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

Goal state before:
pp: case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P✝ : Fin 3 → R
u✝ : R
hu : IsUnit u✝
u : R
P : Fin 3 → R
hP :
  P y ^ 2 + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 3 -
      (P x ^ 3 + W'.a₂ * P x ^ 2 * P z ^ 2 + W'.a₄ * P x * P z ^ 4 + W'.a₆ * P z ^ 6) =
    0
⊢ (u ^ 3 * P y) ^ 2 + W'.a₁ * (u ^ 2 * P x) * (u ^ 3 * P y) * (u * P z) + W'.a₃ * (u ^ 3 * P y) * (u * P z) ^ 3 -
          ((u ^ 2 * P x) ^ 3 + W'.a₂ * (u ^ 2 * P x) ^ 2 * (u * P z) ^ 2 + W'.a₄ * (u ^ 2 * P x) * (u * P z) ^ 4 +
            W'.a₆ * (u * P z) ^ 6) +
        u ^ 6 * 0 -
      (0 +
        u ^ 6 *
          (P y ^ 2 + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 3 -
            (P x ^ 3 + W'.a₂ * P x ^ 2 * P z ^ 2 + W'.a₄ * P x * P z ^ 4 + W'.a₆ * P z ^ 6))) =
    0

Tactic:
ring1

Goal state after:
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.IsShortNF
inst✝ : E'.IsShortNF
ha₄ : E.a₄ ≠ 0
ha₄' : E'.a₄ ≠ 0
this : NeZero 4
u : F
hu : u ^ 4 = E.a₄ / E'.a₄
r : F
hr : 1 * r ^ 3 + E.a₄ * r + (E.a₆ - u ^ 6 * E'.a₆) = 0
hu0 : u ≠ 0
⊢ ↑(Units.mk0 u hu0)⁻¹ * (0 + 2 * 0) = 0

Tactic:
ring1

Goal state after:
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 3
inst✝¹ : E.IsShortNF
inst✝ : E'.IsShortNF
ha₄ : E.a₄ ≠ 0
ha₄' : E'.a₄ ≠ 0
this : NeZero 4
u : F
hu : u ^ 4 = E.a₄ / E'.a₄
r : F
hr : 1 * r ^ 3 + E.a₄ * r + (E.a₆ - u ^ 6 * E'.a₆) = 0
hu0 : u ≠ 0
⊢ ↑(Units.mk0 u hu0)⁻¹ ^ 2 * (0 - 0 * 0 + 0 * r - 0 ^ 2) = 0

Tactic:
ring1

Goal state after:
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 3
inst✝¹ : E.IsShortNF
inst✝ : E'.IsShortNF
ha₄ : E.a₄ ≠ 0
ha₄' : E'.a₄ ≠ 0
this : NeZero 4
u : F
hu : u ^ 4 = E.a₄ / E'.a₄
r : F
hr : 1 * r ^ 3 + E.a₄ * r + (E.a₆ - u ^ 6 * E'.a₆) = 0
hu0 : u ≠ 0
⊢ ↑(Units.mk0 u hu0)⁻¹ ^ 3 * (0 + r * 0 + 2 * 0) = 0

Tactic:
ring1

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

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
⊢ ↑2 * Y ^ (2 - 1) * 0 + (((0 * X + C W'.a₁ * 1) * Y + C W'.a₁ * X * 0) * Z + C W'.a₁ * X * Y * 0) +
        ((0 * Y + C W'.a₃ * 0) * Z ^ 3 + C W'.a₃ * Y * (↑3 * Z ^ (3 - 1) * 0)) -
      (↑3 * X ^ (3 - 1) * 1 +
            ((0 * X ^ 2 + C W'.a₂ * (↑2 * X ^ (2 - 1) * 1)) * Z ^ 2 + C W'.a₂ * X ^ 2 * (↑2 * Z ^ (2 - 1) * 0)) +
          ((0 * X + C W'.a₄ * 1) * Z ^ 4 + C W'.a₄ * X * (↑4 * Z ^ (4 - 1) * 0)) +
        (0 * Z ^ 6 + C W'.a₆ * (↑6 * Z ^ (6 - 1) * 0))) =
    C W'.a₁ * Y * Z - (3 * X ^ 2 + 2 * C W'.a₂ * X * Z ^ 2 + C W'.a₄ * Z ^ 4)

