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 *********************)
21 (* In this file a notion of morphism between two arbitrary real number
22 structures, is introduced together with te proofs that this notion of
23 morphism preserves the basic algebraic structure. *)
25 include "reals/CReals.ma".
27 (* This comes from CReals1.v *)
29 inline procedural "cic:/CoRN/reals/R_morphism/Cauchy_Lim_prop2.con" as definition.
35 alias id "R1" = "cic:/CoRN/reals/R_morphism/morphism/R1.var".
37 alias id "R2" = "cic:/CoRN/reals/R_morphism/morphism/R2.var".
40 Section morphism_details
43 alias id "phi" = "cic:/CoRN/reals/R_morphism/morphism/morphism_details/phi.var".
45 alias id "p1" = "cic:/CoRN/reals/R_morphism/morphism/morphism_details/p1.var".
47 alias id "p2" = "cic:/CoRN/reals/R_morphism/morphism/morphism_details/p2.var".
49 alias id "f1" = "cic:/CoRN/reals/R_morphism/morphism/morphism_details/f1.var".
51 alias id "f2" = "cic:/CoRN/reals/R_morphism/morphism/morphism_details/f2.var".
53 alias id "g1" = "cic:/CoRN/reals/R_morphism/morphism/morphism_details/g1.var".
55 alias id "g2" = "cic:/CoRN/reals/R_morphism/morphism/morphism_details/g2.var".
57 inline procedural "cic:/CoRN/reals/R_morphism/fun_pres_relation.con" as definition.
59 inline procedural "cic:/CoRN/reals/R_morphism/fun_pres_un_fun.con" as definition.
61 inline procedural "cic:/CoRN/reals/R_morphism/fun_pres_bin_fun.con" as definition.
64 Definition fun_pres_partial_fun:=(x:R1;H1:x[#]Zero;H2:(phi x)[#]Zero)
65 (phi (nzinj R1 (i1 (nzpro R1 x H1))))[=](nzinj R2 (i2 (nzpro R2 (phi x) H2))).
68 inline procedural "cic:/CoRN/reals/R_morphism/fun_pres_Lim.con" as definition.
74 inline procedural "cic:/CoRN/reals/R_morphism/Homomorphism.ind".
77 cic:/matita/CoRN-Procedural/reals/R_morphism/map.con
80 (* This might be useful later...
81 Definition Homo_as_CSetoid_fun:=
83 (Build_CSetoid_fun R1 R2 f
84 (fun_strext_imp_wd R1 R2 f (!map_strext f))
89 inline procedural "cic:/CoRN/reals/R_morphism/map_strext_unfolded.con" as lemma.
91 inline procedural "cic:/CoRN/reals/R_morphism/map_wd_unfolded.con" as lemma.
93 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_less_unfolded.con" as lemma.
95 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_plus_unfolded.con" as lemma.
97 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_mult_unfolded.con" as lemma.
99 (* Now we start to derive some useful properties of a Homomorphism *)
101 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_zero.con" as lemma.
103 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_zero_unfolded.con" as lemma.
105 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_minus.con" as lemma.
107 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_minus_unfolded.con" as lemma.
109 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_apartness.con" as lemma.
111 (* Merely a useful special case *)
113 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_ap_zero.con" as lemma.
115 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_one.con" as lemma.
117 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_one_unfolded.con" as lemma.
119 (* I will not use the following lemma *)
121 inline procedural "cic:/CoRN/reals/R_morphism/map_pres_inv_unfolded.con" as lemma.
131 alias id "R1" = "cic:/CoRN/reals/R_morphism/composition/R1.var".
133 alias id "R2" = "cic:/CoRN/reals/R_morphism/composition/R2.var".
135 alias id "R3" = "cic:/CoRN/reals/R_morphism/composition/R3.var".
137 alias id "f" = "cic:/CoRN/reals/R_morphism/composition/f.var".
139 alias id "g" = "cic:/CoRN/reals/R_morphism/composition/g.var".
141 inline procedural "cic:/CoRN/reals/R_morphism/compose.con" as definition.
143 inline procedural "cic:/CoRN/reals/R_morphism/compose_strext.con" as lemma.
145 inline procedural "cic:/CoRN/reals/R_morphism/compose_pres_less.con" as lemma.
