matita/contribs/RELATIONAL/Zah/setoid.ma

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/RELATIONAL/Zah/setoid".
  16
  17 include "NPlus/monoid.ma".
  18 include "Zah/defs.ma".
  19
  20 theorem nplus_total: \forall p,q. \exists r. p + q == r.
  21  intros 2. elim q; clear q;
  22  decompose; auto.
  23 qed.
  24
  25 theorem zeq_intro: \forall z1,z2.
  26                    (\forall n.((\fst z1) + (\snd z2) == n) \to ((\fst z2) + (\snd z1) == n)) \to
  27                    (\forall n.((\fst z2) + (\snd z1) == n) \to ((\fst z1) + (\snd z2) == n)) \to
  28                    z1 = z2.
  29  unfold ZEq. intros. apply iff_intro; intros; auto.
  30 qed.
  31
  32 theorem zeq_refl: \forall z. z = z.
  33  unfold ZEq. intros. auto.
  34 qed.
  35
  36 theorem zeq_sym: \forall z1,z2. z1 = z2 \to z2 = z1.
  37  unfold ZEq. intros. lapply linear (H n). auto.
  38 qed.
  39 (*
  40 theorem zeq_trans: \forall z1,z2. z1 = z2 \to
  41                    \forall z3. z2 = z3 \to z1 = z3.
  42  unfold ZEq. intros.
  43  lapply (nplus_total (\snd z2) (\fst z2)). decompose.
  44
  45
  46
  47  apply iff_intro; intros;
  48  [ lapply (nplus_total (\fst z1) (\snd z3)). decompose
  49  |
  50
  51
  52
  53
  54  lapply (nplus_total (\snd z2) (\fst z2)). decompose.
  55  lapply (nplus_total n a). decompose.
  56  lapply linear (H a1). lapply linear (H1 a1). decompose.
  57
  58
  59
  60 *)