-- Mathlib/LinearAlgebra/Reflection.lean
/-- A formula for $(r_1 r_2)^m z$, where $m$ is a natural number and $z \in M$. -/
lemma reflection_mul_reflection_pow_apply (m : ℕ) (z : M)
    (t : R := f y * g x - 2) (ht : t = f y * g x - 2

Proof state:
case a.a
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
z : M
t : optParam R (f y * g x - 2)
ht : autoParam (t = f y * g x - 2) _auto✝
m : ℕ
S_eval_t_sub_two :
  ∀ (k : ℤ), Polynomial.eval t (S R (k - 2)) = t * Polynomial.eval t (S R (k - 1)) - Polynomial.eval t (S R k)
S_eval_t_sq_add_S_eval_t_sq :
  ∀ (k : ℤ),
    Polynomial.eval t (S R k) ^ 2 + Polynomial.eval t (S R (k + 1)) ^ 2 -
        t * Polynomial.eval t (S R k) * Polynomial.eval t (S R (k + 1)) =
      1
k : ℤ := ↑m / 2
ih :
  ((reflection hf * reflection hg) ^ m) z =
    z +
        (Polynomial.eval t (S R ((↑m - 2) / 2)) *
            (Polynomial.eval t (S R ((↑m - 1) / 2)) + Polynomial.eval t (S R ((↑m - 3) / 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (Polynomial.eval t (S R ((↑m - 1) / 2)) * (Polynomial.eval t (S R k) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
        ((f y * g z - f z) • x - g z • y)
e : ℤ
he : e = 0 ∨ e = 1
⊢ -(f z * 1) - g z * -(f y * 1) +
            Polynomial.eval t (S R ((e - 2) / 2 + k)) *
                (Polynomial.eval t (S R ((e - 1) / 2 + k)) + Polynomial.eval t (S R ((e - 3) / 2 + k))) *
              ((g x * f z - g z) * (-(f y * 1) - 2 * -(f y * 1)) - f z * (1 - 2 * 1 - g x * -(f y * 1))) +
          Polynomial.eval t (S R ((e - 1) / 2 + k)) *
              (Polynomial.eval t (S R (e / 2 + k)) + Polynomial.eval t (S R ((e - 2) / 2 + k))) *
            ((f y * g z - f z) * (1 - 2 * 1 - g x * -(f y * 1)) - g z * (-(f y * 1) - 2 * -(f y * 1))) +
        ((-g z * f y * Polynomial.eval t (S R (e - 1 + k)) + f z * Polynomial.eval t (S R (e - 1 + k))) *
            (t * Polynomial.eval t (S R (e + k - 1)) - Polynomial.eval t (S R (e + k))) +
          (-g z * f y + f z) * 1) -
      (Polynomial.eval t (S R ((e + 1 - 2) / 2 + k)) *
              (Polynomial.eval t (S R ((e + 1 - 1) / 2 + k)) + Polynomial.eval t (S R ((e + 1 - 3) / 2 + k))) *
            -(f z * 1) +
          Polynomial.eval t (S R ((e + 1 - 1) / 2 + k)) *
              (Polynomial.eval t (S R ((e + 1) / 2 + k)) + Polynomial.eval t (S R ((e + 1 - 2) / 2 + k))) *
            ((f y * g z - f z) * 1) +
        ((-g z * f y * Polynomial.eval t (S R (e - 1 + k)) + f z * Polynomial.eval t (S R (e - 1 + k))) *
            Polynomial.eval t (S R (e + k - 2)) +
          (-g z * f y + f z) *
            (Polynomial.eval t (S R (k - 1)) ^ 2 + Polynomial.eval t (S R (k - 1 + 1)) ^ 2 -
              t * Polynomial.eval t (S R (k - 1)) * Polynomial.eval t (S R (k - 1 + 1))))) =
    0

