helm/software/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 = (times x y).
  23
  24 theorem times_Sn_m:
  25 \forall n,m:nat. m+n*m = S n*m.
  26 intros. reflexivity.
  27 qed.
  28
  29 theorem times_n_O: \forall n:nat. O = n*O.
  30 intros.elim n.
  31 simplify.reflexivity.
  32 simplify.assumption.
  33 qed.
  34
  35 theorem times_n_Sm :
  36 \forall n,m:nat. n+(n*m) = n*(S m).
  37 intros.elim n.
  38 simplify.reflexivity.
  39 simplify.apply eq_f.rewrite < H.
  40 transitivity ((n1+m)+n1*m).symmetry.apply assoc_plus.
  41 transitivity ((m+n1)+n1*m).
  42 apply eq_f2.
  43 apply sym_plus.
  44 reflexivity.
  45 apply assoc_plus.
  46 qed.
  47
  48 theorem times_O_to_O: \forall n,m:nat.n*m = O \to n = O \lor m= O.
  49 apply nat_elim2;intros
  50   [left.reflexivity
  51   |right.reflexivity
  52   |apply False_ind.
  53    simplify in H1.
  54    apply (not_eq_O_S ? (sym_eq  ? ? ? H1))
  55   ]
  56 qed.
  57
  58 theorem times_n_SO : \forall n:nat. n = n * S O.
  59 intros.
  60 rewrite < times_n_Sm.
  61 rewrite < times_n_O.
  62 rewrite < plus_n_O.
  63 reflexivity.
  64 qed.
  65
  66 theorem times_SSO_n : \forall n:nat. n + n = S (S O) * n.
  67 intros.
  68 simplify.
  69 rewrite < plus_n_O.
  70 reflexivity.
  71 qed.
  72
  73 alias num (instance 0) = "natural number".
  74 lemma times_SSO: \forall n.2*(S n) = S(S(2*n)).
  75 intro.simplify.rewrite < plus_n_Sm.reflexivity.
  76 qed.
  77
  78 theorem or_eq_eq_S: \forall n.\exists m.
  79 n = (S(S O))*m \lor n = S ((S(S O))*m).
  80 intro.elim n
  81   [apply (ex_intro ? ? O).
  82    left.reflexivity
  83   |elim H.elim H1
  84     [apply (ex_intro ? ? a).
  85      right.apply eq_f.assumption
  86     |apply (ex_intro ? ? (S a)).
  87      left.rewrite > H2.
  88      apply sym_eq.
  89      apply times_SSO
  90     ]
  91   ]
  92 qed.
  93
  94 theorem symmetric_times : symmetric nat times.
  95 unfold symmetric.
  96 intros.elim x.
  97 simplify.apply times_n_O.
  98 (* applyS times_n_Sm. *)
  99 simplify.rewrite > H.apply times_n_Sm.
 100 qed.
 101
 102 variant sym_times : \forall n,m:nat. n*m = m*n \def
 103 symmetric_times.
 104
 105 theorem distributive_times_plus : distributive nat times plus.
 106 unfold distributive.
 107 intros.elim x.
 108 simplify.reflexivity.
 109 simplify.rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
 110 apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
 111 rewrite > assoc_plus.reflexivity.
 112 qed.
 113
 114 variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
 115 \def distributive_times_plus.
 116
 117 theorem associative_times: associative nat times.
 118 unfold associative.
 119 intros.
 120 elim x. simplify.apply refl_eq.
 121 simplify.rewrite < sym_times.
 122 rewrite > distr_times_plus.
 123 rewrite < sym_times.
 124 rewrite < (sym_times (times n y) z).
 125 rewrite < H.apply refl_eq.
 126 qed.
 127
 128 variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
 129 associative_times.
 130
 131 lemma times_times: ∀x,y,z. x*(y*z) = y*(x*z).
 132 intros.
 133 demodulate. reflexivity;
 134 (* autobatch paramodulation. *)
 135 qed.
 136