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/exp".
17 include "nat/div_and_mod.ma".
19 let rec exp n m on m\def
22 | (S p) \Rightarrow (times n (exp n p)) ].
24 theorem exp_plus_times : \forall n,p,q:nat.
25 eq nat (exp n (plus p q)) (times (exp n p) (exp n q)).
27 simplify.rewrite < plus_n_O.reflexivity.
28 simplify.rewrite > H.symmetry.
32 theorem exp_n_O : \forall n:nat. eq nat (S O) (exp n O).
33 intro.simplify.reflexivity.
36 theorem exp_n_SO : \forall n:nat. eq nat n (exp n (S O)).
37 intro.simplify.rewrite < times_n_SO.reflexivity.
40 theorem exp_exp_times : \forall n,p,q:nat.
41 eq nat (exp (exp n p) q) (exp n (times p q)).
43 elim q.simplify.rewrite < times_n_O.simplify.reflexivity.
44 simplify.rewrite > H.rewrite < exp_plus_times.
45 rewrite < times_n_Sm.reflexivity.
48 theorem lt_O_exp: \forall n,m:nat. O < n \to O < exp n m.
49 intros.elim m.simplify.apply le_n.
50 simplify.rewrite > times_n_SO.
51 apply le_times.assumption.assumption.
54 theorem lt_m_exp_nm: \forall n,m:nat. (S O) < n \to m < exp n m.
55 intros.elim m.simplify.reflexivity.
57 apply trans_le ? ((S(S O))*(S n1)).
59 rewrite < plus_n_Sm.apply le_S_S.apply le_S_S.
62 apply le_times.assumption.assumption.
65 theorem exp_to_eq_O: \forall n,m:nat. (S O) < n
66 \to exp n m = (S O) \to m = O.
67 intros.apply antisym_le.apply le_S_S_to_le.
68 rewrite < H1.change with m < exp n m.
69 apply lt_m_exp_nm.assumption.
73 theorem injective_exp_r: \forall n:nat. (S O) < n \to
74 injective nat nat (\lambda m:nat. exp n m).
76 apply nat_elim2 (\lambda x,y.exp n x = exp n y \to x = y).
77 intros.apply sym_eq.apply exp_to_eq_O n.assumption.
78 rewrite < H1.reflexivity.
79 intros.apply exp_to_eq_O n.assumption.assumption.
82 (* esprimere inj_times senza S *)
83 cut \forall a,b:nat.O < n \to n*a=n*b \to a=b.
84 apply Hcut.simplify. apply le_S_S_to_le. apply le_S. assumption.
88 intros.apply False_ind.apply not_le_Sn_O O H3.
90 apply inj_times_r m1.assumption.
93 variant inj_exp_r: \forall p:nat. (S O) < p \to \forall n,m:nat.
94 (exp p n) = (exp p m) \to n = m \def