matita/matita/contribs/RELATIONAL/NPlus/fun.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15
  16
  17 include "NPlus/inv.ma".
  18
  19 (* Functional properties ****************************************************)
  20
  21 theorem nplus_total: ∀p,q. ∃r. p ⊕ q ≍ r.
  22  intros; elim q; clear q;
  23  [ autobatch | decompose; autobatch ].
  24 qed.
  25
  26 theorem nplus_mono: ∀p,q,r1. p ⊕ q ≍ r1 →
  27                     ∀r2. p ⊕ q ≍ r2 → r1 = r2.
  28  intros 4; elim H; clear H q r1;
  29  [ lapply linear nplus_inv_zero_2 to H1
  30  | lapply linear nplus_inv_succ_2 to H3. decompose
  31  ]; destruct; autobatch.
  32 qed.
  33
  34 theorem nplus_inj_1: ∀p1, q, r. p1 ⊕ q ≍ r →
  35                      ∀p2. p2 ⊕ q ≍ r → p2 = p1.
  36  intros 4; elim H; clear H q r;
  37  [ lapply linear nplus_inv_zero_2 to H1
  38  | lapply linear nplus_inv_succ_2_3 to H3
  39  ]; autobatch.
  40 qed.