--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/complex/CComplex".
+
+include "CoRN.ma".
+
+(* $Id: CComplex.v,v 1.8 2004/04/23 10:00:55 lcf Exp $ *)
+
+(*#* printing Re %\ensuremath{\Re}% #ℜ# *)
+
+(*#* printing Im %\ensuremath{\Im}% #ℑ# *)
+
+(*#* printing CC %\ensuremath{\mathbb C}% #<b>C</b># *)
+
+(*#* printing II %\ensuremath{\imath}% #i# *)
+
+(*#* printing [+I*] %\ensuremath{+\imath}% *)
+
+(*#* printing AbsCC %\ensuremath{|\cdot|_{\mathbb C}}% *)
+
+(*#* printing CCX %\ensuremath{\mathbb C[X]}% #<b>C</b>[X]# *)
+
+include "reals/NRootIR.ma".
+
+(*#* * Complex Numbers
+** Algebraic structure
+*)
+
+(* UNEXPORTED
+Section Complex_Numbers
+*)
+
+inline "cic:/CoRN/complex/CComplex/CC_set.ind".
+
+inline "cic:/CoRN/complex/CComplex/cc_ap.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_eq.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_is_CSetoid.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_csetoid.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_plus.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_mult.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_zero.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_one.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_i.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_inv.con".
+
+(* not needed anymore
+Lemma cc_plus_op_proof : (bin_op_wd cc_csetoid cc_plus).
+Unfold bin_op_wd. Unfold bin_fun_wd.
+Intros x1 x2 y1 y2. Elim x1. Elim x2. Elim y1. Elim y2.
+Simpl. Unfold cc_eq. Simpl. Intros.
+Elim H. Clear H. Intros. Elim H0. Clear H0. Intros.
+Split; Algebra.
+Qed.
+
+Lemma cc_mult_op_proof : (bin_op_wd cc_csetoid cc_mult).
+Unfold bin_op_wd. Unfold bin_fun_wd.
+Intros x1 x2 y1 y2. Elim x1. Elim x2. Elim y1. Elim y2.
+Simpl. Unfold cc_eq. Simpl. Intros.
+Elim H. Clear H. Intros. Elim H0. Clear H0. Intros.
+Split; Algebra.
+Qed.
+
+Lemma cc_inv_op_proof : (un_op_wd cc_csetoid cc_inv).
+Unfold un_op_wd. Unfold fun_wd.
+Intros x y. Elim x. Elim y.
+Simpl. Unfold cc_eq. Simpl. Intros.
+Elim H. Clear H. Intros.
+Split; Algebra.
+Qed.
+*)
+
+inline "cic:/CoRN/complex/CComplex/cc_inv_strext.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_plus_strext.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_mult_strext.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_inv_op.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_plus_op.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_mult_op.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_csg_associative.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_cr_mult_associative.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_csemi_grp.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_cm_proof.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_cmonoid.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_cg_proof.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_cr_dist.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_cr_non_triv.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_cgroup.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_cabgroup.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_cr_mult_mon.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_mult_commutes.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_isCRing.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_cring.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_ap_zero.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_inv_aid.con".
+
+(*#*
+If [x [~=] Zero] or [y [~=] Zero], then [x [/] x[^]2 [+] y[^]2 [~=] Zero] or
+[[--]y[/]x[^]2[+]y[^]2 [~=] Zero].
+*)
+
+inline "cic:/CoRN/complex/CComplex/cc_inv_aid2.con".
+
+(*
+REMARK KEPT FOR SENTIMENTAL REASONS...
+
+This definition seems clever. Even though we *cannot* construct an
+element of (NonZeros cc_cring) (a Set) by deciding which part of the
+input (Re or Im) is NonZero (a Prop), we manage to construct the
+actual function.
+*)
+
+inline "cic:/CoRN/complex/CComplex/cc_recip.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_cfield_proof.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_Recip_proof.con".
+
+(* UNEXPORTED
+Opaque cc_recip.
+*)
+
+(* UNEXPORTED
+Opaque cc_inv.
+*)
+
+inline "cic:/CoRN/complex/CComplex/cc_cfield.con".
+
+inline "cic:/CoRN/complex/CComplex/CC.con".
+
+(*#*
+Maps from reals to complex and vice-versa are defined, as well as conjugate,
+absolute value and the imaginary unit [I] *)
+
+inline "cic:/CoRN/complex/CComplex/cc_set_CC.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_IR.con".
+
+inline "cic:/CoRN/complex/CComplex/CC_conj.con".
+
+(* old def
+Definition CC_conj' : CC->CC := [z:CC_set] (CC_set_rec [_:CC_set]CC_set [Re0,Im0:IR] (Build_CC_set Re0 [--]Im0) z).
+*)
+
+inline "cic:/CoRN/complex/CComplex/AbsCC.con".
+
+inline "cic:/CoRN/complex/CComplex/TwoCC_ap_zero.con".
+
+(* UNEXPORTED
+End Complex_Numbers
+*)
+
+(* begin hide *)
+
+(* NOTATION
+Notation CCX := (cpoly_cring CC).
+*)
+
+(* end hide *)
+
+inline "cic:/CoRN/complex/CComplex/II.con".
+
+(* NOTATION
+Infix "[+I*]" := cc_set_CC (at level 48, no associativity).
+*)
+
+(*#* ** Properties of [II] *)
+
+(* UNEXPORTED
+Section I_properties
+*)
+
+inline "cic:/CoRN/complex/CComplex/I_square.con".