147 inline procedural "cic:/CoRN/reals/R_morphism/compose_pres_plus.con" as lemma.
149 inline procedural "cic:/CoRN/reals/R_morphism/compose_pres_mult.con" as lemma.
151 inline procedural "cic:/CoRN/reals/R_morphism/compose_pres_Lim.con" as lemma.
153 inline procedural "cic:/CoRN/reals/R_morphism/Compose.con" as definition.
163 alias id "R1" = "cic:/CoRN/reals/R_morphism/isomorphism/R1.var".
165 alias id "R2" = "cic:/CoRN/reals/R_morphism/isomorphism/R2.var".
171 alias id "R3" = "cic:/CoRN/reals/R_morphism/isomorphism/identity_map/R3.var".
173 alias id "f" = "cic:/CoRN/reals/R_morphism/isomorphism/identity_map/f.var".
175 inline procedural "cic:/CoRN/reals/R_morphism/map_is_id.con" as definition.
181 inline procedural "cic:/CoRN/reals/R_morphism/Isomorphism.ind".
188 Section surjective_map
191 alias id "R1" = "cic:/CoRN/reals/R_morphism/surjective_map/R1.var".
193 alias id "R2" = "cic:/CoRN/reals/R_morphism/surjective_map/R2.var".
195 alias id "f" = "cic:/CoRN/reals/R_morphism/surjective_map/f.var".
197 inline procedural "cic:/CoRN/reals/R_morphism/map_is_surjective.con" as definition.
204 Section simplification
207 alias id "R1" = "cic:/CoRN/reals/R_morphism/simplification/R1.var".
209 alias id "R2" = "cic:/CoRN/reals/R_morphism/simplification/R2.var".
211 alias id "f" = "cic:/CoRN/reals/R_morphism/simplification/f.var".
213 alias id "H1" = "cic:/CoRN/reals/R_morphism/simplification/H1.var".
215 inline procedural "cic:/CoRN/reals/R_morphism/f_well_def.con" as lemma.
221 alias id "H2" = "cic:/CoRN/reals/R_morphism/simplification/with_less/H2.var".
223 inline procedural "cic:/CoRN/reals/R_morphism/less_pres_f.con" as lemma.
225 inline procedural "cic:/CoRN/reals/R_morphism/leEq_pres_f.con" as lemma.
227 inline procedural "cic:/CoRN/reals/R_morphism/f_pres_leEq.con" as lemma.
229 inline procedural "cic:/CoRN/reals/R_morphism/f_pres_apartness.con" as lemma.
239 alias id "H3" = "cic:/CoRN/reals/R_morphism/simplification/with_plus/H3.var".
241 inline procedural "cic:/CoRN/reals/R_morphism/f_pres_Zero.con" as lemma.
243 inline procedural "cic:/CoRN/reals/R_morphism/f_pres_minus.con" as lemma.
245 inline procedural "cic:/CoRN/reals/R_morphism/f_pres_min.con" as lemma.
252 Section with_plus_less
255 alias id "H2" = "cic:/CoRN/reals/R_morphism/simplification/with_plus_less/H2.var".
257 alias id "H3" = "cic:/CoRN/reals/R_morphism/simplification/with_plus_less/H3.var".
259 inline procedural "cic:/CoRN/reals/R_morphism/f_pres_ap_zero.con" as lemma.
262 Section surjectivity_helps
265 alias id "f_surj" = "cic:/CoRN/reals/R_morphism/simplification/with_plus_less/surjectivity_helps/f_surj.var".
267 inline procedural "cic:/CoRN/reals/R_morphism/f_pres_Lim.con" as lemma.
270 End surjectivity_helps
274 Section with_mult_plus_less
277 alias id "H4" = "cic:/CoRN/reals/R_morphism/simplification/with_plus_less/with_mult_plus_less/H4.var".
279 inline procedural "cic:/CoRN/reals/R_morphism/f_pres_one.con" as lemma.
281 inline procedural "cic:/CoRN/reals/R_morphism/f_pres_inv.con" as lemma.
283 inline procedural "cic:/CoRN/reals/R_morphism/simplified_Homomorphism.con" as definition.
286 End with_mult_plus_less