Tactic:
ring1

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

Goal state before:
pp: case a
F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ (W.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W.a₂ * P x * P z ^ 2 + W.a₄ * P z ^ 4)) / P z ^ 4 +
        (W.a₁ * P y / P z ^ 3 * 1 + 2 * W.a₂ * P x / P z ^ 2 * (P z ^ 2 / P z ^ 2) + W.a₄ * (P z ^ 4 / P z ^ 4)) -
      (W.toAffine.a₁ * (P y / P z ^ 3) -
          (3 * (P x / P z ^ 2) ^ 2 + 2 * W.toAffine.a₂ * (P x / P z ^ 2) + W.toAffine.a₄) +
        (W.a₁ * P y / P z ^ 3 * (P z / P z) + 2 * W.a₂ * P x / P z ^ 2 * 1 + W.a₄ * 1)) =
    0

Tactic:
ring1

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

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
⊢ ↑2 * Y ^ (2 - 1) * 1 + (((0 * X + C W'.a₁ * 0) * Y + C W'.a₁ * X * 1) * Z + C W'.a₁ * X * Y * 0) +
        ((0 * Y + C W'.a₃ * 1) * Z ^ 3 + C W'.a₃ * Y * (↑3 * Z ^ (3 - 1) * 0)) -
      (↑3 * X ^ (3 - 1) * 0 +
            ((0 * X ^ 2 + C W'.a₂ * (↑2 * X ^ (2 - 1) * 0)) * Z ^ 2 + C W'.a₂ * X ^ 2 * (↑2 * Z ^ (2 - 1) * 0)) +
          ((0 * X + C W'.a₄ * 0) * Z ^ 4 + C W'.a₄ * X * (↑4 * Z ^ (4 - 1) * 0)) +
        (0 * Z ^ 6 + C W'.a₆ * (↑6 * Z ^ (6 - 1) * 0))) =
    2 * Y + C W'.a₁ * X * Z + C W'.a₃ * Z ^ 3

Tactic:
ring1

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

Goal state before:
pp: case a
F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ (2 * P y + W.a₁ * P x * P z + W.a₃ * P z ^ 3) / P z ^ 3 + (W.a₁ * P x / P z ^ 2 * 1 + W.a₃ * 1) -
      (2 * (P y / P z ^ 3) + W.toAffine.a₁ * (P x / P z ^ 2) + W.toAffine.a₃ +
        (W.a₁ * P x / P z ^ 2 * (P z / P z) + W.a₃ * (P z ^ 3 / P z ^ 3))) =
    0

Tactic:
ring1

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

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
⊢ ↑2 * Y ^ (2 - 1) * 0 + (((0 * X + C W'.a₁ * 0) * Y + C W'.a₁ * X * 0) * Z + C W'.a₁ * X * Y * 1) +
        ((0 * Y + C W'.a₃ * 0) * Z ^ 3 + C W'.a₃ * Y * (↑3 * Z ^ (3 - 1) * 1)) -
      (↑3 * X ^ (3 - 1) * 0 +
            ((0 * X ^ 2 + C W'.a₂ * (↑2 * X ^ (2 - 1) * 0)) * Z ^ 2 + C W'.a₂ * X ^ 2 * (↑2 * Z ^ (2 - 1) * 1)) +
          ((0 * X + C W'.a₄ * 0) * Z ^ 4 + C W'.a₄ * X * (↑4 * Z ^ (4 - 1) * 1)) +
        (0 * Z ^ 6 + C W'.a₆ * (↑6 * Z ^ (6 - 1) * 1))) =
    C W'.a₁ * X * Y + 3 * C W'.a₃ * Y * Z ^ 2 -
      (2 * C W'.a₂ * X ^ 2 * Z + 4 * C W'.a₄ * X * Z ^ 3 + 6 * C W'.a₆ * Z ^ 5)

