Complete proof of Bertrand for n >= 256.

[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 d4e1b2bd991ce58ef77329a8e2bde2dfea7aad7a..4c326ef25dcc19fe187ab98ef8e1cf04e1d584c6 100644 (file)
--- a/helm/software/matita/library/nat/log.ma
+++ b/helm/software/matita/library/nat/log.ma
@@ -12,8 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/log".
-
 include "datatypes/constructors.ma".
 include "nat/minimization.ma".
 include "nat/relevant_equations.ma".
@@ -159,6 +157,50 @@ cases n
   ]
 qed.
 
+theorem log_times_l: \forall p,n,m.O < n \to O < m \to S O < p \to 
+log p n+log p m \le log p (n*m) .
+intros.
+unfold log in ⊢ (? ? (% ? ?)).
+apply f_m_to_le_max
+  [elim H
+    [rewrite > log_SO
+      [simplify.
+       rewrite < plus_n_O.
+       apply le_log_n_n.
+       assumption
+      |assumption
+      ]
+    |elim H1
+      [rewrite > log_SO
+        [rewrite < plus_n_O.
+         rewrite < times_n_SO.
+         apply le_log_n_n.
+         assumption
+        |assumption
+        ]
+      |apply (trans_le ? (S n1 + S n2))
+        [apply le_plus;apply le_log_n_n;assumption
+        |simplify.
+         apply le_S_S.
+         rewrite < plus_n_Sm.
+         change in ⊢ (? % ?) with ((S n1)+n2).
+         rewrite > sym_plus.
+         apply le_plus_r.
+         change with (n1 < n1*S n2).
+         rewrite > times_n_SO in ⊢ (? % ?).
+         apply lt_times_r1
+          [assumption
+          |apply le_S_S.assumption
+          ]
+        ]
+      ]
+    ]
+  |apply le_to_leb_true.
+   rewrite > exp_plus_times.
+   apply le_times;apply le_exp_log;assumption
+  ]
+qed.
+
 theorem log_exp: \forall p,n,m.S O < p \to O < m \to
 log p ((exp p n)*m)=n+log p m.
 intros.
@@ -271,20 +313,42 @@ apply (lt_exp_to_lt p)
   ]
 qed.
 
-theorem le_log: \forall p,n,m. S O < p \to O < n \to n \le m \to 
+lemma le_log_plus: \forall p,n.S O < p \to log p n \leq log p (S n).
+intros;apply (bool_elim ? (leb (p*(exp p n)) (S n)))
+  [simplify;intro;rewrite > H1;simplify;apply (trans_le ? n)
+    [apply le_log_n_n;assumption
+    |apply le_n_Sn]
+  |intro;unfold log;simplify;rewrite > H1;simplify;apply le_max_f_max_g;
+   intros;apply le_to_leb_true;constructor 2;apply leb_true_to_le;assumption]
+qed.
+
+theorem le_log: \forall p,n,m. S O < p \to n \le m \to 
 log p n \le log p m.
+intros.elim H1
+  [constructor 1
+  |apply (trans_le ? ? ? H3);apply le_log_plus;assumption]
+qed.
+         
+theorem log_div: \forall p,n,m. S O < p \to O < m \to m \le n \to
+log p (n/m) \le log p n -log p m.
 intros.
-apply le_S_S_to_le.
-apply (lt_exp_to_lt p)
-  [assumption
-  |apply (le_to_lt_to_lt ? n)
-    [apply le_exp_log.
-     assumption
-    |apply (le_to_lt_to_lt ? m)
+apply le_plus_to_minus_r.
+apply (trans_le ? (log p ((n/m)*m)))
+  [apply log_times_l
+    [apply le_times_to_le_div
       [assumption
-      |apply lt_exp_log.
+      |rewrite < times_n_SO.
        assumption
       ]
+    |assumption
+    |assumption
+    ]
+  |apply le_log
+    [assumption
+    |rewrite > (div_mod n m) in ⊢ (? ? %)
+      [apply le_plus_n_r
+      |assumption
+      ]
     ]
   ]
 qed.
@@ -331,14 +395,12 @@ apply antisymmetric_le
     [rewrite < (log_n_n i) in ⊢ (? % ?)
       [apply le_log
         [apply (trans_lt ? n);assumption
-        |apply (ltn_to_ltO ? ? H1)
         |apply le_n
         ]
       |apply (trans_lt ? n);assumption
       ]
     |apply le_log
       [apply (trans_lt ? n);assumption
-      |apply (ltn_to_ltO ? ? H1)
       |assumption
       ]
     ]
@@ -350,6 +412,7 @@ 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.
@@ -358,9 +421,8 @@ 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 lt_times_to_lt_div.
+     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)))
@@ -369,15 +431,13 @@ rewrite < eq_div_div_div_times
     |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.
@@ -409,3 +469,4 @@ elim n
   |rewrite > true_to_sigma_p_Sn
     [apply (trans_le ? (m/n1+(log p (exp n1 m))))
       [
+*)
\ No newline at end of file