+
+(* UNEXPORTED
+Hint Resolve I_square: algebra.
+*)
+
+inline "cic:/CoRN/complex/CComplex/I_square'.con".
+
+inline "cic:/CoRN/complex/CComplex/I_recip_lft.con".
+
+inline "cic:/CoRN/complex/CComplex/I_recip_rht.con".
+
+inline "cic:/CoRN/complex/CComplex/mult_I.con".
+
+inline "cic:/CoRN/complex/CComplex/I_wd.con".
+
+(*#* ** Properties of [Re] and [Im] *)
+
+inline "cic:/CoRN/complex/CComplex/calculate_norm.con".
+
+inline "cic:/CoRN/complex/CComplex/calculate_Re.con".
+
+inline "cic:/CoRN/complex/CComplex/calculate_Im.con".
+
+inline "cic:/CoRN/complex/CComplex/Re_wd.con".
+
+inline "cic:/CoRN/complex/CComplex/Im_wd.con".
+
+inline "cic:/CoRN/complex/CComplex/Re_resp_plus.con".
+
+inline "cic:/CoRN/complex/CComplex/Re_resp_inv.con".
+
+inline "cic:/CoRN/complex/CComplex/Im_resp_plus.con".
+
+inline "cic:/CoRN/complex/CComplex/Im_resp_inv.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_calculate_square.con".
+
+(* UNEXPORTED
+End I_properties
+*)
+
+(* UNEXPORTED
+Hint Resolve I_square I_square' I_recip_lft I_recip_rht mult_I calculate_norm
+ cc_calculate_square: algebra.
+*)
+
+(* UNEXPORTED
+Hint Resolve I_wd Re_wd Im_wd: algebra_c.
+*)
+
+(*#* ** Properties of conjugation *)
+
+(* UNEXPORTED
+Section Conj_properties
+*)
+
+inline "cic:/CoRN/complex/CComplex/CC_conj_plus.con".
+
+inline "cic:/CoRN/complex/CComplex/CC_conj_mult.con".
+
+(* UNEXPORTED
+Hint Resolve CC_conj_mult: algebra.
+*)
+
+inline "cic:/CoRN/complex/CComplex/CC_conj_strext.con".
+
+inline "cic:/CoRN/complex/CComplex/CC_conj_conj.con".
+
+inline "cic:/CoRN/complex/CComplex/CC_conj_zero.con".
+
+inline "cic:/CoRN/complex/CComplex/CC_conj_one.con".
+
+(* UNEXPORTED
+Hint Resolve CC_conj_one: algebra.
+*)
+
+inline "cic:/CoRN/complex/CComplex/CC_conj_nexp.con".
+
+(* UNEXPORTED
+End Conj_properties
+*)
+
+(* UNEXPORTED
+Hint Resolve CC_conj_plus CC_conj_mult CC_conj_nexp CC_conj_conj
+ CC_conj_zero: algebra.
+*)
+
+(*#* ** Properties of the real axis *)
+
+(* UNEXPORTED
+Section cc_IR_properties
+*)
+
+inline "cic:/CoRN/complex/CComplex/Re_cc_IR.con".
+
+inline "cic:/CoRN/complex/CComplex/Im_cc_IR.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_IR_wd.con".
+
+(* UNEXPORTED
+Hint Resolve cc_IR_wd: algebra_c.
+*)
+
+inline "cic:/CoRN/complex/CComplex/cc_IR_resp_ap.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_IR_mult.con".
+
+(* UNEXPORTED
+Hint Resolve cc_IR_mult: algebra.
+*)
+
+inline "cic:/CoRN/complex/CComplex/cc_IR_mult_lft.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_IR_mult_rht.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_IR_plus.con".
+
+(* UNEXPORTED
+Hint Resolve cc_IR_plus: algebra.
+*)
+
+inline "cic:/CoRN/complex/CComplex/cc_IR_minus.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_IR_zero.con".
+
+(* UNEXPORTED
+Hint Resolve cc_IR_zero: algebra.
+*)
+
+inline "cic:/CoRN/complex/CComplex/cc_IR_one.con".
+
+(* UNEXPORTED
+Hint Resolve cc_IR_one: algebra.
+*)
+
+inline "cic:/CoRN/complex/CComplex/cc_IR_nring.con".
+
+inline "cic:/CoRN/complex/CComplex/cc_IR_nexp.con".
+
+(* UNEXPORTED
+End cc_IR_properties
+*)
+
+(* UNEXPORTED
+Hint Resolve Re_cc_IR Im_cc_IR: algebra.
+*)
+
+(* UNEXPORTED
+Hint Resolve cc_IR_wd: algebra_c.
+*)
+
+(* UNEXPORTED
+Hint Resolve cc_IR_mult cc_IR_nexp cc_IR_mult_lft cc_IR_mult_rht cc_IR_plus
+ cc_IR_minus: algebra.
+*)
+
+(* UNEXPORTED
+Hint Resolve cc_IR_nring cc_IR_zero: algebra.
+*)
+
+(*#* ** [CC] has characteristic zero *)
+
+include "tactics/Transparent_algebra.ma".
+
+inline "cic:/CoRN/complex/CComplex/char0_CC.con".
+
+include "tactics/Opaque_algebra.ma".
+
+inline "cic:/CoRN/complex/CComplex/poly_apzero_CC.con".
+
+inline "cic:/CoRN/complex/CComplex/poly_CC_extensional.con".
+
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