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/orders_op".
17 include "higher_order_defs/functions.ma".
18 include "nat/times.ma".
19 include "nat/orders.ma".
22 theorem monotonic_le_plus_r:
23 \forall n:nat.monotonic nat le (\lambda m.plus n m).
24 simplify.intros.elim n.
26 simplify.apply le_S_S.assumption.
29 theorem le_plus_r: \forall p,n,m:nat. le n m \to le (plus p n) (plus p m)
30 \def monotonic_le_plus_r.
32 theorem monotonic_le_plus_l:
33 \forall m:nat.monotonic nat le (\lambda n.plus n m).
35 rewrite < sym_plus.rewrite < sym_plus m.
36 apply le_plus_r.assumption.
39 theorem le_plus_l: \forall p,n,m:nat. le n m \to le (plus n p) (plus m p)
40 \def monotonic_le_plus_l.
42 theorem le_plus: \forall n1,n2,m1,m2:nat. le n1 n2 \to le m1 m2
43 \to le (plus n1 m1) (plus n2 m2).
45 apply trans_le ? (plus n2 m1).
46 apply le_plus_l.assumption.
47 apply le_plus_r.assumption.
50 theorem le_plus_n :\forall n,m:nat. le m (plus n m).
51 intros.change with le (plus O m) (plus n m).
52 apply le_plus_l.apply le_O_n.
55 theorem eq_plus_to_le: \forall n,m,p:nat.n=plus m p
63 theorem monotonic_le_times_r:
64 \forall n:nat.monotonic nat le (\lambda m.times n m).
65 simplify.intros.elim n.
66 simplify.apply le_O_n.
67 simplify.apply le_plus.
72 theorem le_times_r: \forall p,n,m:nat. le n m \to le (times p n) (times p m)
73 \def monotonic_le_times_r.
75 theorem monotonic_le_times_l:
76 \forall m:nat.monotonic nat le (\lambda n.times n m).
78 rewrite < sym_times.rewrite < sym_times m.
79 apply le_times_r.assumption.
82 theorem le_times_l: \forall p,n,m:nat. le n m \to le (times n p) (times m p)
83 \def monotonic_le_times_l.
85 theorem le_times: \forall n1,n2,m1,m2:nat. le n1 n2 \to le m1 m2
86 \to le (times n1 m1) (times n2 m2).
88 apply trans_le ? (times n2 m1).
89 apply le_times_l.assumption.
90 apply le_times_r.assumption.
93 theorem le_times_n: \forall n,m:nat.le (S O) n \to le m (times n m).
94 intros.elim H.simplify.
95 elim (plus_n_O ?).apply le_n.
96 simplify.rewrite < sym_plus.apply le_plus_n.