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 74a3be71f63d202ed5118663088ac238688d142d..25c81c0697bdd167b48883261e1fa5af5c62c0aa 100644 (file)
--- a/helm/software/matita/library/nat/exp.ma
+++ b/helm/software/matita/library/nat/exp.ma
@@ -40,6 +40,13 @@ theorem exp_n_SO : \forall n:nat. n = n \sup (S O).
 intro.simplify.rewrite < times_n_SO.reflexivity.
 qed.
 
+theorem exp_SO_n : \forall n:nat. S O = (S O) \sup n.
+intro.elim n
+  [reflexivity
+  |simplify.rewrite < plus_n_O.assumption
+  ]
+qed.
+
 theorem exp_SSO: \forall n. exp n (S(S O)) = n*n.
 intro.simplify.
 rewrite < times_n_SO.
@@ -146,6 +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.
@@ -170,6 +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.