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/times.ma".
18 include "nat/orders.ma".
19 include "higher_order_defs/functions.ma".
21 let rec exp n m on m\def
24 | (S p) \Rightarrow (times n (exp n p)) ].
26 theorem exp_plus_times : \forall n,p,q:nat.
27 eq nat (exp n (plus p q)) (times (exp n p) (exp n q)).
29 simplify.rewrite < plus_n_O.reflexivity.
30 simplify.rewrite > H.symmetry.
34 theorem exp_n_O : \forall n:nat. eq nat (S O) (exp n O).
35 intro.simplify.reflexivity.
38 theorem exp_n_SO : \forall n:nat. eq nat n (exp n (S O)).
39 intro.simplify.rewrite < times_n_SO.reflexivity.
42 theorem exp_exp_times : \forall n,p,q:nat.
43 eq nat (exp (exp n p) q) (exp n (times p q)).
45 elim q.simplify.rewrite < times_n_O.simplify.reflexivity.
46 simplify.rewrite > H.rewrite < exp_plus_times.
47 rewrite < times_n_Sm.reflexivity.
50 theorem lt_O_exp: \forall n,m:nat. O < n \to O < exp n m.
51 intros.elim m.simplify.apply le_n.
52 simplify.rewrite > times_n_SO.
53 apply le_times.assumption.assumption.
56 theorem lt_m_exp_nm: \forall n,m:nat. (S O) < n \to m < exp n m.
57 intros.elim m.simplify.reflexivity.
59 apply trans_le ? ((S(S O))*(S n1)).
61 rewrite < plus_n_Sm.apply le_S_S.apply le_S_S.
64 apply le_times.assumption.assumption.
67 theorem exp_to_eq_O: \forall n,m:nat. (S O) < n
68 \to exp n m = (S O) \to m = O.
69 intros.apply antisym_le.apply le_S_S_to_le.
70 rewrite < H1.change with m < exp n m.
71 apply lt_m_exp_nm.assumption.
75 theorem injective_exp_r: \forall n:nat. (S O) < n \to
76 injective nat nat (\lambda m:nat. exp n m).
78 apply nat_elim2 (\lambda x,y.exp n x = exp n y \to x = y).
79 intros.apply sym_eq.apply exp_to_eq_O n.assumption.
80 rewrite < H1.reflexivity.
81 intros.apply exp_to_eq_O n.assumption.assumption.
84 (* esprimere inj_times senza S *)
85 cut \forall a,b:nat.O < n \to n*a=n*b \to a=b.
86 apply Hcut.simplify. apply le_S_S_to_le. apply le_S. assumption.
90 intros.apply False_ind.apply not_le_Sn_O O H3.
91 intros.apply inj_times_r m1.assumption.
94 variant inj_exp_r: \forall p:nat. (S O) < p \to \forall n,m:nat.
95 (exp p n) = (exp p m) \to n = m \def