Definition id_phi_N (x:N) : N := x.

(* Specifications: *)
Lemma pow_N_pow_N : forall x n, pow_N x (id_phi_N n) == pow_N x n.
Lemma ring_mul_0_l : forall x, 0 * x == 0.
Lemma ring_mul_0_r : forall x, x * 0 == 0.
Lemma ring_opp_mul_l : forall x y, -(x * y) == -x * y.
Lemma ring_opp_mul_r : forall x y, -(x * y) == x * -y.
Lemma ring_opp_add : forall x y, -(x + y) == -x + -y.
Lemma ring_opp_opp : forall x, - -x == x.
Lemma ring_sub_ext : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 - y1 == x2 - y2.