(** This file is part of the Coquelicot formalization of real analysis in Coq: http://coquelicot.saclay.inria.fr/  Copyright (C) 2011-2015 Sylvie Boldo #<br /># Copyright (C) 2011-2015 Catherine Lelay #<br /># Copyright (C) 2011-2015 Guillaume Melquiond  This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.  This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the COPYING file for more details. *)
(** This file defines complex numbers [C] as [R * R]. Operations are given, and [C] is proved to be a field, a normed module, and a complete space. *)
(** * The set of complex numbers *)
Definition C := (R * R)%type.
Definition RtoC (x : R) : C := (x,0).
Coercion RtoC : R >-> C.

(* Specifications: *)
Lemma RtoC_inj : forall (x y : R),
RtoC x = RtoC y -> x = y.
Lemma Ceq_dec (z1 z2 : C) : { z1 = z2 } + { z1 <> z2 }.
Definition Ci : C := (0,1).