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 include "nat/plus.ma".
  16
  17 let rec times n m \def
  18  match n with
  19  [ O \Rightarrow O
  20  | (S p) \Rightarrow m+(times p m) ].
  21
  22 interpretation "natural times" 'times x y = (cic:/matita/nat/times/times.con x y).
  23
  24 theorem times_n_O: \forall n:nat. O = n*O.
  25 intros.elim n.
  26 simplify.reflexivity.
  27 simplify.assumption.
  28 qed.
  29
  30 theorem times_n_Sm :
  31 \forall n,m:nat. n+(n*m) = n*(S m).
  32 intros.elim n.
  33 simplify.reflexivity.
  34 simplify.apply eq_f.rewrite < H.
  35 transitivity ((n1+m)+n1*m).symmetry.apply assoc_plus.
  36 transitivity ((m+n1)+n1*m).
  37 apply eq_f2.
  38 apply sym_plus.
  39 reflexivity.
  40 apply assoc_plus.
  41 qed.
  42
  43 theorem times_O_to_O: \forall n,m:nat.n*m = O \to n = O \lor m= O.
  44 apply nat_elim2;intros
  45   [left.reflexivity
  46   |right.reflexivity
  47   |apply False_ind.
  48    simplify in H1.
  49    apply (not_eq_O_S ? (sym_eq  ? ? ? H1))
  50   ]
  51 qed.
  52
  53 theorem times_n_SO : \forall n:nat. n = n * S O.
  54 intros.
  55 rewrite < times_n_Sm.
  56 rewrite < times_n_O.
  57 rewrite < plus_n_O.
  58 reflexivity.
  59 qed.
  60
  61 theorem times_SSO_n : \forall n:nat. n + n = S (S O) * n.
  62 intros.
  63 simplify.
  64 rewrite < plus_n_O.
  65 reflexivity.
  66 qed.
  67
  68 alias num (instance 0) = "natural number".
  69 lemma times_SSO: \forall n.2*(S n) = S(S(2*n)).
  70 intro.simplify.rewrite < plus_n_Sm.reflexivity.
  71 qed.
  72
  73 theorem or_eq_eq_S: \forall n.\exists m.
  74 n = (S(S O))*m \lor n = S ((S(S O))*m).
  75 intro.elim n
  76   [apply (ex_intro ? ? O).
  77    left.reflexivity
  78   |elim H.elim H1
  79     [apply (ex_intro ? ? a).
  80      right.apply eq_f.assumption
  81     |apply (ex_intro ? ? (S a)).
  82      left.rewrite > H2.
  83      apply sym_eq.
  84      apply times_SSO
  85     ]
  86   ]
  87 qed.
  88
  89 theorem symmetric_times : symmetric nat times.
  90 unfold symmetric.
  91 intros.elim x.
  92 simplify.apply times_n_O.
  93 simplify.rewrite > H.apply times_n_Sm.
  94 qed.
  95
  96 variant sym_times : \forall n,m:nat. n*m = m*n \def
  97 symmetric_times.
  98
  99 theorem distributive_times_plus : distributive nat times plus.
 100 unfold distributive.
 101 intros.elim x.
 102 simplify.reflexivity.
 103 simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
 104 apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
 105 rewrite > assoc_plus.reflexivity.
 106 qed.
 107
 108 variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
 109 \def distributive_times_plus.
 110
 111 theorem associative_times: associative nat times.
 112 unfold associative.intros.
 113 elim x.simplify.apply refl_eq.
 114 simplify.rewrite < sym_times.
 115 rewrite > distr_times_plus.
 116 rewrite < sym_times.
 117 rewrite < (sym_times (times n y) z).
 118 rewrite < H.apply refl_eq.
 119 qed.
 120
 121 variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
 122 associative_times.