X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Flog.ma;h=39d9b2ecdfae7ecc55209fd8e979581fd28dbbc9;hb=10f29fdd78ee089a9a94446207b543d33d6c851c;hp=04ab67d5b6b5e2d7281260721459bb754161a360;hpb=42f80e6de0618ef919d72557e179072a0c8dc771;p=helm.git

diff --git a/helm/software/matita/library/nat/log.ma b/helm/software/matita/library/nat/log.ma
index 04ab67d5b..39d9b2ecd 100644
--- 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).
@@ -157,6 +159,50 @@ 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.
@@ -214,6 +260,20 @@ 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
@@ -232,20 +292,65 @@ intros.elim m
   ]
 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.
+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
-  |apply (le_to_lt_to_lt ? n)
-    [apply le_exp_log.
+  |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
-    |apply (le_to_lt_to_lt ? m)
+    ]
+  ]
+qed.
+
+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_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.
@@ -292,16 +397,78 @@ 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
       ]
     ]
   ]
 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.
+     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.
+
+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))))
+      [
+*)
\ No newline at end of file