fixed some notational collisions

[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..c9f2c6984ee6d31aeb1ddc8ab9b96b936b19a9e0 100644 (file)
--- a/helm/software/matita/library/nat/exp.ma
+++ b/helm/software/matita/library/nat/exp.ma
@@ -12,8 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/exp".
-
 include "nat/div_and_mod.ma".
 include "nat/lt_arith.ma".
 
@@ -153,6 +151,15 @@ 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 +184,11 @@ 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.
@@ -193,6 +205,23 @@ elim (le_to_or_lt_eq n m)
     ]
   ]
 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.