X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fexp.ma;h=49efe8525504d4f431e81de36ea702f7c9d46f76;hb=d6c41ecd6b4d6944becbef2f7f6a625c51d8867c;hp=c69be6b5fda43ece413ac03cbfab7103a08aa0c1;hpb=6423f1b6e3056883016598e454c55cab1004dfd2;p=helm.git

diff --git a/helm/software/matita/library/nat/exp.ma b/helm/software/matita/library/nat/exp.ma
index c69be6b5f..49efe8525 100644
--- a/helm/software/matita/library/nat/exp.ma
+++ b/helm/software/matita/library/nat/exp.ma
@@ -40,6 +40,12 @@ theorem exp_n_SO : \forall n:nat. n = n \sup (S O).
 intro.simplify.rewrite < times_n_SO.reflexivity.
 qed.
 
+theorem exp_SSO: \forall n. exp n (S(S O)) = n*n.
+intro.simplify.
+rewrite < times_n_SO.
+reflexivity.
+qed.
+
 theorem exp_exp_times : \forall n,p,q:nat. 
 (n \sup p) \sup q = n \sup (p * q).
 intros.
@@ -113,7 +119,7 @@ apply nat_elim2
     ]
   ]
 qed.
- 
+
 theorem lt_exp: \forall n,m,p:nat. S O < p \to n < m \to exp p n < exp p m.
 apply nat_elim2
   [intros.
@@ -140,6 +146,46 @@ apply nat_elim2
   ]
 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.
+apply nat_elim2;intros
+  [apply le_O_n
+  |apply False_ind.
+   apply (le_to_not_lt ? ? H1).
+   simplify.
+   rewrite > times_n_SO.
+   apply lt_to_le_to_lt_times
+    [assumption
+    |apply lt_O_exp.apply lt_to_le.assumption
+    |apply lt_O_exp.apply lt_to_le.assumption
+    ]
+  |simplify in H2.
+   apply le_S_S.
+   apply H
+    [assumption
+    |apply (le_times_to_le a)
+      [apply lt_to_le.assumption|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.
+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_le a)
+    [assumption
+    |apply lt_to_le.
+     assumption
+    ]
+  ]
+qed.