helm/matita/library/nat/orders_op.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 set "baseuri" "cic:/matita/nat/orders_op".
  16
  17 include "higher_order_defs/functions.ma".
  18 include "nat/times.ma".
  19 include "nat/orders.ma".
  20
  21 (* plus *)
  22 theorem monotonic_le_plus_r:
  23 \forall n:nat.monotonic nat le (\lambda m.n+m).
  24 simplify.intros.elim n.
  25 simplify.assumption.
  26 simplify.apply le_S_S.assumption.
  27 qed.
  28
  29 theorem le_plus_r: \forall p,n,m:nat. n \leq m \to p+n \leq p+m
  30 \def monotonic_le_plus_r.
  31
  32 theorem monotonic_le_plus_l:
  33 \forall m:nat.monotonic nat le (\lambda n.n+m).
  34 simplify.intros.
  35 rewrite < sym_plus.rewrite < sym_plus m.
  36 apply le_plus_r.assumption.
  37 qed.
  38
  39 theorem le_plus_l: \forall p,n,m:nat. n \leq m \to n+p \leq m+p
  40 \def monotonic_le_plus_l.
  41
  42 theorem le_plus: \forall n1,n2,m1,m2:nat. n1 \leq n2  \to m1 \leq m2
  43 \to n1+m1 \leq n2+m2.
  44 intros.
  45 apply trans_le ? (n2+m1).
  46 apply le_plus_l.assumption.
  47 apply le_plus_r.assumption.
  48 qed.
  49
  50 theorem le_plus_n :\forall n,m:nat. m \leq n+m.
  51 intros.change with O+m \leq n+m.
  52 apply le_plus_l.apply le_O_n.
  53 qed.
  54
  55 theorem eq_plus_to_le: \forall n,m,p:nat.n=m+p \to m \leq n.
  56 intros.rewrite > H.
  57 rewrite < sym_plus.
  58 apply le_plus_n.
  59 qed.
  60
  61 (* times *)
  62 theorem monotonic_le_times_r:
  63 \forall n:nat.monotonic nat le (\lambda m.n*m).
  64 simplify.intros.elim n.
  65 simplify.apply le_O_n.
  66 simplify.apply le_plus.
  67 assumption.
  68 assumption.
  69 qed.
  70
  71 theorem le_times_r: \forall p,n,m:nat. n \leq m \to p*n \leq p*m
  72 \def monotonic_le_times_r.
  73
  74 theorem monotonic_le_times_l:
  75 \forall m:nat.monotonic nat le (\lambda n.n*m).
  76 simplify.intros.
  77 rewrite < sym_times.rewrite < sym_times m.
  78 apply le_times_r.assumption.
  79 qed.
  80
  81 theorem le_times_l: \forall p,n,m:nat. n \leq m \to n*p \leq m*p
  82 \def monotonic_le_times_l.
  83
  84 theorem le_times: \forall n1,n2,m1,m2:nat. n1 \leq n2  \to m1 \leq m2
  85 \to n1*m1 \leq n2*m2.
  86 intros.
  87 apply trans_le ? (n2*m1).
  88 apply le_times_l.assumption.
  89 apply le_times_r.assumption.
  90 qed.
  91
  92 theorem le_times_n: \forall n,m:nat.S O \leq n \to m \leq n*m.
  93 intros.elim H.simplify.
  94 elim (plus_n_O ?).apply le_n.
  95 simplify.rewrite < sym_plus.apply le_plus_n.
  96 qed.
  97
  98