Tactic:
ring1

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

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
P : Fin 3 → R
⊢ 6 *
      (P y ^ 2 + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 3 -
        (P x ^ 3 + W'.a₂ * P x ^ 2 * P z ^ 2 + W'.a₄ * P x * P z ^ 4 + W'.a₆ * P z ^ 6)) =
    2 * P x * (W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z ^ 2 + W'.a₄ * P z ^ 4)) +
        3 * P y * (2 * P y + W'.a₁ * P x * P z + W'.a₃ * P z ^ 3) +
      P z *
        (W'.a₁ * P x * P y + 3 * W'.a₃ * P y * P z ^ 2 -
          (2 * W'.a₂ * P x ^ 2 * P z + 4 * W'.a₄ * P x * P z ^ 3 + 6 * W'.a₆ * P z ^ 5))

Tactic:
ring1

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

Goal state before:
pp: case pos
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=
  fun {n} ↦ natDegree_coeff_preΨ' W n
n : ℕ
h✝ : Even n.succ
⊢ ↑n ^ 2 + 2 * ↑n * 1 = ↑n ^ 2 + 2 * ↑n * 1 + 1 - 4 + 3

Tactic:
ring1

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

Goal state before:
pp: case neg
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=
  fun {n} ↦ natDegree_coeff_preΨ' W n
n : ℕ
h✝ : ¬Even n.succ
⊢ ↑n ^ 2 + 2 * ↑n * 1 = ↑n ^ 2 + 2 * ↑n * 1 + 1 - 1 + 0

Tactic:
ring1

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

Goal state before:
pp: case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P✝ : Fin 3 → R
u✝ : R
hu✝ : IsUnit u✝
u : R
hu : IsUnit u
P : Fin 3 → R
hP✝ : W'.Nonsingular (u • P)
hP : W'.Equation (u • P)
hP' :
  W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z ^ 2 + W'.a₄ * P z ^ 4) = 0 ∧
    2 * P y + W'.a₁ * P x * P z + W'.a₃ * P z ^ 3 = 0 ∧
      W'.a₁ * P x * P y + 3 * W'.a₃ * P y * P z ^ 2 -
          (2 * W'.a₂ * P x ^ 2 * P z + 4 * W'.a₄ * P x * P z ^ 3 + 6 * W'.a₆ * P z ^ 5) =
        0
⊢ W'.a₁ * (u ^ 3 * P y) * (u * P z) -
          (3 * (u ^ 2 * P x) ^ 2 + 2 * W'.a₂ * (u ^ 2 * P x) * (u * P z) ^ 2 + W'.a₄ * (u * P z) ^ 4) +
        u ^ 4 * 0 -
      (0 + u ^ 4 * (W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z ^ 2 + W'.a₄ * P z ^ 4))) =
    0

Tactic:
ring1

Goal state after:
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
k : I
fki : k ⟶ i
ak : (j : ↥s) → ↑(D.map fki ⁻¹ᵁ 𝒱 j) ⟶ X
hak : ∀ (j : ↥s), ak j ≫ f = (D.map fki ⁻¹ᵁ 𝒱 j).ι ≫ t.app k
hak' : ∀ (j : ↥s), c.π.app k ∣_ D.map fki ⁻¹ᵁ 𝒱 j ≫ ak j = (c.π.app k ⁻¹ᵁ D.map fki ⁻¹ᵁ 𝒱 j).ι ≫ a
⊢ ∃ l flk,
    ∀ (j₁ j₂ : ↥s),
      (D.obj l).homOfLE ⋯ ≫ D.map flk ∣_ D.map fki ⁻¹ᵁ 𝒱 j₁ ≫ ak j₁ =
        (D.obj l).homOfLE ⋯ ≫ D.map flk ∣_ D.map fki ⁻¹ᵁ 𝒱 j₂ ≫ ak j₂

