Module: Mathlib.LinearAlgebra.Reflection

Goal state before:
pp: case neg
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
a✝ :
  ∀ (n : ℕ),
    ((reflection hf * reflection hg) ^ ↑n) z =
      z +
          (Polynomial.eval t (S R ((↑n - 2) / 2)) *
              (Polynomial.eval t (S R ((↑n - 1) / 2)) + Polynomial.eval t (S R ((↑n - 3) / 2)))) •
            ((g x * f z - g z) • y - f z • x) +
        (Polynomial.eval t (S R ((↑n - 1) / 2)) *
            (Polynomial.eval t (S R (↑n / 2)) + Polynomial.eval t (S R ((↑n - 2) / 2)))) •
          ((f y * g z - f z) • x - g z • y)
m : ℕ
ht' : t = g x * f y - 2
aux : ∀ (a b : ℤ), autoParam (a + b = -3) reflection_mul_reflection_zpow_apply._auto_1_2 → a / 2 = -(b / 2) - 2
⊢ z +
        (Polynomial.eval t (S R ((↑m - 1) / 2)) *
            (Polynomial.eval t (S R (↑m / 2)) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (Polynomial.eval t (S R ((↑m - 2) / 2)) *
          (Polynomial.eval t (S R ((↑m - 1) / 2)) + Polynomial.eval t (S R ((↑m - 3) / 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 - 2) / 2) - 2)) + Polynomial.eval t (S R (-(↑m / 2) - 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (Polynomial.eval t (S R (-((↑m - 2) / 2) - 2)) *
          (Polynomial.eval t (S R (-((↑m - 3) / 2) - 2)) + Polynomial.eval t (S R (-((↑m - 1) / 2) - 2)))) •
        ((f y * g z - f z) • x - g z • y)

Tactic:
simp only [S_neg_sub_two, Polynomial.eval_neg]

Goal state after:
case neg
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
a✝ :
  ∀ (n : ℕ),
    ((reflection hf * reflection hg) ^ ↑n) z =
      z +
          (Polynomial.eval t (S R ((↑n - 2) / 2)) *
              (Polynomial.eval t (S R ((↑n - 1) / 2)) + Polynomial.eval t (S R ((↑n - 3) / 2)))) •
            ((g x * f z - g z) • y - f z • x) +
        (Polynomial.eval t (S R ((↑n - 1) / 2)) *
            (Polynomial.eval t (S R (↑n / 2)) + Polynomial.eval t (S R ((↑n - 2) / 2)))) •
          ((f y * g z - f z) • x - g z • y)
m : ℕ
ht' : t = g x * f y - 2
aux : ∀ (a b : ℤ), autoParam (a + b = -3) reflection_mul_reflection_zpow_apply._auto_1_2 → a / 2 = -(b / 2) - 2
⊢ z +
        (Polynomial.eval t (S R ((↑m - 1) / 2)) *
            (Polynomial.eval t (S R (↑m / 2)) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (Polynomial.eval t (S R ((↑m - 2) / 2)) *
          (Polynomial.eval t (S R ((↑m - 1) / 2)) + Polynomial.eval t (S R ((↑m - 3) / 2)))) •
        ((f y * g z - f z) • x - g z • y) =
    z +
        (-Polynomial.eval t (S R ((↑m - 1) / 2)) *
            (-Polynomial.eval t (S R ((↑m - 2) / 2)) + -Polynomial.eval t (S R (↑m / 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (-Polynomial.eval t (S R ((↑m - 2) / 2)) *
          (-Polynomial.eval t (S R ((↑m - 3) / 2)) + -Polynomial.eval t (S R ((↑m - 1) / 2)))) •
        ((f y * g z - f z) • x - g z • y)