-- Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
lemma XYIdeal_mul_XYIdeal {x₁ x₂ y₁ y₂ : F} (h₁ : W.Equation x₁ y₁) (h₂ : W.Equation x₂ y₂)
    (hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)) :
    XIdeal W (W.addX x₁ x₂ <| W.slope x₁ x₂ y₁ y₂) * (XYIdeal W x₁ (C y₁) * XYIdeal W x₂ (C y₂)) =
      YIdeal W (linePolynomial x₁ y₁ <| W.slope x₁ x₂ y₁ y₂) *
        XYIdeal W (W.addX x₁ x₂ <| W.slope x₁ x₂ y₁ y₂)
          (C <| W.addY x₁ x₂ y₁ <| W.slope x₁ x₂ y₁ y₂)

Proof state:
F : Type u
inst✝ : Field F
W : Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Equation x₁ y₁
h₂ : W.Equation x₂ y₂
hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)
⊢ XIdeal W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) * (XYIdeal W x₁ (C y₁) * XYIdeal W x₂ (C y₂)) =
    YIdeal W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) *
      XYIdeal W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (C (W.addY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂)))

Tactic:
have sup_rw : ∀ a b c d : Ideal W.CoordinateRing, a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c :=
  fun _ _ c _ => by rw [← sup_assoc, sup_comm c, sup_sup_sup_comm, ← sup_assoc]

Result:
F : Type u
inst✝ : Field F
W : Affine 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
⊢ XIdeal W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) * (XYIdeal W x₁ (C y₁) * XYIdeal W x₂ (C y₂)) =
    YIdeal W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) *
      XYIdeal W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (C (W.addY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂)))

Proof state:
case a.a
F : Type u
inst✝ : Field F
W : Affine F
x₁ y₁ : F
h₁ : W.Equation x₁ y₁
sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c
h₂ : W.Equation x₁ y₁
hy : y₁ ≠ W.negY x₁ y₁
y : F := (y₁ - W.negY x₁ y₁) ^ 2
hxy : (y₁ - W.negY x₁ y₁) ^ 2 ≠ 0
⊢ C (C 1) * (Y - (-Y - C (C W.a₁ * X + C W.a₃))) * (Y - (-Y - C (C W.a₁ * X + C W.a₃))) +
        -4 * C (C (x₁ ^ 3 + W.a₂ * x₁ ^ 2 + W.a₄ * x₁ + W.a₆)) -
      (C (C ((y₁ - (-y₁ - W.a₁ * x₁ - W.a₃)) ^ 2)) * 1 +
              C (C (1 * 4)) * (Y ^ 2 + C (C W.a₁ * X + C W.a₃) * Y - C (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆)) +
            C
              ((C (1 * 4) * X ^ 2 + C (1 * 4 * x₁ + 1 * (W.a₁ ^ 2 + 4 * W.a₂)) * X +
                  C (1 * 4 * x₁ ^ 2 + 1 * (W.a₁ ^ 2 + 4 * W.a₂) * x₁ + 1 * 2 * (2 * W.a₄ + W.a₁ * W.a₃))) *
                (X - C x₁)) +
          C (C ((y₁ - (-y₁ - W.a₁ * x₁ - W.a₃)) ^ 2)) * 0 * C (X - C x₁) +
        -4 * C (C (y₁ ^ 2 + W.a₁ * x₁ * y₁ + W.a₃ * y₁))) =
    0

