1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/nat/le_arith".
17 include "nat/times.ma".
18 include "nat/orders.ma".
21 theorem monotonic_le_plus_r:
22 \forall n:nat.monotonic nat le (\lambda m.plus n m).
23 simplify.intros.elim n.
25 simplify.apply le_S_S.assumption.
28 theorem le_plus_r: \forall p,n,m:nat. le n m \to le (plus p n) (plus p m)
29 \def monotonic_le_plus_r.
31 theorem monotonic_le_plus_l:
32 \forall m:nat.monotonic nat le (\lambda n.plus n m).
34 rewrite < sym_plus.rewrite < sym_plus m.
35 apply le_plus_r.assumption.
38 theorem le_plus_l: \forall p,n,m:nat. le n m \to le (plus n p) (plus m p)
39 \def monotonic_le_plus_l.
41 theorem le_plus: \forall n1,n2,m1,m2:nat. le n1 n2 \to le m1 m2
42 \to le (plus n1 m1) (plus n2 m2).
44 apply trans_le ? (plus n2 m1).
45 apply le_plus_l.assumption.
46 apply le_plus_r.assumption.
49 theorem le_plus_n :\forall n,m:nat. le m (plus n m).
50 intros.change with le (plus O m) (plus n m).
51 apply le_plus_l.apply le_O_n.
54 theorem eq_plus_to_le: \forall n,m,p:nat.eq nat n (plus m p)
62 theorem monotonic_le_times_r:
63 \forall n:nat.monotonic nat le (\lambda m.times n m).
64 simplify.intros.elim n.
65 simplify.apply le_O_n.
66 simplify.apply le_plus.
71 theorem le_times_r: \forall p,n,m:nat. le n m \to le (times p n) (times p m)
72 \def monotonic_le_times_r.
74 theorem monotonic_le_times_l:
75 \forall m:nat.monotonic nat le (\lambda n.times n m).
77 rewrite < sym_times.rewrite < sym_times m.
78 apply le_times_r.assumption.
81 theorem le_times_l: \forall p,n,m:nat. le n m \to le (times n p) (times m p)
82 \def monotonic_le_times_l.
84 theorem le_times: \forall n1,n2,m1,m2:nat. le n1 n2 \to le m1 m2
85 \to le (times n1 m1) (times n2 m2).
87 apply trans_le ? (times n2 m1).
88 apply le_times_l.assumption.
89 apply le_times_r.assumption.
92 theorem le_times_n: \forall n,m:nat.le (S O) n \to le m (times n m).
93 intros.elim H.simplify.
94 elim (plus_n_O ?).apply le_n.
95 simplify.rewrite < sym_plus.apply le_plus_n.