Tactic:
subst ht

Result:
case a.a
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
z : M
m : ℕ
k : ℤ := ↑m / 2
e : ℤ
he : e = 0 ∨ e = 1
S_eval_t_sub_two :
  ∀ (k : ℤ),
    Polynomial.eval (f y * g x - 2) (S R (k - 2)) =
      (f y * g x - 2) * Polynomial.eval (f y * g x - 2) (S R (k - 1)) - Polynomial.eval (f y * g x - 2) (S R k)
S_eval_t_sq_add_S_eval_t_sq :
  ∀ (k : ℤ),
    Polynomial.eval (f y * g x - 2) (S R k) ^ 2 + Polynomial.eval (f y * g x - 2) (S R (k + 1)) ^ 2 -
        (f y * g x - 2) * Polynomial.eval (f y * g x - 2) (S R k) * Polynomial.eval (f y * g x - 2) (S R (k + 1)) =
      1
ih :
  ((reflection hf * reflection hg) ^ m) z =
    z +
        (Polynomial.eval (f y * g x - 2) (S R ((↑m - 2) / 2)) *
            (Polynomial.eval (f y * g x - 2) (S R ((↑m - 1) / 2)) +
              Polynomial.eval (f y * g x - 2) (S R ((↑m - 3) / 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (Polynomial.eval (f y * g x - 2) (S R ((↑m - 1) / 2)) *
          (Polynomial.eval (f y * g x - 2) (S R k) + Polynomial.eval (f y * g x - 2) (S R ((↑m - 2) / 2)))) •
        ((f y * g z - f z) • x - g z • y)
⊢ -(f z * 1) - g z * -(f y * 1) +
            Polynomial.eval (f y * g x - 2) (S R ((e - 2) / 2 + k)) *
                (Polynomial.eval (f y * g x - 2) (S R ((e - 1) / 2 + k)) +
                  Polynomial.eval (f y * g x - 2) (S R ((e - 3) / 2 + k))) *
              ((g x * f z - g z) * (-(f y * 1) - 2 * -(f y * 1)) - f z * (1 - 2 * 1 - g x * -(f y * 1))) +
          Polynomial.eval (f y * g x - 2) (S R ((e - 1) / 2 + k)) *
              (Polynomial.eval (f y * g x - 2) (S R (e / 2 + k)) +
                Polynomial.eval (f y * g x - 2) (S R ((e - 2) / 2 + k))) *
            ((f y * g z - f z) * (1 - 2 * 1 - g x * -(f y * 1)) - g z * (-(f y * 1) - 2 * -(f y * 1))) +
        ((-g z * f y * Polynomial.eval (f y * g x - 2) (S R (e - 1 + k)) +
              f z * Polynomial.eval (f y * g x - 2) (S R (e - 1 + k))) *
            ((f y * g x - 2) * Polynomial.eval (f y * g x - 2) (S R (e + k - 1)) -
              Polynomial.eval (f y * g x - 2) (S R (e + k))) +
          (-g z * f y + f z) * 1) -
      (Polynomial.eval (f y * g x - 2) (S R ((e + 1 - 2) / 2 + k)) *
              (Polynomial.eval (f y * g x - 2) (S R ((e + 1 - 1) / 2 + k)) +
                Polynomial.eval (f y * g x - 2) (S R ((e + 1 - 3) / 2 + k))) *
            -(f z * 1) +
          Polynomial.eval (f y * g x - 2) (S R ((e + 1 - 1) / 2 + k)) *
              (Polynomial.eval (f y * g x - 2) (S R ((e + 1) / 2 + k)) +
                Polynomial.eval (f y * g x - 2) (S R ((e + 1 - 2) / 2 + k))) *
            ((f y * g z - f z) * 1) +
        ((-g z * f y * Polynomial.eval (f y * g x - 2) (S R (e - 1 + k)) +
              f z * Polynomial.eval (f y * g x - 2) (S R (e - 1 + k))) *
            Polynomial.eval (f y * g x - 2) (S R (e + k - 2)) +
          (-g z * f y + f z) *
            (Polynomial.eval (f y * g x - 2) (S R (k - 1)) ^ 2 + Polynomial.eval (f y * g x - 2) (S R (k - 1 + 1)) ^ 2 -
              (f y * g x - 2) * Polynomial.eval (f y * g x - 2) (S R (k - 1)) *
                Polynomial.eval (f y * g x - 2) (S R (k - 1 + 1))))) =
    0