-- Lean 4 proof step from Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange (goal)
Goal:
R : Type u
inst : CommRing R
W_1 : WeierstrassCurve R
C_1 C C' : VariableChange R
W : WeierstrassCurve R
⊢ { a₁ := (↑(C.u * C'.u)⁻¹ : R) * (W.a₁ + 2 * ((↑C'.u : R) * C.s + C'.s)),
        a₂ :=
          HPow.hPow (β := ℕ) (↑(C.u * C'.u)⁻¹ : R) 2 *
            (W.a₂ - ((↑C'.u : R) * C.s + C'.s) * W.a₁ + 3 * (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) -
              HPow.hPow (β := ℕ) ((↑C'.u : R) * C.s + C'.s) 2),
        a₃ :=
          HPow.hPow (β := ℕ) (↑(C.u * C'.u)⁻¹ : R) 3 *
            (W.a₃ + (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) * W.a₁ +
              2 * (C.t * HPow.hPow (β := ℕ) (↑C'.u : R) 3 + C.r * C'.s * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.t)),
        a₄ :=
          HPow.hPow (β := ℕ) (↑(C.u * C'.u)⁻¹ : R) 4 *
            (W.a₄ - ((↑C'.u : R) * C.s + C'.s) * W.a₃ + 2 * (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) * W.a₂ -
                  (C.t * HPow.hPow (β := ℕ) (↑C'.u : R) 3 + C.r * C'.s * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.t +
                      (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) * ((↑C'.u : R) * C.s + C'.s)) *
                    W.a₁ +
                3 * HPow.hPow (β := ℕ) (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) 2 -
              2 * ((↑C'.u : R) * C.s + C'.s) *
                (C.t * HPow.hPow (β := ℕ) (↑C'.u : R) 3 + C.r * C'.s * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.t)),
        a₆ :=
          HPow.hPow (β := ℕ) (↑(C.u * C'.u)⁻¹ : R) 6 *
            (W.a₆ + (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) * W.a₄ +
                      HPow.hPow (β := ℕ) (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) 2 * W.a₂ +
                    HPow.hPow (β := ℕ) (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) 3 -
                  (C.t * HPow.hPow (β := ℕ) (↑C'.u : R) 3 + C.r * C'.s * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.t) *
                    W.a₃ -
                HPow.hPow (β := ℕ)
                  (C.t * HPow.hPow (β := ℕ) (↑C'.u : R) 3 + C.r * C'.s * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.t) 2 -
              (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) *
                  (C.t * HPow.hPow (β := ℕ) (↑C'.u : R) 3 + C.r * C'.s * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.t) *
                W.a₁) :
        WeierstrassCurve R }.a₄ =
    { a₁ := (↑C.u⁻¹ : R) * ((↑C'.u⁻¹ : R) * (W.a₁ + 2 * C'.s) + 2 * C.s),
        a₂ :=
          HPow.hPow (β := ℕ) (↑C.u⁻¹ : R) 2 *
            (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 2 * (W.a₂ - C'.s * W.a₁ + 3 * C'.r - HPow.hPow (β := ℕ) C'.s 2) -
                  C.s * ((↑C'.u⁻¹ : R) * (W.a₁ + 2 * C'.s)) +
                3 * C.r -
              HPow.hPow (β := ℕ) C.s 2),
        a₃ :=
          HPow.hPow (β := ℕ) (↑C.u⁻¹ : R) 3 *
            (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 3 * (W.a₃ + C'.r * W.a₁ + 2 * C'.t) +
                C.r * ((↑C'.u⁻¹ : R) * (W.a₁ + 2 * C'.s)) +
              2 * C.t),
        a₄ :=
          HPow.hPow (β := ℕ) (↑C.u⁻¹ : R) 4 *
            (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 4 *