Tactic:
C_simp

Result:
case a.a
F : Type u
inst✝ : Field F
W : Affine F
x₁ y₁ : F
h₁ : W.Equation x₁ y₁
sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c
h₂ : W.Equation x₁ y₁
hy : y₁ ≠ W.negY x₁ y₁
y : F := (y₁ - W.negY x₁ y₁) ^ 2
hxy : (y₁ - W.negY x₁ y₁) ^ 2 ≠ 0
⊢ 1 * (Y - (-Y - (C (C W.a₁) * C X + C (C W.a₃)))) * (Y - (-Y - (C (C W.a₁) * C X + C (C W.a₃)))) +
        -4 * (C (C x₁) ^ 3 + C (C W.a₂) * C (C x₁) ^ 2 + C (C W.a₄) * C (C x₁) + C (C W.a₆)) -
      ((C (C y₁) - (-C (C y₁) - C (C W.a₁) * C (C x₁) - C (C W.a₃))) ^ 2 * 1 +
              1 * 4 *
                (Y ^ 2 + (C (C W.a₁) * C X + C (C W.a₃)) * Y -
                  (C X ^ 3 + C (C W.a₂) * C X ^ 2 + C (C W.a₄) * C X + C (C W.a₆))) +
            (1 * 4 * C X ^ 2 + (1 * 4 * C (C x₁) + 1 * (C (C W.a₁) ^ 2 + 4 * C (C W.a₂))) * C X +
                (1 * 4 * C (C x₁) ^ 2 + 1 * (C (C W.a₁) ^ 2 + 4 * C (C W.a₂)) * C (C x₁) +
                  1 * 2 * (2 * C (C W.a₄) + C (C W.a₁) * C (C W.a₃)))) *
              (C X - C (C x₁)) +
          (C (C y₁) - (-C (C y₁) - C (C W.a₁) * C (C x₁) - C (C W.a₃))) ^ 2 * 0 * (C X - C (C x₁)) +
        -4 * (C (C y₁) ^ 2 + C (C W.a₁) * C (C x₁) * C (C y₁) + C (C W.a₃) * C (C y₁))) =
    0

Proof state:
case pos
F : Type u
inst✝ : Field F
W : Affine 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:
have hy : y₁ ≠ W.negY x₂ y₂ := fun h => hxy ⟨hx, h⟩

Result:
case pos
F : Type u
inst✝ : Field F
W : Affine 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
hy : y₁ ≠ W.negY x₂ y₂
⊢ ∃ 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

Proof state:
F : Type u
inst✝ : Field F
W : Affine 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
⊢ span
        {XClass W x₁ * XClass W x₂, XClass W x₁ * YClass W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)),
          YClass W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) * XClass W x₂,
          YClass W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) *
            YClass W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂))} *
      span {XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))} =
    YIdeal W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) *
      (span {(mk W) (W.negPolynomial - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)))} ⊔
        span {XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))})

Tactic:
simp_rw [span_insert, sup_rw, Ideal.sup_mul, span_singleton_mul_span_singleton]

Result:
F : Type u
inst✝ : Field F
W : Affine 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
⊢ span {XClass W x₁ * XClass W x₂ * XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))} ⊔
          span
            {YClass W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) *
                  YClass W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) *
                XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))} ⊔
        span
          {XClass W x₁ * YClass W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) *
              XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))} ⊔
      span
        {YClass W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) * XClass W x₂ *
            XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))} =
    YIdeal W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) *
      (span {(mk W) (W.negPolynomial - C (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)))} ⊔
        span {XClass W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂))})

Proof state:
case neg
F : Type u
inst✝ : Field F
W : Affine 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 ≠ 0
⊢ (C (C x₁) - C (C x₂)) * 0 * (Y - W.negPolynomial) =
    (C (C x₁) - C (C x₂)) * 1 + 1 * (C X - C (C x₁)) + 1 * -1 * (C X - C (C x₂))

Tactic:
ring1

Result:
no goals

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

Tactic:
let y := (y₁ - W.negY x₁ y₁) ^ 2

Result:
case pos
F : Type u
inst✝ : Field F
W : Affine F
x₁ y₁ : F
h₁ : W.Equation x₁ y₁
sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c
h₂ : W.Equation x₁ y₁
hxy : ¬(x₁ = x₁ ∧ y₁ = W.negY x₁ y₁)
hy : y₁ ≠ W.negY x₁ y₁
y : F := (y₁ - W.negY x₁ y₁) ^ 2
⊢ ∃ x,
    (∃ a a_1 a_2, a_2 * (Y - W.negPolynomial) = x + a * C (X - C x₁) + a_1 * C (X - C x₁)) ∧
      (AdjoinRoot.mk W.polynomial) x = 1