X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Flibrary%2Fnat%2Fexp.ma;h=193478c0fd6706ddc075aa2d6402051f553489e8;hb=c3bba4af040f8040e5eae07e70690c52f8c614f8;hp=11d84f74ca7deb3b676007693f72b585c0547435;hpb=7f2444c2670cadafddd8785b687ef312158376b0;p=helm.git

diff --git a/matita/library/nat/exp.ma b/matita/library/nat/exp.ma
index 11d84f74c..193478c0f 100644
--- a/matita/library/nat/exp.ma
+++ b/matita/library/nat/exp.ma
@@ -15,6 +15,7 @@
 set "baseuri" "cic:/matita/nat/exp".
 
 include "nat/div_and_mod.ma".
+include "nat/lt_arith.ma".
 
 let rec exp n m on m\def 
  match m with 
@@ -54,7 +55,7 @@ apply le_times.assumption.assumption.
 qed.
 
 theorem lt_m_exp_nm: \forall n,m:nat. (S O) < n \to m < n \sup m.
-intros.elim m.simplify.unfold lt.reflexivity.
+intros.elim m.simplify.unfold lt.apply le_n.
 simplify.unfold lt.
 apply (trans_le ? ((S(S O))*(S n1))).
 simplify.
@@ -95,3 +96,76 @@ qed.
 variant inj_exp_r: \forall p:nat. (S O) < p \to \forall n,m:nat.
 p \sup n = p \sup m \to n = m \def
 injective_exp_r.
+
+theorem le_exp: \forall n,m,p:nat. O < p \to n \le m \to exp p n \le exp p m.
+apply nat_elim2
+  [intros.
+   apply lt_O_exp.assumption
+  |intros.
+   apply False_ind.
+   apply (le_to_not_lt ? ? ? H1).
+   apply le_O_n
+  |intros.
+   simplify.
+   apply le_times
+    [apply le_n
+    |apply H[assumption|apply le_S_S_to_le.assumption]
+    ]
+  ]
+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.
+   apply (lt_O_n_elim ? H1).intro.
+   simplify.unfold lt.
+   rewrite > times_n_SO.
+   apply le_times
+    [assumption
+    |apply lt_O_exp.
+     apply (trans_lt ? (S O))[apply le_n|assumption]
+    ]
+  |intros.
+   apply False_ind.
+   apply (le_to_not_lt ? ? ? H1).
+   apply le_O_n
+  |intros.simplify.
+   apply lt_times_r1
+    [apply (trans_lt ? (S O))[apply le_n|assumption]
+    |apply H
+      [apply H1
+      |apply le_S_S_to_le.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.
+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.
+
+
+     
+   
+   
+   
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