1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/complex/CComplex".
21 (* $Id: CComplex.v,v 1.8 2004/04/23 10:00:55 lcf Exp $ *)
23 (*#* printing Re %\ensuremath{\Re}% #ℜ# *)
25 (*#* printing Im %\ensuremath{\Im}% #ℑ# *)
27 (*#* printing CC %\ensuremath{\mathbb C}% #<b>C</b># *)
29 (*#* printing II %\ensuremath{\imath}% #i# *)
31 (*#* printing [+I*] %\ensuremath{+\imath}% *)
33 (*#* printing AbsCC %\ensuremath{|\cdot|_{\mathbb C}}% *)
35 (*#* printing CCX %\ensuremath{\mathbb C[X]}% #<b>C</b>[X]# *)
37 include "reals/NRootIR.ma".
39 (*#* * Complex Numbers
40 ** Algebraic structure
44 Section Complex_Numbers
47 inline "cic:/CoRN/complex/CComplex/CC_set.ind".
49 inline "cic:/CoRN/complex/CComplex/cc_ap.con".
51 inline "cic:/CoRN/complex/CComplex/cc_eq.con".
53 inline "cic:/CoRN/complex/CComplex/cc_is_CSetoid.con".
55 inline "cic:/CoRN/complex/CComplex/cc_csetoid.con".
57 inline "cic:/CoRN/complex/CComplex/cc_plus.con".
59 inline "cic:/CoRN/complex/CComplex/cc_mult.con".
61 inline "cic:/CoRN/complex/CComplex/cc_zero.con".
63 inline "cic:/CoRN/complex/CComplex/cc_one.con".
65 inline "cic:/CoRN/complex/CComplex/cc_i.con".
67 inline "cic:/CoRN/complex/CComplex/cc_inv.con".
70 Lemma cc_plus_op_proof : (bin_op_wd cc_csetoid cc_plus).
71 Unfold bin_op_wd. Unfold bin_fun_wd.
72 Intros x1 x2 y1 y2. Elim x1. Elim x2. Elim y1. Elim y2.
73 Simpl. Unfold cc_eq. Simpl. Intros.
74 Elim H. Clear H. Intros. Elim H0. Clear H0. Intros.
78 Lemma cc_mult_op_proof : (bin_op_wd cc_csetoid cc_mult).
79 Unfold bin_op_wd. Unfold bin_fun_wd.
80 Intros x1 x2 y1 y2. Elim x1. Elim x2. Elim y1. Elim y2.
81 Simpl. Unfold cc_eq. Simpl. Intros.
82 Elim H. Clear H. Intros. Elim H0. Clear H0. Intros.
86 Lemma cc_inv_op_proof : (un_op_wd cc_csetoid cc_inv).
87 Unfold un_op_wd. Unfold fun_wd.
88 Intros x y. Elim x. Elim y.
89 Simpl. Unfold cc_eq. Simpl. Intros.
90 Elim H. Clear H. Intros.
95 inline "cic:/CoRN/complex/CComplex/cc_inv_strext.con".
97 inline "cic:/CoRN/complex/CComplex/cc_plus_strext.con".
99 inline "cic:/CoRN/complex/CComplex/cc_mult_strext.con".
101 inline "cic:/CoRN/complex/CComplex/cc_inv_op.con".
103 inline "cic:/CoRN/complex/CComplex/cc_plus_op.con".
105 inline "cic:/CoRN/complex/CComplex/cc_mult_op.con".
107 inline "cic:/CoRN/complex/CComplex/cc_csg_associative.con".
109 inline "cic:/CoRN/complex/CComplex/cc_cr_mult_associative.con".
111 inline "cic:/CoRN/complex/CComplex/cc_csemi_grp.con".
113 inline "cic:/CoRN/complex/CComplex/cc_cm_proof.con".
115 inline "cic:/CoRN/complex/CComplex/cc_cmonoid.con".
