helm/software/matita/library/nat/times.ma

   1
   2 (**************************************************************************)
   3 (*       __                                                               *)
   4 (*      ||M||                                                             *)
   5 (*      ||A||       A project by Andrea Asperti                           *)
   6 (*      ||T||                                                             *)
   7 (*      ||I||       Developers:                                           *)
   8 (*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
   9 (*      ||A||       E.Tassi, S.Zacchiroli                                 *)
  10 (*      \   /                                                             *)
  11 (*       \ /        This file is distributed under the terms of the       *)
  12 (*        v         GNU Lesser General Public License Version 2.1         *)
  13 (*                                                                        *)
  14 (**************************************************************************)
  15
  16 include "nat/plus.ma".
  17
  18 let rec times n m \def
  19  match n with
  20  [ O \Rightarrow O
  21  | (S p) \Rightarrow m+(times p m) ].
  22
  23 interpretation "natural times" 'times x y = (times x y).
  24
  25 theorem times_Sn_m:
  26 \forall n,m:nat. m+n*m = S n*m.
  27 intros. reflexivity.
  28 qed.
  29
  30 theorem times_n_O: \forall n:nat. O = n*O.
  31 intros.elim n.
  32 simplify.reflexivity.
  33 simplify.assumption.
  34 qed.
  35
  36 theorem times_n_Sm :
  37 \forall n,m:nat. n+(n*m) = n*(S m).
  38 intros.elim n.
  39 simplify.reflexivity.
  40 simplify.
  41 demodulate all.
  42 (*
  43 apply eq_f.rewrite < H.
  44 transitivity ((n1+m)+n1*m).symmetry.apply assoc_plus.
  45 transitivity ((m+n1)+n1*m).
  46 apply eq_f2.
  47 apply sym_plus.
  48 reflexivity.
  49 apply assoc_plus.
  50 *)
  51 qed.
  52
  53 theorem times_O_to_O: \forall n,m:nat.n*m = O \to n = O \lor m= O.
  54 apply nat_elim2;intros
  55   [left.reflexivity
  56   |right.reflexivity
  57   |apply False_ind.
  58    simplify in H1.
  59    apply (not_eq_O_S ? (sym_eq  ? ? ? H1))
  60   ]
  61 qed.
  62
  63 theorem times_n_SO : \forall n:nat. n = n * S O.
  64 intros. demodulate. reflexivity. (*
  65 rewrite < times_n_Sm.
  66 rewrite < times_n_O.
  67 rewrite < plus_n_O.
  68 reflexivity.*)
  69 qed.
  70
  71 theorem times_SSO_n : \forall n:nat. n + n = S (S O) * n.
  72 intros.
  73 simplify.
  74 rewrite < plus_n_O.
  75 reflexivity.
  76 qed.
  77
  78 alias num (instance 0) = "natural number".
  79 lemma times_SSO: \forall n.2*(S n) = S(S(2*n)).
  80 intro.simplify.rewrite < plus_n_Sm.reflexivity.
  81 qed.
  82
  83 theorem or_eq_eq_S: \forall n.\exists m.
  84 n = (S(S O))*m \lor n = S ((S(S O))*m).
  85 intro.elim n
  86   [apply (ex_intro ? ? O).
  87    left.reflexivity
  88   |elim H.elim H1
  89     [apply (ex_intro ? ? a).
  90      right.apply eq_f.assumption
  91     |apply (ex_intro ? ? (S a)).
  92      left.rewrite > H2.
  93      apply sym_eq.
  94      apply times_SSO
  95     ]
  96   ]
  97 qed.
  98
  99 theorem symmetric_times : symmetric nat times.
 100 unfold symmetric.
 101 intros.elim x;
 102  [ simplify. apply times_n_O.
 103  | demodulate. reflexivity. (*
 104 (* applyS times_n_Sm. *)
 105 simplify.rewrite > H.apply times_n_Sm.*)]
 106 qed.
 107
 108 variant sym_times : \forall n,m:nat. n*m = m*n \def
 109 symmetric_times.
 110
 111 theorem distributive_times_plus : distributive nat times plus.
 112 unfold distributive.
 113 intros.elim x.
 114 simplify.reflexivity.
 115 simplify.
 116 demodulate all.
 117 (*
 118 rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
 119 apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
 120 rewrite > assoc_plus.reflexivity. *)
 121 qed.
 122
 123 variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
 124 \def distributive_times_plus.
 125
 126 theorem associative_times: associative nat times.
 127 unfold associative.
 128 intros.
 129 elim x. simplify.apply refl_eq.
 130 simplify.
 131 demodulate all.
 132 (*
 133 rewrite < sym_times.
 134 rewrite > distr_times_plus.
 135 rewrite < sym_times.
 136 rewrite < (sym_times (times n y) z).
 137 rewrite < H.apply refl_eq.
 138 *)
 139 qed.
 140
 141 variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
 142 associative_times.
 143
 144 lemma times_times: ∀x,y,z. x*(y*z) = y*(x*z).
 145 intros.
 146 demodulate. reflexivity.
 147 (* autobatch paramodulation. *)
 148 qed.
 149