := 0 } • E).a₂ = E'.a₂
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₂
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ 1 ^ 2 * (E.a₂ - s * 1 + 3 * 0 - s ^ 2) = E'.a₂

Tactic:
linear_combination -hs + E.a₂ * CharP.cast_eq_zero F 2

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

Goal state before:
pp: case h.a₃
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ({ u := 1, r := 0, s := s, t := 0 } • E).a₃ = E'.a₃

Tactic:
[

Goal state after:
case h.a₃
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ({ u := 1, r := 0, s := s, t := 0 } • E).a₃ = E'.a₃
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₄
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ({ u := 1, r := 0, s := s, t := 0 } • E).a₄ = E'.a₄

Tactic:
[

Goal state after:
case h.a₄
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ({ u := 1, r := 0, s := s, t := 0 } • E).a₄ = E'.a₄
Module: Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ (W.toShortNFOfCharThree • W).a₂ = W.b₂

Tactic:
[

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 3
⊢ (W.toShortNFOfCharThree • W).a₂ = W.b₂
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₆
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ({ u := 1, r := 0, s := s, t := 0 } • E).a₆ = E'.a₆

Tactic:
[

Goal state after:
case h.a₆
F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
s : F
hs : 1 * s ^ 2 + 1 * s + (E.a₂ + E'.a₂) = 0
⊢ ({ u := 1, r := 0, s := s, t := 0 } • E).a₆ = E'.a₆
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₁
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJEqZeroNF
inst✝ : E'.IsCharTwoJEqZeroNF
ha₃ : E.a₃ ≠ 0
ha₃' : E'.a₃ ≠ 0
this : NeZero 3
u : F
hu : u ^ 3 = E.a₃ / E'.a₃
s : F
hs : 1 * s ^ 4 + E.a₃ * s + (E.a₄ - u ^ 4 * E'.a₄) = 0
t : F
ht : 1 * t ^ 2 + E.a₃ * t + (s ^ 6 + E.a₄ * s ^ 2 + E.a₆ - u ^ 6 * E'.a₆) = 0
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := s ^ 2, s := s, t := t } • E).a₁ = E'.a₁

Tactic:
[

Goal state after:
case h.a₁
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJEqZeroNF
inst✝ : E'.IsCharTwoJEqZeroNF
ha₃ : E.a₃ ≠ 0
ha₃' : E'.a₃ ≠ 0
this : NeZero 3
u : F
hu : u ^ 3 = E.a₃ / E'.a₃
s : F
hs : 1 * s ^ 4 + E.a₃ * s + (E.a₄ - u ^ 4 * E'.a₄) = 0
t : F
ht : 1 * t ^ 2 + E.a₃ * t + (s ^ 6 + E.a₄ * s ^ 2 + E.a₆ - u ^ 6 * E'.a₆) = 0
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := s ^ 2, s := s, t := t } • E).a₁ = E'.a₁
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₂
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJEqZeroNF
inst✝ : E'.IsCharTwoJEqZeroNF
ha₃ : E.a₃ ≠ 0
ha₃' : E'.a₃ ≠ 0
this : NeZero 3
u : F
hu : u ^ 3 = E.a₃ / E'.a₃
s : F
hs : 1 * s ^ 4 + E.a₃ * s + (E.a₄ - u ^ 4 * E'.a₄) = 0
t : F
ht : 1 * t ^ 2 + E.a₃ * t + (s ^ 6 + E.a₄ * s ^ 2 + E.a₆ - u ^ 6 * E'.a₆) = 0
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := s ^ 2, s := s, t := t } • E).a₂ = E'.a₂

Tactic:
[

Goal state after:
case h.a₂
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJEqZeroNF
inst✝ : E'.IsCharTwoJEqZeroNF
ha₃ : E.a₃ ≠ 0
ha₃' : E'.a₃ ≠ 0
this : NeZero 3
u : F
hu : u ^ 3 = E.a₃ / E'.a₃
s : F
hs : 1 * s ^ 4 + E.a₃ * s + (E.a₄ - u ^ 4 * E'.a₄) = 0
t : F
ht : 1 * t ^ 2 + E.a₃ * t + (s ^ 6 + E.a₄ * s ^ 2 + E.a₆ - u ^ 6 * E'.a₆) = 0
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := s ^ 2, s := s, t := t } • E).a₂ = E'.a₂
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₃
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJEqZeroNF
inst✝ : E'.IsCharTwoJEqZeroNF
ha₃ : E.a₃ ≠ 0
ha₃' : E'.a₃ ≠ 0
this : NeZero 3
u : F
hu : u ^ 3 = E.a₃ / E'.a₃
s : F
hs : 1 * s ^ 4 + E.a₃ * s + (E.a₄ - u ^ 4 * E'.a₄) = 0
t : F
ht : 1 * t ^ 2 + E.a₃ * t + (s ^ 6 + E.a₄ * s ^ 2 + E.a₆ - u ^ 6 * E'.a₆) = 0
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := s ^ 2, s := s, t := t } • E).a₃ = E'.a₃

Tactic:
[

Goal state after:
case h.a₃
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJEqZeroNF
inst✝ : E'.IsCharTwoJEqZeroNF
ha₃ : E.a₃ ≠ 0
ha₃' : E'.a₃ ≠ 0
this : NeZero 3
u : F
hu : u ^ 3 = E.a₃ / E'.a₃
s : F
hs : 1 * s ^ 4 + E.a₃ * s + (E.a₄ - u ^ 4 * E'.a₄) = 0
t : F
ht : 1 * t ^ 2 + E.a₃ * t + (s ^ 6 + E.a₄ * s ^ 2 + E.a₆ - u ^ 6 * E'.a₆) = 0
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := s ^ 2, s := s, t := t } • E).a₃ = E'.a₃
Module: Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms

