upper bound for logarithmic summation

[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 193478c0fd6706ddc075aa2d6402051f553489e8..25c81c0697bdd167b48883261e1fa5af5c62c0aa 100644 (file)
--- a/helm/software/matita/library/nat/exp.ma
+++ b/helm/software/matita/library/nat/exp.ma
@@ -40,6 +40,19 @@ 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.
@@ -140,6 +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.
@@ -164,8 +186,52 @@ 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.
+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 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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