matita/library/nat/times.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/nat/times".
  16
  17 include "nat/plus.ma".
  18
  19 let rec times n m \def
  20  match n with
  21  [ O \Rightarrow O
  22  | (S p) \Rightarrow m+(times p m) ].
  23
  24 (*CSC: the URI must disappear: there is a bug now *)
  25 interpretation "natural times" 'times x y = (cic:/matita/nat/times/times.con x y).
  26
  27 theorem times_n_O: \forall n:nat. O = n*O.
  28 intros.elim n.
  29 simplify.reflexivity.
  30 simplify.assumption.
  31 qed.
  32
  33 theorem times_n_Sm :
  34 \forall n,m:nat. n+(n*m) = n*(S m).
  35 intros.elim n.
  36 simplify.reflexivity.
  37 simplify.apply eq_f.rewrite < H.
  38 transitivity ((n1+m)+n1*m).symmetry.apply assoc_plus.
  39 transitivity ((m+n1)+n1*m).
  40 apply eq_f2.
  41 apply sym_plus.
  42 reflexivity.
  43 apply assoc_plus.
  44 qed.
  45
  46 theorem times_n_SO : \forall n:nat. n = n * S O.
  47 intros.
  48 rewrite < times_n_Sm.
  49 rewrite < times_n_O.
  50 rewrite < plus_n_O.
  51 reflexivity.
  52 qed.
  53
  54 theorem times_SSO_n : \forall n:nat. n + n = S (S O) * n.
  55 intros.
  56 simplify.
  57 rewrite < plus_n_O.
  58 reflexivity.
  59 qed.
  60
  61 theorem symmetric_times : symmetric nat times.
  62 unfold symmetric.
  63 intros.elim x.
  64 simplify.apply times_n_O.
  65 simplify.rewrite > H.apply times_n_Sm.
  66 qed.
  67
  68 variant sym_times : \forall n,m:nat. n*m = m*n \def
  69 symmetric_times.
  70
  71 theorem distributive_times_plus : distributive nat times plus.
  72 unfold distributive.
  73 intros.elim x.
  74 simplify.reflexivity.
  75 simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
  76 apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
  77 rewrite > assoc_plus.reflexivity.
  78 qed.
  79
  80 variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
  81 \def distributive_times_plus.
  82
  83 theorem associative_times: associative nat times.
  84 unfold associative.intros.
  85 elim x.simplify.apply refl_eq.
  86 simplify.rewrite < sym_times.
  87 rewrite > distr_times_plus.
  88 rewrite < sym_times.
  89 rewrite < (sym_times (times n y) z).
  90 rewrite < H.apply refl_eq.
  91 qed.
  92
  93 variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
  94 associative_times.