Goal state before:
pp: case a₂
R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 2
ha₁ : W.a₁ = 0
⊢ (W.toCharTwoJEqZeroNF • W).a₂ = 0

Tactic:
[

Goal state after:
case a₂
R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : CharP R 2
ha₁ : W.a₁ = 0
⊢ (W.toCharTwoJEqZeroNF • W).a₂ = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case of_j_ne_zero.of_j_ne_zero
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
heq : E.j = E'.j
C : VariableChange F
inst✝¹ : (C • E).IsCharTwoJNeZeroNF
C' : VariableChange F
inst✝ : (C' • E').IsCharTwoJNeZeroNF
⊢ ∃ C, C • E = E'

Tactic:
[

Goal state after:
case of_j_ne_zero.of_j_ne_zero
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 2
heq : E.j = E'.j
C : VariableChange F
inst✝¹ : (C • E).IsCharTwoJNeZeroNF
C' : VariableChange F
inst✝ : (C' • E').IsCharTwoJNeZeroNF
⊢ ∃ C, C • E = E'
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝⁵ : 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✝⁴ : IsCofiltered I
inst✝³ : LocallyOfFinitePresentation f
inst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)
inst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)
a : c.pt ⟶ X
ha : c.π ≫ t = (Functor.const I).map (a ≫ f)
𝒰 : IsOpenCover fun V ↦ (↑V).1
i : I
s : Finset { x // x.1 ∈ c.pt.affineOpens ∧ x.1 ≤ (Opens.comap (TopCat.Hom.hom a.base)) (↑x.2).1 }
𝒱 : ↥s → (D.obj i).Opens
h𝒱 : IsOpenCover 𝒱
h𝒱𝒰 : ∀ (j : ↥s), IsAffineOpen (𝒱 j) ∧ (↑↑j).1 = c.π.app i ⁻¹ᵁ 𝒱 j
i' : I
fi'i : i' ⟶ i
hi' : ∀ (j : ↥s), D.map fi'i ⁻¹ᵁ 𝒱 j ≤ t.app i' ⁻¹ᵁ ↑(↑(↑↑j).2).2
𝒱' : ↥s → (D.obj i').Opens := fun x ↦ D.map fi'i ⁻¹ᵁ 𝒱 x
h𝒱'𝒰 : ∀ (j : ↥s), c.π.app i' ⁻¹ᵁ 𝒱' j = (↑↑j).1
x✝ : ∀ (j : ↥s) (k : Over i'), IsAffine ((opensDiagram D i' (𝒱' j)).obj k)
t𝒱 : (j : ↥s) → opensDiagram D i' (𝒱' j) ⟶ (Functor.const (Over i')).obj ↑↑(↑(↑↑j).2).2 :=
  fun j ↦ { app := fun k ↦ Hom.resLE (t.app k.left) (↑(↑(↑↑j).2).2) (D.map k.hom ⁻¹ᵁ 𝒱' j) ⋯, naturality := ⋯ }
this : ∀ (j : ↥s), IsAffine ↑(↑(↑↑j).2).1
j : ↥s
⊢ (opensCone D c i' (𝒱' j)).π ≫ t𝒱 j =
    (Functor.const (Over i')).map
      (Hom.resLE a (↑(↑↑j).2).1 (c.π.app i' ⁻¹ᵁ 𝒱' j) ⋯ ≫ Hom.resLE f (↑(↑(↑↑j).2).2) (↑(↑↑j).2).1 ⋯)

Tactic:
by
      ext k
      simp [t𝒱, Hom.resLE_comp_resLE, show c.π.app k.left ≫ t.app k.left = a ≫ f from
        congr(($ha).app k.left)]

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
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
heq : E.j = E'.j
h : E.a₂ ^ 3 ≠ 0 ∧ E.a₆ ≠ 0
⊢ ∃ C, C • E = E'

Tactic:
pow_ne_zero_iff three_ne_zero

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
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
heq : E.j = E'.j
h : E.a₂ ≠ 0 ∧ E.a₆ ≠ 0
⊢ ∃ C, C • E = E'
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
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
heq : E.j = E'.j
ha₂ : E.a₂ ≠ 0
ha₆ : E.a₆ ≠ 0
h : E'.a₂ ^ 3 ≠ 0 ∧ E'.a₆ ≠ 0
⊢ ∃ C, C • E = E'

Tactic:
pow_ne_zero_iff three_ne_zero

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
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
heq : E.j = E'.j
ha₂ : E.a₂ ≠ 0
ha₆ : E.a₆ ≠ 0
h : E'.a₂ ≠ 0 ∧ E'.a₆ ≠ 0
⊢ ∃ C, C • E = E'
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₁
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
heq : E.j = E'.j
ha₂ : E.a₂ ≠ 0
ha₆ : E.a₆ ≠ 0
ha₂' : E'.a₂ ≠ 0
ha₆' : E'.a₆ ≠ 0
this : NeZero 2
u : F
hu : u ^ 2 = E.a₂ / E'.a₂
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := 0, s := 0, t := 0 } • E).a₁ = E'.a₁

Tactic:
[

Goal state after:
case h.a₁
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsCharThreeJNeZeroNF
inst✝ : E'.IsCharThreeJNeZeroNF
heq : E.j = E'.j
ha₂ : E.a₂ ≠ 0
ha₆ : E.a₆ ≠ 0
ha₂' : E'.a₂ ≠ 0
ha₆' : E'.a₆ ≠ 0
this : NeZero 2
u : F
hu : u ^ 2 = E.a₂ / E'.a₂
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := 0, s := 0, t := 0 } • E).a₁ = E'.a₁