Module: Mathlib.LinearAlgebra.Reflection

Goal state before:
pp: case succ
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 : R
ht : t = f y * g x - 2
m : ℕ
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 (↑m / 2)) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
        ((f y * g z - f z) • x - g z • y)
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
⊢ (reflection hf * reflection hg)
      (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 (↑m / 2)) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
          ((f y * g z - f z) • x - g z • y)) =
    z +
        (Polynomial.eval t (S R ((↑(m + 1) - 2) / 2)) *
            (Polynomial.eval t (S R ((↑(m + 1) - 1) / 2)) + Polynomial.eval t (S R ((↑(m + 1) - 3) / 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (Polynomial.eval t (S R ((↑(m + 1) - 1) / 2)) *
          (Polynomial.eval t (S R (↑(m + 1) / 2)) + Polynomial.eval t (S R ((↑(m + 1) - 2) / 2)))) •
        ((f y * g z - f z) • x - g z • y)

Tactic:
LinearEquiv.mul_apply

Goal state after:
case succ
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 : R
ht : t = f y * g x - 2
m : ℕ
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 (↑m / 2)) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
        ((f y * g z - f z) • x - g z • y)
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
⊢ (reflection hf)
      ((reflection hg)
        (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 (↑m / 2)) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
            ((f y * g z - f z) • x - g z • y))) =
    z +
        (Polynomial.eval t (S R ((↑(m + 1) - 2) / 2)) *
            (Polynomial.eval t (S R ((↑(m + 1) - 1) / 2)) + Polynomial.eval t (S R ((↑(m + 1) - 3) / 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (Polynomial.eval t (S R ((↑(m + 1) - 1) / 2)) *
          (Polynomial.eval t (S R (↑(m + 1) / 2)) + Polynomial.eval t (S R ((↑(m + 1) - 2) / 2)))) •
        ((f y * g z - f z) • x - g z • y)