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
  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_O_to_O: \forall n,m:nat.n*m = O \to n = O \lor m= O.
  47 apply nat_elim2;intros
  48   [left.reflexivity
  49   |right.reflexivity
  50   |apply False_ind.
  51    simplify in H1.
  52    apply (not_eq_O_S ? (sym_eq  ? ? ? H1))
  53   ]
  54 qed.
  55
  56 theorem times_n_SO : \forall n:nat. n = n * S O.
  57 intros.
  58 rewrite < times_n_Sm.
  59 rewrite < times_n_O.
  60 rewrite < plus_n_O.
  61 reflexivity.
  62 qed.
  63
  64 theorem times_SSO_n : \forall n:nat. n + n = S (S O) * n.
  65 intros.
  66 simplify.
  67 rewrite < plus_n_O.
  68 reflexivity.
  69 qed.
  70
  71 theorem symmetric_times : symmetric nat times.
  72 unfold symmetric.
  73 intros.elim x.
  74 simplify.apply times_n_O.
  75 simplify.rewrite > H.apply times_n_Sm.
  76 qed.
  77
  78 variant sym_times : \forall n,m:nat. n*m = m*n \def
  79 symmetric_times.
  80
  81 theorem distributive_times_plus : distributive nat times plus.
  82 unfold distributive.
  83 intros.elim x.
  84 simplify.reflexivity.
  85 simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
  86 apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
  87 rewrite > assoc_plus.reflexivity.
  88 qed.
  89
  90 variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
  91 \def distributive_times_plus.
  92
  93 theorem associative_times: associative nat times.
  94 unfold associative.intros.
  95 elim x.simplify.apply refl_eq.
  96 simplify.rewrite < sym_times.
  97 rewrite > distr_times_plus.
  98 rewrite < sym_times.
  99 rewrite < (sym_times (times n y) z).
 100 rewrite < H.apply refl_eq.
 101 qed.
 102
 103 variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
 104 associative_times.