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.
  40 demodulate all steps=3.
  41 (*
  42 apply eq_f.rewrite < H.
  43 transitivity ((n1+m)+n1*m).symmetry.apply assoc_plus.
  44 transitivity ((m+n1)+n1*m).
  45 apply eq_f2.
  46 apply sym_plus.
  47 reflexivity.
  48 apply assoc_plus.
  49 *)
  50 qed.
  51
  52 theorem times_O_to_O: \forall n,m:nat.n*m = O \to n = O \lor m= O.
  53 apply nat_elim2;intros
  54   [left.reflexivity
  55   |right.reflexivity
  56   |apply False_ind.
  57    simplify in H1.
  58    apply (not_eq_O_S ? (sym_eq  ? ? ? H1))
  59   ]
  60 qed.
  61
  62 theorem times_n_SO : \forall n:nat. n = n * S O.
  63 intros. demodulate. reflexivity. (*
  64 rewrite < times_n_Sm.
  65 rewrite < times_n_O.
  66 rewrite < plus_n_O.
  67 reflexivity.*)
  68 qed.
  69
  70 theorem times_SSO_n : \forall n:nat. n + n = S (S O) * n.
  71 intros.
  72 simplify.
  73 rewrite < plus_n_O.
  74 reflexivity.
  75 qed.
  76
  77 alias num (instance 0) = "natural number".
  78 lemma times_SSO: \forall n.2*(S n) = S(S(2*n)).
  79 intro.simplify.rewrite < plus_n_Sm.reflexivity.
  80 qed.
  81
  82 theorem or_eq_eq_S: \forall n.\exists m.
  83 n = (S(S O))*m \lor n = S ((S(S O))*m).
  84 intro.elim n
  85   [apply (ex_intro ? ? O).
  86    left.reflexivity
  87   |elim H.elim H1
  88     [apply (ex_intro ? ? a).
  89      right.apply eq_f.assumption
  90     |apply (ex_intro ? ? (S a)).
  91      left.rewrite > H2.
  92      apply sym_eq.
  93      apply times_SSO
  94     ]
  95   ]
  96 qed.
  97
  98 theorem symmetric_times : symmetric nat times.
  99 unfold symmetric.
 100 intros.elim x;
 101  [ simplify. apply times_n_O.
 102  | demodulate. reflexivity. (*
 103 (* applyS times_n_Sm. *)
 104 simplify.rewrite > H.apply times_n_Sm.*)]
 105 qed.
 106
 107 variant sym_times : \forall n,m:nat. n*m = m*n \def
 108 symmetric_times.
 109
 110 theorem distributive_times_plus : distributive nat times plus.
 111 unfold distributive.
 112 intros.elim x.
 113 simplify.reflexivity.
 114 simplify.
 115 demodulate all steps=16.
 116 (*
 117 rewrite > H. rewrite > assoc_plus.rewrite > assoc_plus.
 118 apply eq_f.rewrite < assoc_plus. rewrite < (sym_plus ? z).
 119 rewrite > assoc_plus.reflexivity. *)
 120 qed.
 121
 122 variant distr_times_plus: \forall n,m,p:nat. n*(m+p) = n*m + n*p
 123 \def distributive_times_plus.
 124
 125 theorem associative_times: associative nat times.
 126 unfold associative.
 127 intros.
 128 elim x. simplify.apply refl_eq.
 129 simplify.
 130 demodulate all steps=4.
 131 (*
 132 rewrite < sym_times.
 133 rewrite > distr_times_plus.
 134 rewrite < sym_times.
 135 rewrite < (sym_times (times n y) z).
 136 rewrite < H.apply refl_eq.
 137 *)
 138 qed.
 139
 140 variant assoc_times: \forall n,m,p:nat. (n*m)*p = n*(m*p) \def
 141 associative_times.
 142
 143 lemma times_times: ∀x,y,z. x*(y*z) = y*(x*z).
 144 intros.
 145 demodulate. reflexivity.
 146 (* autobatch paramodulation. *)
 147 qed.
 148