                        (W.a₄ - C'.s * W.a₃ + 2 * C'.r * W.a₂ - (C'.t + C'.r * C'.s) * W.a₁ +
                            3 * HPow.hPow (β := ℕ) C'.r 2 -
                          2 * C'.s * C'.t) -
                      C.s * (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 3 * (W.a₃ + C'.r * W.a₁ + 2 * C'.t)) +
                    2 * C.r *
                      (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 2 *
                        (W.a₂ - C'.s * W.a₁ + 3 * C'.r - HPow.hPow (β := ℕ) C'.s 2)) -
                  (C.t + C.r * C.s) * ((↑C'.u⁻¹ : R) * (W.a₁ + 2 * C'.s)) +
                3 * HPow.hPow (β := ℕ) C.r 2 -
              2 * C.s * C.t),
        a₆ :=
          HPow.hPow (β := ℕ) (↑C.u⁻¹ : R) 6 *
            (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 6 *
                          (W.a₆ + C'.r * W.a₄ + HPow.hPow (β := ℕ) C'.r 2 * W.a₂ + HPow.hPow (β := ℕ) C'.r 3 -
                                C'.t * W.a₃ -
                              HPow.hPow (β := ℕ) C'.t 2 -
                            C'.r * C'.t * W.a₁) +
                        C.r *
                          (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 4 *
                            (W.a₄ - C'.s * W.a₃ + 2 * C'.r * W.a₂ - (C'.t + C'.r * C'.s) * W.a₁ +
                                3 * HPow.hPow (β := ℕ) C'.r 2 -
                              2 * C'.s * C'.t)) +
                      HPow.hPow (β := ℕ) C.r 2 *
                        (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 2 *
                          (W.a₂ - C'.s * W.a₁ + 3 * C'.r - HPow.hPow (β := ℕ) C'.s 2)) +
                    HPow.hPow (β := ℕ) C.r 3 -
                  C.t * (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 3 * (W.a₃ + C'.r * W.a₁ + 2 * C'.t)) -
                HPow.hPow (β := ℕ) C.t 2 -
              C.r * C.t * ((↑C'.u⁻¹ : R) * (W.a₁ + 2 * C'.s))) }.a₄
have a₄ : HPow.hPow (β := ℕ) ((↑C.u⁻¹ : R) * (↑C'.u⁻¹ : R)) 4 *
    (W.a₄ - ((↑C'.u : R) * C.s + C'.s) * W.a₃ + 2 * (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) * W.a₂ -
          (C.t * HPow.hPow (β := ℕ) (↑C'.u : R) 3 + C.r * C'.s * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.t +
              (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) * ((↑C'.u : R) * C.s + C'.s)) *
            W.a₁ +
        3 * HPow.hPow (β := ℕ) (C.r * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.r) 2 -
      2 * ((↑C'.u : R) * C.s + C'.s) *
        (C.t * HPow.hPow (β := ℕ) (↑C'.u : R) 3 + C.r * C'.s * HPow.hPow (β := ℕ) (↑C'.u : R) 2 + C'.t)) =
  HPow.hPow (β := ℕ) (↑C.u⁻¹ : R) 4 *
    (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 4 *
                (W.a₄ - C'.s * W.a₃ + 2 * C'.r * W.a₂ - (C'.t + C'.r * C'.s) * W.a₁ + 3 * HPow.hPow (β := ℕ) C'.r 2 -
                  2 * C'.s * C'.t) -
              C.s * (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 3 * (W.a₃ + C'.r * W.a₁ + 2 * C'.t)) +
            2 * C.r *
              (HPow.hPow (β := ℕ) (↑C'.u⁻¹ : R) 2 * (W.a₂ - C'.s * W.a₁ + 3 * C'.r - HPow.hPow (β := ℕ) C'.s 2)) -
          (C.t + C.r * C.s) * ((↑C'.u⁻¹ : R) * (W.a₁ + 2 * C'.s)) +
        3 * HPow.hPow (β := ℕ) C.r 2 -
      2 * C.s * C.t)
Tactic: simp only [mul_inv, Units.val_mul]