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 (*CSC: the URI must disappear: there is a bug now *)
  25 interpretation "natural plus" 'plus x y = (cic:/matita/library_autobatch/nat/plus/plus.con x y).
  26
  27 theorem plus_n_O: \forall n:nat. n = n+O.
  28 intros.elim n
  29 [ autobatch
  30   (*simplify.
  31   reflexivity*)
  32 | autobatch
  33   (*simplify.
  34   apply eq_f.
  35   assumption.*)
  36 ]
  37 qed.
  38
  39 theorem plus_n_Sm : \forall n,m:nat. S (n+m) = n+(S m).
  40 intros.elim n
  41 [ autobatch
  42   (*simplify.
  43   reflexivity.*)
  44 | simplify.
  45   autobatch
  46   (*
  47   apply eq_f.
  48   assumption.*)]
  49 qed.
  50
  51 theorem sym_plus: \forall n,m:nat. n+m = m+n.
  52 intros.elim n
  53 [ autobatch
  54   (*simplify.
  55   apply plus_n_O.*)
  56 | simplify.
  57   autobatch
  58   (*rewrite > H.
  59   apply plus_n_Sm.*)]
  60 qed.
  61
  62 theorem associative_plus : associative nat plus.
  63 unfold associative.intros.elim x
  64 [autobatch
  65  (*simplify.
  66  reflexivity.*)
  67 |simplify.
  68  autobatch
  69  (*apply eq_f.
  70  assumption.*)
  71 ]
  72 qed.
  73
  74 theorem assoc_plus : \forall n,m,p:nat. (n+m)+p = n+(m+p)
  75 \def associative_plus.
  76
  77 theorem injective_plus_r: \forall n:nat.injective nat nat (\lambda m.n+m).
  78 intro.simplify.intros 2.elim n
  79 [ exact H
  80 | autobatch
  81   (*apply H.apply inj_S.apply H1.*)
  82 ]
  83 qed.
  84
  85 theorem inj_plus_r: \forall p,n,m:nat. p+n = p+m \to n=m
  86 \def injective_plus_r.
  87
  88 theorem injective_plus_l: \forall m:nat.injective nat nat (\lambda n.n+m).
  89 intro.simplify.intros.autobatch.
  90 (*apply (injective_plus_r m).
  91 rewrite < sym_plus.
  92 rewrite < (sym_plus y).
  93 assumption.*)
  94 qed.
  95
  96 theorem inj_plus_l: \forall p,n,m:nat. n+p = m+p \to n=m
  97 \def injective_plus_l.