Tactic:
by
    let 𝒱' := (D.map fki ⁻¹ᵁ 𝒱 ·)
    have (j₁ j₂ : s) : ∃ (l : I) (flk : l ⟶ k), Scheme.homOfLE _ inf_le_left ≫ D.map flk ∣_ _ ≫
        ak j₁ = Scheme.homOfLE _ inf_le_right ≫ D.map flk ∣_ _ ≫ ak j₂ := by
      have _ (x) : CompactSpace ↥((opensDiagram D k (𝒱' j₁ ⊓ 𝒱' j₂)).obj x) :=
        isCompact_iff_compactSpace.mp (QuasiCompact.isCompact_preimage _ (𝒱' j₁ ⊓ 𝒱' j₂).isOpen
          (((h𝒱𝒰 _).1.preimage _).isCompact.inter_of_isOpen ((h𝒱𝒰 _).1.preimage _).isCompact
            (D.map fki ⁻¹ᵁ 𝒱 j₁).2 (D.map fki ⁻¹ᵁ 𝒱 j₂).2))
      obtain ⟨⟨l, ⟨⟨⟩⟩, flk⟩, ⟨flk', ⟨⟨⟨⟩⟩⟩, h⟩, e⟩ :=
        Scheme.exists_hom_comp_eq_comp_of_locallyOfFiniteType _
          (opensDiagramι .. ≫ (Over.forget k).whiskerLeft t) f _
          (isLimitOpensCone D c hc k (𝒱' j₁ ⊓ 𝒱' j₂)) (i := .mk (𝟙 k))
          (Scheme.homOfLE _ (by simp [𝒱']) ≫ ak j₁) (Scheme.homOfLE _ (by simp [𝒱']) ≫ ak j₂)
          (by simp [hak]) (by simp [hak]) (by simp; simp [Hom.resLE, hak'])
      obtain rfl : flk = flk' := by simpa using! h.symm
      refine ⟨l, flk, by simpa [← Scheme.Hom.resLE_eq_morphismRestrict] using! e⟩
    choose l flk hflk using this
    obtain ⟨l', fl'k, fl'l, hl'⟩ := IsCofiltered.wideCospan (I := s × s) fun x ↦ flk x.1 x.2
    refine ⟨l', fl'k, fun j₁ j₂ ↦ ?_⟩
    have H : (D.map fl'k ≫ D.map fki) ⁻¹ᵁ (𝒱 j₁ ⊓ 𝒱 j₂) ≤
        (D.map (fl'l (j₁, j₂)) ≫ D.map (flk j₁ j₂) ≫ D.map fki) ⁻¹ᵁ (𝒱 j₁ ⊓ 𝒱 j₂) := by
      simp only [← Functor.map_comp, reassoc_of% hl']; rfl
    simpa [← Scheme.Hom.resLE_eq_morphismRestrict, Scheme.Hom.resLE_comp_resLE_assoc,
      ← Functor.map_comp, hl'] using! congr((D.map (fl'l (j₁, j₂))).resLE _ _ H ≫ $(hflk j₁ j₂))

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

Goal state before:
pp: case pos
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=
  fun {n} ↦ natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1) ^ 2 - 1 = 2 * expDegree (n + 1) + if Even (n + 1) then 3 else 0
h✝ : Even (n + 1)
⊢ (↑n + 1) ^ 2 = ((↑n + 1) / 2) ^ 2 * 4