117 inline "cic:/CoRN/complex/CComplex/cc_cg_proof.con".
119 inline "cic:/CoRN/complex/CComplex/cc_cr_dist.con".
121 inline "cic:/CoRN/complex/CComplex/cc_cr_non_triv.con".
123 inline "cic:/CoRN/complex/CComplex/cc_cgroup.con".
125 inline "cic:/CoRN/complex/CComplex/cc_cabgroup.con".
127 inline "cic:/CoRN/complex/CComplex/cc_cr_mult_mon.con".
129 inline "cic:/CoRN/complex/CComplex/cc_mult_commutes.con".
131 inline "cic:/CoRN/complex/CComplex/cc_isCRing.con".
133 inline "cic:/CoRN/complex/CComplex/cc_cring.con".
135 inline "cic:/CoRN/complex/CComplex/cc_ap_zero.con".
137 inline "cic:/CoRN/complex/CComplex/cc_inv_aid.con".
140 If [x [~=] Zero] or [y [~=] Zero], then [x [/] x[^]2 [+] y[^]2 [~=] Zero] or
141 [[--]y[/]x[^]2[+]y[^]2 [~=] Zero].
144 inline "cic:/CoRN/complex/CComplex/cc_inv_aid2.con".
147 REMARK KEPT FOR SENTIMENTAL REASONS...
149 This definition seems clever. Even though we *cannot* construct an
150 element of (NonZeros cc_cring) (a Set) by deciding which part of the
151 input (Re or Im) is NonZero (a Prop), we manage to construct the
155 inline "cic:/CoRN/complex/CComplex/cc_recip.con".
157 inline "cic:/CoRN/complex/CComplex/cc_cfield_proof.con".
159 inline "cic:/CoRN/complex/CComplex/cc_Recip_proof.con".
169 inline "cic:/CoRN/complex/CComplex/cc_cfield.con".
171 inline "cic:/CoRN/complex/CComplex/CC.con".
174 Maps from reals to complex and vice-versa are defined, as well as conjugate,
175 absolute value and the imaginary unit [I] *)
177 inline "cic:/CoRN/complex/CComplex/cc_set_CC.con".
179 inline "cic:/CoRN/complex/CComplex/cc_IR.con".
181 inline "cic:/CoRN/complex/CComplex/CC_conj.con".
184 Definition CC_conj' : CC->CC := [z:CC_set] (CC_set_rec [_:CC_set]CC_set [Re0,Im0:IR] (Build_CC_set Re0 [--]Im0) z).
187 inline "cic:/CoRN/complex/CComplex/AbsCC.con".
189 inline "cic:/CoRN/complex/CComplex/TwoCC_ap_zero.con".
198 Notation CCX := (cpoly_cring CC).
203 inline "cic:/CoRN/complex/CComplex/II.con".
206 Infix "[+I*]" := cc_set_CC (at level 48, no associativity).
209 (*#* ** Properties of [II] *)
215 inline "cic:/CoRN/complex/CComplex/I_square.con".
218 Hint Resolve I_square: algebra.
221 inline "cic:/CoRN/complex/CComplex/I_square'.con".
223 inline "cic:/CoRN/complex/CComplex/I_recip_lft.con".
225 inline "cic:/CoRN/complex/CComplex/I_recip_rht.con".
227 inline "cic:/CoRN/complex/CComplex/mult_I.con".
229 inline "cic:/CoRN/complex/CComplex/I_wd.con".
231 (*#* ** Properties of [Re] and [Im] *)
233 inline "cic:/CoRN/complex/CComplex/calculate_norm.con".
235 inline "cic:/CoRN/complex/CComplex/calculate_Re.con".
237 inline "cic:/CoRN/complex/CComplex/calculate_Im.con".
239 inline "cic:/CoRN/complex/CComplex/Re_wd.con".
241 inline "cic:/CoRN/complex/CComplex/Im_wd.con".
