Definition C_AbsRing_mixin :=
AbsRing.Mixin _ _ Cmod_0 Cmod_m1 Cmod_triangle (fun x y => Req_le _ _ (Cmod_mult x y)) Cmod_eq_0.
Canonical C_AbsRing :=
AbsRing.Pack C (AbsRing.Class _ _ C_AbsRing_mixin) C.

(* Specifications: *)
Lemma Cmod_ge_0 :
forall x, 0 <= Cmod x.
Lemma Cmod_gt_0 :
forall (x : C), x <> 0 <-> 0 < Cmod x.
Lemma Cmod_norm :
forall x : C, Cmod x = (@norm R_AbsRing _ x).
Lemma Cmod_R :
forall x : R, Cmod x = Rabs x.
Lemma Cmod_inv :
forall x : C, x <> 0 -> Cmod (/ x) = Rinv (Cmod x).
Lemma Cmod_div (x y : C) : y <> 0 ->
Cmod (x / y) = Rdiv (Cmod x) (Cmod y).
Lemma Cmult_neq_0 (z1 z2 : C) : z1 <> 0 -> z2 <> 0 -> z1 * z2 <> 0.
Lemma Cminus_eq_contra : forall r1 r2 : C, r1 <> r2 -> r1 - r2 <> 0.
Lemma C_field_theory : field_theory (RtoC 0) (RtoC 1) Cplus Cmult Cminus Copp Cdiv Cinv eq.
Add Field C_field_field : C_field_theory.