X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fexp.ma;h=c6e2a008ba565a043360d1bad656b1fe037b7141;hb=dc74ed7c4af1aa9b90fc5d2f0a86bd7825696e71;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..c6e2a008b 100644
--- 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".
 
@@ -22,7 +20,7 @@ let rec exp n m on m\def
  [ O \Rightarrow (S O)
  | (S p) \Rightarrow (times n (exp n p)) ].
 
-interpretation "natural exponent" 'exp a b = (cic:/matita/nat/exp/exp.con a b).
+interpretation "natural exponent" 'exp a b = (exp a b).
 
 theorem exp_plus_times : \forall n,p,q:nat. 
 n \sup (p + q) = (n \sup p) * (n \sup q).
@@ -40,6 +38,19 @@ 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.
+reflexivity.
+qed.
+
 theorem exp_exp_times : \forall n,p,q:nat. 
 (n \sup p) \sup q = n \sup (p * q).
 intros.
@@ -113,7 +124,7 @@ 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.
@@ -140,7 +151,102 @@ 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.
+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 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.
+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.
+
+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.
+intros.elim p
+  [simplify.reflexivity
+  |simplify.
+   rewrite > assoc_times.
+   rewrite < assoc_times in ⊢ (? ? (? ? %) ?).
+   rewrite < sym_times in ⊢ (? ? (? ? (? % ?)) ?).
+   rewrite > assoc_times in ⊢ (? ? (? ? %) ?).
+   rewrite < assoc_times.
+   rewrite < H.
+   reflexivity
+  ]
+qed.
+
+theorem monotonic_exp1: \forall n.
+monotonic nat le (\lambda x.(exp x n)).
+unfold monotonic. intros.
+simplify.elim n
+  [apply le_n
+  |simplify.
+   apply le_times;assumption
+  ]
+qed.
+  
    
    
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