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/algebra/Basics".
19 (* $Id: Basics.v,v 1.7 2004/04/09 15:58:31 lcf Exp $ *)
21 (*#* printing alpha %\ensuremath{\alpha}% #α# *)
23 (*#* printing beta %\ensuremath{\beta}% #β# *)
25 (*#* printing delta %\ensuremath{\delta}% #δ# *)
27 (*#* printing eps %\ensuremath{\varepsilon}% #ε# *)
29 (*#* printing phi %\ensuremath{\phi}% #φ# *)
31 (*#* printing eta %\ensuremath{\eta}% #η# *)
33 (*#* printing omega %\ensuremath{\omega}% #ω# *)
35 (*#* printing nat %\ensuremath{\mathbb N}% #<b>N</b># *)
37 (*#* printing Z %\ensuremath{\mathbb Z}% #<b>Z</b># *)
60 This is random stuff that should be in the Coq basic library.
63 inline cic:/CoRN/algebra/Basics/lt_le_dec.con.
65 inline cic:/CoRN/algebra/Basics/lt_z_two.con.
67 inline cic:/CoRN/algebra/Basics/le_pred.con.
69 inline cic:/CoRN/algebra/Basics/lt_mult_right.con.
71 inline cic:/CoRN/algebra/Basics/le_mult_right.con.
73 (*#* The power function does not exist in the standard library *)
75 inline cic:/CoRN/algebra/Basics/power.con.
77 (*#* Factorial function. Does not exist in Arith.
78 Needed for special operations on polynomials. *)
80 inline cic:/CoRN/algebra/Basics/fac.con.
82 inline cic:/CoRN/algebra/Basics/nat_fac_gtzero.con.
84 (* needed for computational behavior of "Inversion" tactic *)
94 (* Following only needed in finite, but tha's now obsolete
96 Lemma deMorgan_or_and: (A,B,X:Prop)((A\/B)->X)->(A->X)/\(B->X).
100 Lemma deMorgan_and_or: (A,B,X:Prop)(A->X)/\(B->X)->(A\/B->X).
104 Lemma deMorgan_ex_all:
105 (A:Set)(P:A->Prop)(X:Prop)((Ex P)->X)->(a:A)(P a)->X.
106 Intros. Apply H; Exists a; Assumption.
109 Lemma deMorgan_all_ex:
110 (A:Set)(P:A->Prop)(X:Prop)((a:A)(P a)->X)->(Ex P)->X.
111 Intros. Elim H0; Assumption.
114 Implicit Arguments Off.
116 Three lemmas for proving properties about definitions made with case
117 distinction to a sumbool, i.e. [{A} + {B}].
119 Lemma sumbool_rec_or : (A,B:Prop)(S:Set)(l,r:S)(s:{A}+{B})
120 (sumbool_rec A B [_:{A}+{B}]S [x:A]l [x:B]r s) = l \/
121 (sumbool_rec A B [_:{A}+{B}]S [x:A]l [x:B]r s) = r.
123 Intros. Left. Reflexivity.
124 Intros. Right. Reflexivity.
128 inline cic:/CoRN/algebra/Basics/not_r_sumbool_rec.con.
130 inline cic:/CoRN/algebra/Basics/not_l_sumbool_rec.con.
135 Set Implicit Arguments.
139 Unset Strict Implicit.
144 (*#* **Some results about [Z]
146 We consider the injection [inject_nat] from [nat] to [Z] as a
153 inline cic:/CoRN/algebra/Basics/POS_anti_convert.con.
155 inline cic:/CoRN/algebra/Basics/NEG_anti_convert.con.
157 inline cic:/CoRN/algebra/Basics/lt_O_positive_to_nat.con.
159 inline cic:/CoRN/algebra/Basics/anti_convert_pred_convert.con.
161 inline cic:/CoRN/algebra/Basics/p_is_some_anti_convert.con.
163 inline cic:/CoRN/algebra/Basics/convert_is_POS.con.
165 inline cic:/CoRN/algebra/Basics/min_convert_is_NEG.con.
167 inline cic:/CoRN/algebra/Basics/inject_nat_convert.con.
169 inline cic:/CoRN/algebra/Basics/Z_exh.con.
171 inline cic:/CoRN/algebra/Basics/nats_Z_ind.con.
173 inline cic:/CoRN/algebra/Basics/pred_succ_Z_ind.con.
175 inline cic:/CoRN/algebra/Basics/Zmult_minus_distr_r.con.
177 inline cic:/CoRN/algebra/Basics/Zodd_Zeven_min1.con.
182 Set Implicit Arguments.
186 Unset Strict Implicit.
191 inline cic:/CoRN/algebra/Basics/caseZ_diff.con.
200 Unset Implicit Arguments.
205 inline cic:/CoRN/algebra/Basics/caseZ_diff_O.con.
207 inline cic:/CoRN/algebra/Basics/caseZ_diff_Pos.con.
209 inline cic:/CoRN/algebra/Basics/caseZ_diff_Neg.con.
211 inline cic:/CoRN/algebra/Basics/proper_caseZ_diff.con.
213 inline cic:/CoRN/algebra/Basics/diff_Z_ind.con.
215 inline cic:/CoRN/algebra/Basics/Zlt_reg_mult_l.con.
217 inline cic:/CoRN/algebra/Basics/Zlt_opp.con.
219 inline cic:/CoRN/algebra/Basics/Zlt_conv_mult_l.con.
221 inline cic:/CoRN/algebra/Basics/Zgt_not_eq.con.
223 inline cic:/CoRN/algebra/Basics/Zmult_absorb.con.
226 Section Well_foundedT.
229 inline cic:/CoRN/algebra/Basics/A.var.
231 inline cic:/CoRN/algebra/Basics/R.var.
233 (*#* The accessibility predicate is defined to be non-informative *)
235 inline cic:/CoRN/algebra/Basics/Acc.ind.
245 inline cic:/CoRN/algebra/Basics/A.var.
247 inline cic:/CoRN/algebra/Basics/well_founded.con.
254 Implicit Arguments Acc [A].
261 inline cic:/CoRN/algebra/Basics/A.var.
263 inline cic:/CoRN/algebra/Basics/R.var.
269 inline cic:/CoRN/algebra/Basics/P.var.
271 inline cic:/CoRN/algebra/Basics/F.var.
273 inline cic:/CoRN/algebra/Basics/Acc_inv.con.
275 inline cic:/CoRN/algebra/Basics/Acc_iter.con.
281 inline cic:/CoRN/algebra/Basics/Rwf.var.
283 inline cic:/CoRN/algebra/Basics/well_founded_induction_type.con.
293 inline cic:/CoRN/algebra/Basics/A.var.
295 inline cic:/CoRN/algebra/Basics/f.var.
297 inline cic:/CoRN/algebra/Basics/ltof.con.
299 inline cic:/CoRN/algebra/Basics/well_founded_ltof.con.
301 inline cic:/CoRN/algebra/Basics/induction_ltof2T.con.
311 inline cic:/CoRN/algebra/Basics/lt_wf_rect.con.