Tactic:
ring1

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

Goal state before:
pp: case neg
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=
  fun {n} ↦ natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1) ^ 2 - 1 = 2 * expDegree (n + 1) + if Even (n + 1) then 3 else 0
h✝ : ¬Even (n + 1)
⊢ (↑n + 1) ^ 2 = (↑n + 1) ^ 2 * 1

Tactic:
ring1

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

Goal state before:
pp: case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P✝ : Fin 3 → R
u✝ : R
hu✝ : IsUnit u✝
u : R
hu : IsUnit u
P : Fin 3 → R
hP✝ : W'.Nonsingular (u • P)
hP : W'.Equation (u • P)
hP' :
  W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z ^ 2 + W'.a₄ * P z ^ 4) = 0 ∧
    2 * P y + W'.a₁ * P x * P z + W'.a₃ * P z ^ 3 = 0 ∧
      W'.a₁ * P x * P y + 3 * W'.a₃ * P y * P z ^ 2 -
          (2 * W'.a₂ * P x ^ 2 * P z + 4 * W'.a₄ * P x * P z ^ 3 + 6 * W'.a₆ * P z ^ 5) =
        0
⊢ 2 * (u ^ 3 * P y) + W'.a₁ * (u ^ 2 * P x) * (u * P z) + W'.a₃ * (u * P z) ^ 3 + u ^ 3 * 0 -
      (0 + u ^ 3 * (2 * P y + W'.a₁ * P x * P z + W'.a₃ * P z ^ 3)) =
    0

Tactic:
ring1

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

Goal state before:
pp: case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P✝ : Fin 3 → R
u✝ : R
hu✝ : IsUnit u✝
u : R
hu : IsUnit u
P : Fin 3 → R
hP✝ : W'.Nonsingular (u • P)
hP : W'.Equation (u • P)
hP' :
  W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z ^ 2 + W'.a₄ * P z ^ 4) = 0 ∧
    2 * P y + W'.a₁ * P x * P z + W'.a₃ * P z ^ 3 = 0 ∧
      W'.a₁ * P x * P y + 3 * W'.a₃ * P y * P z ^ 2 -
          (2 * W'.a₂ * P x ^ 2 * P z + 4 * W'.a₄ * P x * P z ^ 3 + 6 * W'.a₆ * P z ^ 5) =
        0
⊢ W'.a₁ * (u ^ 2 * P x) * (u ^ 3 * P y) + 3 * W'.a₃ * (u ^ 3 * P y) * (u * P z) ^ 2 -
          (2 * W'.a₂ * (u ^ 2 * P x) ^ 2 * (u * P z) + 4 * W'.a₄ * (u ^ 2 * P x) * (u * P z) ^ 3 +
            6 * W'.a₆ * (u * P z) ^ 5) +
        u ^ 5 * 0 -
      (0 +
        u ^ 5 *
          (W'.a₁ * P x * P y + 3 * W'.a₃ * P y * P z ^ 2 -
            (2 * W'.a₂ * P x ^ 2 * P z + 4 * W'.a₄ * P x * P z ^ 3 + 6 * W'.a₆ * P z ^ 5))) =
    0

Tactic:
ring1

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

Goal state before:
pp: case a
R : Type r
inst✝¹ : CommRing R
W' : Jacobian R
inst✝ : Nontrivial R
h : 3 * 1 ^ 2 = 0 ∧ 2 * 1 = 0 ∧ W'.a₁ * 1 * 1 = 0
⊢ 1 + (0 + 2 * 1) - (0 + (3 * 1 ^ 2 + 0)) = 0

