. rw [weierstrass]
  apply nzero_mul_left_cancel (e.c4 ^ 3) _ _ (pow_ne_zero _ hc4)
  rw [mul_zero]
  simp only [div_eq_mul_inv]
  rw [show c4 e ^ 3 * (((b2 e * b5 e + 3 * b7 e) * (c4 e)⁻¹) ^ 2 +
        e.a1 * ((18 * b6 e - b2 e * b4 e) * (c4 e)⁻¹) * ((b2 e * b5 e + 3 * b7 e) * (c4 e)⁻¹) +
      e.a3 * ((b2 e * b5 e + 3 * b7 e) * (c4 e)⁻¹) -
      (((18 * b6 e - b2 e * b4 e) * (c4 e)⁻¹) ^ 3 + e.a2 * ((18 * b6 e - b2 e * b4 e) * (c4 e)⁻¹) ^ 2 +
      e.a4 * ((18 * b6 e - b2 e * b4 e) * (c4 e)⁻¹) + e.a6)) =
    (c4 e * (c4 e)⁻¹ * c4 e * (c4 e)⁻¹ * c4 e * ((b2 e * b5 e + 3 * b7 e)) ^ 2 +
      c4 e * (c4 e)⁻¹ * c4 e * (c4 e)⁻¹ * c4 e * e.a1 * ((18 * b6 e - b2 e * b4 e)) * ((b2 e * b5 e + 3 * b7 e)) +
    c4 e * (c4 e)⁻¹ * c4 e * c4 e * e.a3 * ((b2 e * b5 e + 3 * b7 e)) -
    (c4 e * (c4 e)⁻¹ * c4 e * (c4 e)⁻¹ * c4 e * (c4 e)⁻¹ * ((18 * b6 e - b2 e * b4 e)) ^ 3 +
    c4 e * (c4 e)⁻¹ * c4 e * (c4 e)⁻¹ * c4 e * e.a2 * ((18 * b6 e - b2 e * b4 e)) ^ 2 +
      c4 e * (c4 e)⁻¹ * c4 e * c4 e * e.a4 * ((18 * b6 e - b2 e * b4 e)) +
    c4 e * c4 e * c4 e * e.a6)) by ring]
  simp only [mul_inv_cancel hc4, one_mul]
  rw [b5, b7, c4, b2, b4, b6]
  rw [← mul_zero (e.a1^6 + 12*e.a1^4*e.a2 + 48*e.a1^2*e.a2^2 - 36*e.a1^3*e.a3 + 64*e.a2^3
    - 144*e.a1*e.a2*e.a3 - 72*e.a1^2*e.a4 + 216*e.a3^2 - 288*e.a2*e.a4 + 864*e.a6),
    ← h, discr_eq_neg_singular]
  ring
case neg.refine_1
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ weierstrass e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0

case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dx e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0

case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dy e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0
case neg.refine_2
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dx e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0

case neg.refine_3
K : Type u
inst✝¹ : Field K
inst✝ : ECTate.PerfectRing K
e : Model K
h : discr e = 0
hc4 : ¬c4 e = 0
⊢ dweierstrass_dy e ((18 * b6 e - b2 e * b4 e) / c4 e, (b2 e * b5 e + 3 * b7 e) / c4 e) = 0