243 inline "cic:/CoRN/complex/CComplex/Re_resp_plus.con".
245 inline "cic:/CoRN/complex/CComplex/Re_resp_inv.con".
247 inline "cic:/CoRN/complex/CComplex/Im_resp_plus.con".
249 inline "cic:/CoRN/complex/CComplex/Im_resp_inv.con".
251 inline "cic:/CoRN/complex/CComplex/cc_calculate_square.con".
258 Hint Resolve I_square I_square' I_recip_lft I_recip_rht mult_I calculate_norm
259 cc_calculate_square: algebra.
263 Hint Resolve I_wd Re_wd Im_wd: algebra_c.
266 (*#* ** Properties of conjugation *)
269 Section Conj_properties
272 inline "cic:/CoRN/complex/CComplex/CC_conj_plus.con".
274 inline "cic:/CoRN/complex/CComplex/CC_conj_mult.con".
277 Hint Resolve CC_conj_mult: algebra.
280 inline "cic:/CoRN/complex/CComplex/CC_conj_strext.con".
282 inline "cic:/CoRN/complex/CComplex/CC_conj_conj.con".
284 inline "cic:/CoRN/complex/CComplex/CC_conj_zero.con".
286 inline "cic:/CoRN/complex/CComplex/CC_conj_one.con".
289 Hint Resolve CC_conj_one: algebra.
292 inline "cic:/CoRN/complex/CComplex/CC_conj_nexp.con".
299 Hint Resolve CC_conj_plus CC_conj_mult CC_conj_nexp CC_conj_conj
300 CC_conj_zero: algebra.
303 (*#* ** Properties of the real axis *)
306 Section cc_IR_properties
309 inline "cic:/CoRN/complex/CComplex/Re_cc_IR.con".
311 inline "cic:/CoRN/complex/CComplex/Im_cc_IR.con".
313 inline "cic:/CoRN/complex/CComplex/cc_IR_wd.con".
316 Hint Resolve cc_IR_wd: algebra_c.
319 inline "cic:/CoRN/complex/CComplex/cc_IR_resp_ap.con".
321 inline "cic:/CoRN/complex/CComplex/cc_IR_mult.con".
324 Hint Resolve cc_IR_mult: algebra.
327 inline "cic:/CoRN/complex/CComplex/cc_IR_mult_lft.con".
329 inline "cic:/CoRN/complex/CComplex/cc_IR_mult_rht.con".
331 inline "cic:/CoRN/complex/CComplex/cc_IR_plus.con".
334 Hint Resolve cc_IR_plus: algebra.
337 inline "cic:/CoRN/complex/CComplex/cc_IR_minus.con".
339 inline "cic:/CoRN/complex/CComplex/cc_IR_zero.con".
342 Hint Resolve cc_IR_zero: algebra.
345 inline "cic:/CoRN/complex/CComplex/cc_IR_one.con".
348 Hint Resolve cc_IR_one: algebra.
351 inline "cic:/CoRN/complex/CComplex/cc_IR_nring.con".
353 inline "cic:/CoRN/complex/CComplex/cc_IR_nexp.con".
360 Hint Resolve Re_cc_IR Im_cc_IR: algebra.
364 Hint Resolve cc_IR_wd: algebra_c.
368 Hint Resolve cc_IR_mult cc_IR_nexp cc_IR_mult_lft cc_IR_mult_rht cc_IR_plus
369 cc_IR_minus: algebra.
373 Hint Resolve cc_IR_nring cc_IR_zero: algebra.
376 (*#* ** [CC] has characteristic zero *)
378 include "tactics/Transparent_algebra.ma".
380 inline "cic:/CoRN/complex/CComplex/char0_CC.con".
382 include "tactics/Opaque_algebra.ma".
384 inline "cic:/CoRN/complex/CComplex/poly_apzero_CC.con".
386 inline "cic:/CoRN/complex/CComplex/poly_CC_extensional.con".