Tactic:
ring1

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

Goal state before:
pp: case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
a b : R
h :
  b ^ 2 + W'.toAffine.a₁ * a * b + W'.toAffine.a₃ * b -
      (a ^ 3 + W'.toAffine.a₂ * a ^ 2 + W'.toAffine.a₄ * a + W'.toAffine.a₆) =
    0
ha : W'.a₁ * b - (3 * a ^ 2 + 2 * W'.a₂ * a + W'.a₄) = 0
hb : 2 * b + W'.toAffine.a₁ * a + W'.toAffine.a₃ = 0
⊢ W'.a₁ * a * b + 3 * W'.a₃ * b - (2 * W'.a₂ * a ^ 2 + 4 * W'.a₄ * a + 6 * W'.a₆) +
        (6 * 0 + 2 * a * (W'.a₁ * b - (3 * a ^ 2 + 2 * W'.a₂ * a + W'.a₄)) +
          3 * b * (2 * b + W'.toAffine.a₁ * a + W'.toAffine.a₃)) -
      (0 +
        (6 *
              (b ^ 2 + W'.toAffine.a₁ * a * b + W'.toAffine.a₃ * b -
                (a ^ 3 + W'.toAffine.a₂ * a ^ 2 + W'.toAffine.a₄ * a + W'.toAffine.a₆)) +
            2 * a * 0 +
          3 * b * 0)) =
    0

Tactic:
ring1

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

Goal state before:
pp: case pos
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=
  fun {n} ↦ natDegree_coeff_preΨ' W n
n : ℕ
h✝ : Even n
⊢ (↑n + 1 + 1) ^ 2 = 1 + ((↑n + 1 + 1) ^ 2 - 4) + 3

Tactic:
ring1

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

Goal state before:
pp: case neg
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=
  fun {n} ↦ natDegree_coeff_preΨ' W n
n : ℕ
h✝ : ¬Even n
⊢ (↑n + 1 + 1) ^ 2 = 1 + ((↑n + 1 + 1) ^ 2 - 1) + 0

Tactic:
ring1

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

Goal state before:
pp: case pos
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=
  fun {n} ↦ natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * expDegree (n + 2) + if Even (n + 1) then 0 else 3
h✝ : Even n
⊢ 2 * (↑n + 1 + 1) ^ 2 = (↑n + 2 + 1) ^ 2 - 1 + ((↑n + 1) ^ 2 - 1) + 2 * 0

Tactic:
ring1

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

Goal state before:
pp: case neg
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=
  fun {n} ↦ natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * expDegree (n + 2) + if Even (n + 1) then 0 else 3
h✝ : ¬Even n
⊢ 2 * (↑n + 1 + 1) ^ 2 = (↑n + 2 + 1) ^ 2 - 4 + ((↑n + 1) ^ 2 - 4) + 2 * 3

Tactic:
ring1

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

Goal state before:
pp: case pos
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=
  fun {n} ↦ natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' : (n + 1 + 1) ^ 2 = expDegree (n + 3) + expDegree (n + 1) + if Even (n + 1) then 3 else 0
h✝ : Even n
⊢ 1 = 1 * ((↑n + 2) / 2) ^ 2 * 4 - (↑n + 3) * (↑n + 1) * 1

Tactic:
ring1

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

Goal state before:
pp: case neg
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=
  fun {n} ↦ natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' : (n + 1 + 1) ^ 2 = expDegree (n + 3) + expDegree (n + 1) + if Even (n + 1) then 3 else 0
h✝ : ¬Even n
⊢ 1 = 1 * (↑n + 2) ^ 2 * 1 - (↑n + 3) / 2 * ((↑n + 1) / 2) * 4

Tactic:
ring1

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

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
P : Fin 3 → R
u : R
⊢ -(u ^ 3 * P y) - W'.a₁ * (u ^ 2 * P x) * (u * P z) - W'.a₃ * (u * P z) ^ 3 =
    u ^ 3 * (-P y - W'.a₁ * P x * P z - W'.a₃ * P z ^ 3)

