Bertrand's conjecture (weak), some work in progress

[helm.git] / helm / software / matita / library / nat / exp.ma

diff --git a/helm/software/matita/library/nat/exp.ma b/helm/software/matita/library/nat/exp.ma
index c94e713b1b0381684f8ec00528a7ae5f6ce04b77..25c81c0697bdd167b48883261e1fa5af5c62c0aa 100644 (file)
--- a/helm/software/matita/library/nat/exp.ma
+++ b/helm/software/matita/library/nat/exp.ma
@@ -153,6 +153,15 @@ apply nat_elim2
   ]
 qed.
 
+theorem lt_exp1: \forall n,m,p:nat. O < p \to n < m \to exp n p < exp m p.
+intros.
+elim H
+  [rewrite < exp_n_SO.rewrite < exp_n_SO.assumption
+  |simplify.
+   apply lt_times;assumption
+  ]
+qed.
+
 theorem le_exp_to_le: 
 \forall a,n,m. S O < a \to exp a n \le exp a m \to n \le m.
 intro.
@@ -177,6 +186,11 @@ apply nat_elim2;intros
   ]
 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.