helm/software/matita/contribs/library_auto/auto/nat/plus.ma

   1 (**************************************************************************)
   2 (*       ___                                                                *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
   8 (*      ||A||       E.Tassi, S.Zacchiroli                                 *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU Lesser General Public License Version 2.1         *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 set "baseuri" "cic:/matita/library_autobatch/nat/plus".
  16
  17 include "auto/nat/nat.ma".
  18
  19 let rec plus n m \def
  20  match n with
  21  [ O \Rightarrow m
  22  | (S p) \Rightarrow S (plus p m) ].
  23
  24 interpretation "natural plus" 'plus x y = (plus x y).
  25
  26 theorem plus_n_O: \forall n:nat. n = n+O.
  27 intros.elim n
  28 [ autobatch
  29   (*simplify.
  30   reflexivity*)
  31 | autobatch
  32   (*simplify.
  33   apply eq_f.
  34   assumption.*)
  35 ]
  36 qed.
  37
  38 theorem plus_n_Sm : \forall n,m:nat. S (n+m) = n+(S m).
  39 intros.elim n
  40 [ autobatch
  41   (*simplify.
  42   reflexivity.*)
  43 | simplify.
  44   autobatch
  45   (*
  46   apply eq_f.
  47   assumption.*)]
  48 qed.
  49
  50 theorem sym_plus: \forall n,m:nat. n+m = m+n.
  51 intros.elim n
  52 [ autobatch
  53   (*simplify.
  54   apply plus_n_O.*)
  55 | simplify.
  56   autobatch
  57   (*rewrite > H.
  58   apply plus_n_Sm.*)]
  59 qed.
  60
  61 theorem associative_plus : associative nat plus.
  62 unfold associative.intros.elim x
  63 [autobatch
  64  (*simplify.
  65  reflexivity.*)
  66 |simplify.
  67  autobatch
  68  (*apply eq_f.
  69  assumption.*)
  70 ]
  71 qed.
  72
  73 theorem assoc_plus : \forall n,m,p:nat. (n+m)+p = n+(m+p)
  74 \def associative_plus.
  75
  76 theorem injective_plus_r: \forall n:nat.injective nat nat (\lambda m.n+m).
  77 intro.simplify.intros 2.elim n
  78 [ exact H
  79 | autobatch
  80   (*apply H.apply inj_S.apply H1.*)
  81 ]
  82 qed.
  83
  84 theorem inj_plus_r: \forall p,n,m:nat. p+n = p+m \to n=m
  85 \def injective_plus_r.
  86
  87 theorem injective_plus_l: \forall m:nat.injective nat nat (\lambda n.n+m).
  88 intro.simplify.intros.autobatch.
  89 (*apply (injective_plus_r m).
  90 rewrite < sym_plus.
  91 rewrite < (sym_plus y).
  92 assumption.*)
  93 qed.
  94
  95 theorem inj_plus_l: \forall p,n,m:nat. n+p = m+p \to n=m
  96 \def injective_plus_l.