(** 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.
(** ** Constants and usual functions *)
(** 0 and 1 for complex are defined as [RtoC 0] and [RtoC 1] *)
Definition Ci : C := (0,1).
(** *** Arithmetic operations *)
Definition Cplus (x y : C) : C := (fst x + fst y, snd x + snd y).
Definition Copp (x : C) : C := (-fst x, -snd x).
Definition Cminus (x y : C) : C := Cplus x (Copp y).
Definition Cmult (x y : C) : C := (fst x * fst y - snd x * snd y, fst x * snd y + snd x * fst y).
Definition Cinv (x : C) : C := (fst x / (fst x ^ 2 + snd x ^ 2), - snd x / (fst x ^ 2 + snd x ^ 2)).
Definition Cdiv (x y : C) : C := Cmult x (Cinv y).
Delimit Scope C_scope with C.
Local Open Scope C_scope.
Infix "+" := Cplus : C_scope.
Notation "- x" := (Copp x) : C_scope.
Infix "-" := Cminus : C_scope.
Infix "*" := Cmult : C_scope.
Notation "/ x" := (Cinv x) : C_scope.
Infix "/" := Cdiv : C_scope.
(** *** Other usual functions *)
Definition Re (z : C) : R := fst z.
Definition Im (z : C) : R := snd z.
Definition Cmod (x : C) : R := sqrt (fst x ^ 2 + snd x ^ 2).
Definition Cconj (x : C) : C := (fst x, (- snd x)%R).
(** ** C is a field *)
Definition C_AbelianGroup_mixin :=
AbelianGroup.Mixin _ _ _ _ Cplus_comm Cplus_assoc Cplus_0_r Cplus_opp_r.
Canonical C_AbelianGroup :=
AbelianGroup.Pack C C_AbelianGroup_mixin C.
Definition C_Ring_mixin :=
Ring.Mixin _ _ _ Cmult_assoc Cmult_1_r Cmult_1_l Cmult_plus_distr_r Cmult_plus_distr_l.
Canonical C_Ring :=
Ring.Pack C (Ring.Class _ _ C_Ring_mixin) C.
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.
Add Field C_field_field : C_field_theory.
(** * C is a NormedModule *)
Canonical C_UniformSpace :=
UniformSpace.Pack C (UniformSpace.class (prod_UniformSpace _ _)) C.
(** on C (with the balls of R^2) *)
Canonical C_ModuleSpace :=
ModuleSpace.Pack C_Ring C (ModuleSpace.class _ (Ring_ModuleSpace C_Ring)) C.
Canonical C_NormedModuleAux :=
NormedModuleAux.Pack C_AbsRing C (NormedModuleAux.Class C_AbsRing _ (ModuleSpace.class _ C_ModuleSpace) (UniformSpace.class C_UniformSpace)) C.
Definition C_NormedModule_mixin :=
NormedModule.Mixin _ C_NormedModuleAux _ _ Cmod_triangle (fun x y => Req_le _ _ (Cmod_mult x y))
C_NormedModule_mixin_compat1 C_NormedModule_mixin_compat2 Cmod_eq_0.
Canonical C_NormedModule :=
NormedModule.Pack C_AbsRing C (NormedModule.Class _ _ _ C_NormedModule_mixin) C.
(** on R *)
Canonical C_R_ModuleSpace :=
ModuleSpace.Pack R_Ring C (ModuleSpace.class _ (prod_ModuleSpace R_Ring R_ModuleSpace R_ModuleSpace)) C.
Canonical C_R_NormedModuleAux :=
NormedModuleAux.Pack R_AbsRing C (NormedModuleAux.Class R_AbsRing _ (ModuleSpace.class _ C_R_ModuleSpace) (UniformSpace.class _)) C.
Canonical C_R_NormedModule :=
NormedModule.Pack R_AbsRing C (NormedModule.class _ (prod_NormedModule _ _ _)) C.
(** * C is a CompleteSpace *)
Definition C_complete_lim (F : (C -> Prop) -> Prop) :=
(R_complete_lim (fun P => F (fun z => P (Re z))), R_complete_lim (fun P => F (fun z => P (Im z)))).

Lemma C_complete_ax1 :
forall F : (C -> Prop) -> Prop,
ProperFilter F ->
(forall eps : posreal, exists x : C, F (ball x eps)) ->
forall eps : posreal, F (ball (C_complete_lim F) eps).