Restructuring.

[helm.git] / helm / software / matita / library / nat / log.ma

diff --git a/helm/software/matita/library/nat/log.ma b/helm/software/matita/library/nat/log.ma
index 04ab67d5b6b5e2d7281260721459bb754161a360..d4e1b2bd991ce58ef77329a8e2bde2dfea7aad7a 100644 (file)
--- a/helm/software/matita/library/nat/log.ma
+++ b/helm/software/matita/library/nat/log.ma
@@ -18,6 +18,8 @@ include "datatypes/constructors.ma".
 include "nat/minimization.ma".
 include "nat/relevant_equations.ma".
 include "nat/primes.ma".
+include "nat/iteration2.ma".
+include "nat/div_and_mod_diseq.ma".
 
 definition log \def \lambda p,n.
 max n (\lambda x.leb (exp p x) n).
@@ -214,6 +216,20 @@ split
   ]
 qed.
 
+theorem eq_log_exp: \forall p,n.S O < p \to
+log p (exp p n)=n.
+intros.
+rewrite > times_n_SO in ⊢ (? ? (? ? %) ?).
+rewrite > log_exp
+  [rewrite > log_SO
+    [rewrite < plus_n_O.reflexivity
+    |assumption
+    ]
+  |assumption
+  |apply le_n
+  ]
+qed.
+
 theorem log_exp1: \forall p,n,m.S O < p \to
 log p (exp n m) \le m*S(log p n).
 intros.elim m
@@ -232,6 +248,29 @@ intros.elim m
   ]
 qed.
     
+theorem log_exp2: \forall p,n,m.S O < p \to O < n \to
+m*(log p n) \le log p (exp n m).
+intros.
+apply le_S_S_to_le.
+apply (lt_exp_to_lt p)
+  [assumption
+  |rewrite > sym_times.
+   rewrite < exp_exp_times.
+   apply (le_to_lt_to_lt ? (exp n m))
+    [elim m
+      [simplify.apply le_n
+      |simplify.apply le_times
+        [apply le_exp_log.
+         assumption
+        |assumption
+        ]
+      ]
+    |apply lt_exp_log.
+     assumption
+    ]
+  ]
+qed.
+
 theorem le_log: \forall p,n,m. S O < p \to O < n \to n \le m \to 
 log p n \le log p m.
 intros.
@@ -305,3 +344,68 @@ apply antisymmetric_le
     ]
   ]
 qed.
+
+theorem exp_n_O: \forall n. O < n \to exp O n = O.
+intros.apply (lt_O_n_elim ? H).intros.
+simplify.reflexivity.
+qed.
+
+theorem tech1: \forall n,i.O < n \to
+(exp (S n) (S(S i)))/(exp n (S i)) \le ((exp n i) + (exp (S n) (S i)))/(exp n i).
+intros.
+simplify in ⊢ (? (? ? %) ?).
+rewrite < eq_div_div_div_times
+  [apply monotonic_div
+    [apply lt_O_exp.assumption
+    |apply le_S_S_to_le.
+     apply lt_times_to_lt_div
+      [assumption
+      |change in ⊢ (? % ?) with ((exp (S n) (S i)) + n*(exp (S n) (S i))).
+      
+      
+  |apply (trans_le ? ((n)\sup(i)*(S n)\sup(S i)/(n)\sup(S i)))
+    [apply le_times_div_div_times.
+     apply lt_O_exp.assumption
+    |apply le_times_to_le_div2
+      [apply lt_O_exp.assumption
+      |simplify.
+*)
+(* falso 
+theorem tech1: \forall a,b,n,m.O < m \to
+n/m \le b \to (a*n)/m \le a*b.
+intros.
+apply le_times_to_le_div2
+  [assumption
+  |
+*)
+
+theorem tech2: \forall n,m. O < n \to 
+(exp (S n) m) / (exp n m) \le (n + m)/n.
+intros.
+elim m
+  [rewrite < plus_n_O.simplify.
+   rewrite > div_n_n.apply le_n
+  |apply le_times_to_le_div
+    [assumption
+    |apply (trans_le ? (n*(S n)\sup(S n1)/(n)\sup(S n1)))
+      [apply le_times_div_div_times.
+       apply lt_O_exp
+      |simplify in ⊢ (? (? ? %) ?).
+       rewrite > sym_times in ⊢ (? (? ? %) ?). 
+       rewrite < eq_div_div_div_times
+        [apply le_times_to_le_div2
+          [assumption
+          |
+      
+      
+theorem le_log_sigma_p:\forall n,m,p. O < m \to S O < p \to
+log p (exp n m) \le sigma_p n (\lambda i.true) (\lambda i. (m / i)).
+intros.
+elim n
+  [rewrite > exp_n_O
+    [simplify.apply le_n
+    |assumption
+    ]
+  |rewrite > true_to_sigma_p_Sn
+    [apply (trans_le ? (m/n1+(log p (exp n1 m))))
+      [