Tactic:
ring1

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

Goal state before:
pp: case a
F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ (-P y - W.a₁ * P x * P z - W.a₃ * P z ^ 3) / P z ^ 3 + (-W.a₁ * P x / P z ^ 2 * 1 + W.a₃ * (P z ^ 3 / P z ^ 3)) -
      (-(P y / P z ^ 3) - W.toAffine.a₁ * (P x / P z ^ 2) - W.toAffine.a₃ +
        (-W.a₁ * P x / P z ^ 2 * (P z / P z) + W.a₃ * 1)) =
    0

Tactic:
ring1

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

Goal state before:
pp: case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
hx : P x * Q z ^ 2 = Q x * P z ^ 2
⊢ (P y * Q z ^ 3 - Q y * P z ^ 3) * (P y * Q z ^ 3 - (-Q y - W'.a₁ * Q x * Q z - W'.a₃ * Q z ^ 3) * P z ^ 3) +
        (Q z ^ 6 * 0 +
            P z ^ 6 *
              (Q y ^ 2 + W'.a₁ * Q x * Q y * Q z + W'.a₃ * Q y * Q z ^ 3 -
                (Q x ^ 3 + W'.a₂ * Q x ^ 2 * Q z ^ 2 + W'.a₄ * Q x * Q z ^ 4 + W'.a₆ * Q z ^ 6)) +
          (P x ^ 2 * Q z ^ 4 + P x * Q x * P z ^ 2 * Q z ^ 2 + Q x ^ 2 * P z ^ 4 - W'.a₁ * P y * P z * Q z ^ 4 +
                  W'.a₂ * P x * P z ^ 2 * Q z ^ 4 +
                W'.a₂ * Q x * P z ^ 4 * Q z ^ 2 +
              W'.a₄ * P z ^ 4 * Q z ^ 4) *
            (Q x * P z ^ 2)) -
      (0 +
        (Q z ^ 6 *
              (P y ^ 2 + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 3 -
                (P x ^ 3 + W'.a₂ * P x ^ 2 * P z ^ 2 + W'.a₄ * P x * P z ^ 4 + W'.a₆ * P z ^ 6)) +
            P z ^ 6 * 0 +
          (P x ^ 2 * Q z ^ 4 + P x * Q x * P z ^ 2 * Q z ^ 2 + Q x ^ 2 * P z ^ 4 - W'.a₁ * P y * P z * Q z ^ 4 +
                  W'.a₂ * P x * P z ^ 2 * Q z ^ 4 +
                W'.a₂ * Q x * P z ^ 4 * Q z ^ 2 +
              W'.a₄ * P z ^ 4 * Q z ^ 4) *
            (P x * Q z ^ 2))) =
    0

Tactic:
ring1

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

Goal state before:
pp: case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hx : P x * Q z ^ 2 = Q x * P z ^ 2
⊢ P y * Q z ^ 3 - Q y * P z ^ 3 + (P y * Q z ^ 3 - (-Q y - W'.a₁ * Q x * Q z - W'.a₃ * Q z ^ 3) * P z ^ 3) +
        -W'.a₁ * P z * Q z * (Q x * P z ^ 2) -
      ((P y - (-P y - W'.a₁ * P x * P z - W'.a₃ * P z ^ 3)) * Q z ^ 3 + -W'.a₁ * P z * Q z * (P x * Q z ^ 2)) =
    0

Tactic:
ring1

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

Goal state before:
pp: case a
R : Type r
inst✝¹ : CommRing R
W' : Jacobian R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy' : P y * Q z ^ 3 - W'.negY Q * P z ^ 3 = 0
hy : P y = W'.negY P
⊢ P y * Q z ^ 3 + ((P y - negY ?m.190 P) * Q z ^ 3 + Q z ^ 3 * W'.negY P + (P y * Q z ^ 3 - W'.negY Q * P z ^ 3)) -
      (Q y * P z ^ 3 +
        (P y * Q z ^ 3 - Q y * P z ^ 3 + (P y * Q z ^ 3 - negY ?m.190 Q * P z ^ 3) + Q z ^ 3 * P y + 0)) =
    0

Tactic:
ring1

Goal state after: