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".
21 (* $Id: Basics.v,v 1.7 2004/04/09 15:58:31 lcf Exp $ *)
23 (*#* printing alpha %\ensuremath{\alpha}% #α# *)
25 (*#* printing beta %\ensuremath{\beta}% #β# *)
27 (*#* printing delta %\ensuremath{\delta}% #δ# *)
29 (*#* printing eps %\ensuremath{\varepsilon}% #ε# *)
31 (*#* printing phi %\ensuremath{\phi}% #φ# *)
33 (*#* printing eta %\ensuremath{\eta}% #η# *)
35 (*#* printing omega %\ensuremath{\omega}% #ω# *)
37 (*#* printing nat %\ensuremath{\mathbb N}% #<b>N</b># *)
39 (*#* printing Z %\ensuremath{\mathbb Z}% #<b>Z</b># *)
41 include "algebra/ListType.ma".
44 This is random stuff that should be in the Coq basic library.
47 inline "cic:/CoRN/algebra/Basics/lt_le_dec.con".
49 inline "cic:/CoRN/algebra/Basics/lt_z_two.con".
51 inline "cic:/CoRN/algebra/Basics/le_pred.con".
53 inline "cic:/CoRN/algebra/Basics/lt_mult_right.con".
55 inline "cic:/CoRN/algebra/Basics/le_mult_right.con".
57 (*#* The power function does not exist in the standard library *)
59 inline "cic:/CoRN/algebra/Basics/power.con".
61 (*#* Factorial function. Does not exist in Arith.
62 Needed for special operations on polynomials. *)
64 inline "cic:/CoRN/algebra/Basics/fac.con".
66 inline "cic:/CoRN/algebra/Basics/nat_fac_gtzero.con".
68 (* needed for computational behavior of "Inversion" tactic *)
79 Notation Pair := (pair (B:=_)).
83 Notation Proj1 := (proj1 (B:=_)).
87 Notation Proj2 := (proj2 (B:=_)).
90 (* Following only needed in finite, but tha's now obsolete
92 Lemma deMorgan_or_and: (A,B,X:Prop)((A\/B)->X)->(A->X)/\(B->X).
96 Lemma deMorgan_and_or: (A,B,X:Prop)(A->X)/\(B->X)->(A\/B->X).
100 Lemma deMorgan_ex_all:
101 (A:Set)(P:A->Prop)(X:Prop)((Ex P)->X)->(a:A)(P a)->X.
102 Intros. Apply H; Exists a; Assumption.
105 Lemma deMorgan_all_ex:
106 (A:Set)(P:A->Prop)(X:Prop)((a:A)(P a)->X)->(Ex P)->X.
107 Intros. Elim H0; Assumption.
110 Implicit Arguments Off.
112 Three lemmas for proving properties about definitions made with case
113 distinction to a sumbool, i.e. [{A} + {B}].
115 Lemma sumbool_rec_or : (A,B:Prop)(S:Set)(l,r:S)(s:{A}+{B})
116 (sumbool_rec A B [_:{A}+{B}]S [x:A]l [x:B]r s) = l \/
117 (sumbool_rec A B [_:{A}+{B}]S [x:A]l [x:B]r s) = r.
119 Intros. Left. Reflexivity.
120 Intros. Right. Reflexivity.
124 inline "cic:/CoRN/algebra/Basics/not_r_sumbool_rec.con".
126 inline "cic:/CoRN/algebra/Basics/not_l_sumbool_rec.con".
131 Set Implicit Arguments.
135 Unset Strict Implicit.
140 (*#* **Some results about [Z]
142 We consider the injection [inject_nat] from [nat] to [Z] as a
147 coercion cic:/Coq/ZArith/BinInt/Z_of_nat.con 0 (* compounds *).
151 inline "cic:/CoRN/algebra/Basics/POS_anti_convert.con".
153 inline "cic:/CoRN/algebra/Basics/NEG_anti_convert.con".
155 inline "cic:/CoRN/algebra/Basics/lt_O_positive_to_nat.con".
157 inline "cic:/CoRN/algebra/Basics/anti_convert_pred_convert.con".
159 inline "cic:/CoRN/algebra/Basics/p_is_some_anti_convert.con".
161 inline "cic:/CoRN/algebra/Basics/convert_is_POS.con".
163 inline "cic:/CoRN/algebra/Basics/min_convert_is_NEG.con".
165 inline "cic:/CoRN/algebra/Basics/inject_nat_convert.con".
167 inline "cic:/CoRN/algebra/Basics/Z_exh.con".
169 inline "cic:/CoRN/algebra/Basics/nats_Z_ind.con".
171 inline "cic:/CoRN/algebra/Basics/pred_succ_Z_ind.con".
173 inline "cic:/CoRN/algebra/Basics/Zmult_minus_distr_r.con".
175 inline "cic:/CoRN/algebra/Basics/Zodd_Zeven_min1.con".
180 Set Implicit Arguments.
184 Unset Strict Implicit.
189 inline "cic:/CoRN/algebra/Basics/caseZ_diff.con".
198 Unset Implicit Arguments.
203 inline "cic:/CoRN/algebra/Basics/caseZ_diff_O.con".
205 inline "cic:/CoRN/algebra/Basics/caseZ_diff_Pos.con".
207 inline "cic:/CoRN/algebra/Basics/caseZ_diff_Neg.con".
209 inline "cic:/CoRN/algebra/Basics/proper_caseZ_diff.con".
211 inline "cic:/CoRN/algebra/Basics/diff_Z_ind.con".
213 inline "cic:/CoRN/algebra/Basics/Zlt_reg_mult_l.con".
215 inline "cic:/CoRN/algebra/Basics/Zlt_opp.con".
217 inline "cic:/CoRN/algebra/Basics/Zlt_conv_mult_l.con".
219 inline "cic:/CoRN/algebra/Basics/Zgt_not_eq.con".
221 inline "cic:/CoRN/algebra/Basics/Zmult_absorb.con".
224 Section Well_foundedT
227 alias id "A" = "cic:/CoRN/algebra/Basics/Well_foundedT/A.var".
229 alias id "R" = "cic:/CoRN/algebra/Basics/Well_foundedT/R.var".
231 (*#* The accessibility predicate is defined to be non-informative *)
233 inline "cic:/CoRN/algebra/Basics/Acc.ind".
243 alias id "A" = "cic:/CoRN/algebra/Basics/AccT/A.var".
245 inline "cic:/CoRN/algebra/Basics/well_founded.con".
252 Implicit Arguments Acc [A].
259 alias id "A" = "cic:/CoRN/algebra/Basics/IndT/A.var".
261 alias id "R" = "cic:/CoRN/algebra/Basics/IndT/R.var".
267 alias id "P" = "cic:/CoRN/algebra/Basics/IndT/AccIter/P.var".
269 alias id "F" = "cic:/CoRN/algebra/Basics/IndT/AccIter/F.var".
271 inline "cic:/CoRN/algebra/Basics/Acc_inv.con".
273 inline "cic:/CoRN/algebra/Basics/Acc_iter.con".
279 alias id "Rwf" = "cic:/CoRN/algebra/Basics/IndT/Rwf.var".
281 inline "cic:/CoRN/algebra/Basics/well_founded_induction_type.con".
291 alias id "A" = "cic:/CoRN/algebra/Basics/InductionT/A.var".
293 alias id "f" = "cic:/CoRN/algebra/Basics/InductionT/f.var".
295 inline "cic:/CoRN/algebra/Basics/ltof.con".
297 inline "cic:/CoRN/algebra/Basics/well_founded_ltof.con".
299 inline "cic:/CoRN/algebra/Basics/induction_ltof2T.con".
309 inline "cic:/CoRN/algebra/Basics/lt_wf_rect.con".