(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/exp".
-
include "nat/div_and_mod.ma".
include "nat/lt_arith.ma".
]
qed.
+theorem le_exp_to_le1 : \forall n,m,p.O < p \to exp n p \leq exp m p \to n \leq m.
+intros;apply not_lt_to_le;intro;apply (lt_to_not_le ? ? ? H1);
+apply lt_exp1;assumption.
+qed.
+
theorem lt_exp_to_lt:
\forall a,n,m. S O < a \to exp a n < exp a m \to n < m.
intros.
]
]
qed.
+
+theorem lt_exp_to_lt1:
+\forall a,n,m. O < a \to exp n a < exp m a \to n < m.
+intros.
+elim (le_to_or_lt_eq n m)
+ [assumption
+ |apply False_ind.
+ apply (lt_to_not_eq ? ? H1).
+ rewrite < H2.
+ reflexivity
+ |apply (le_exp_to_le1 ? ? a)
+ [assumption
+ |apply lt_to_le.
+ assumption
+ ]
+ ]
+qed.
theorem times_exp:
\forall n,m,p. exp n p * exp m p = exp (n*m) p.