X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Flibrary%2Fnat%2Flog.ma;fp=matita%2Flibrary%2Fnat%2Flog.ma;h=4c326ef25dcc19fe187ab98ef8e1cf04e1d584c6;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

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+(**************************************************************************)
+(*       ___	                                                            *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        Matita is distributed under the terms of the          *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+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).
+
+theorem le_exp_log: \forall p,n. O < n \to
+exp p (log p n) \le n.
+intros.
+apply leb_true_to_le.
+unfold log.
+apply (f_max_true (\lambda x.leb (exp p x) n)).
+apply (ex_intro ? ? O).
+split
+  [apply le_O_n
+  |apply le_to_leb_true.simplify.assumption
+  ]
+qed.
+
+theorem log_SO: \forall n. S O < n \to log n (S O) = O.
+intros.
+apply sym_eq.apply le_n_O_to_eq.
+apply (le_exp_to_le n)
+  [assumption
+  |simplify in ⊢ (? ? %).
+   apply le_exp_log.
+   apply le_n
+  ]
+qed.
+
+theorem lt_to_log_O: \forall n,m. O < m \to m < n \to log n m = O.
+intros.
+apply sym_eq.apply le_n_O_to_eq.
+apply le_S_S_to_le.
+apply (lt_exp_to_lt n)
+  [apply (le_to_lt_to_lt ? m);assumption
+  |simplify in ⊢ (? ? %).
+   rewrite < times_n_SO.
+   apply (le_to_lt_to_lt ? m)
+    [apply le_exp_log.assumption
+    |assumption
+    ]
+  ]
+qed.
+
+theorem lt_log_n_n: \forall p, n. S O < p \to O < n \to log p n < n.
+intros.
+cut (log p n \le n)
+  [elim (le_to_or_lt_eq ? ? Hcut)
+    [assumption
+    |absurd (exp p n \le n)
+      [rewrite < H2 in ⊢ (? (? ? %) ?).
+       apply le_exp_log.
+       assumption
+      |apply lt_to_not_le.
+       apply lt_m_exp_nm.
+       assumption
+      ]
+    ]
+  |unfold log.apply le_max_n
+  ]
+qed.
+
+theorem lt_O_log: \forall p,n. O < n \to p \le n \to O < log p n.
+intros.
+unfold log.
+apply not_lt_to_le.
+intro.
+apply (leb_false_to_not_le ? ? ? H1).
+rewrite > (exp_n_SO p).
+apply (lt_max_to_false ? ? ? H2).
+assumption.
+qed.
+
+theorem le_log_n_n: \forall p,n. S O < p \to log p n \le n.
+intros.
+cases n
+  [apply le_n
+  |apply lt_to_le.
+   apply lt_log_n_n
+    [assumption|apply lt_O_S]
+  ]
+qed.
+
+theorem lt_exp_log: \forall p,n. S O < p \to n < exp p (S (log p n)).
+intros.cases n
+  [simplify.rewrite < times_n_SO.apply lt_to_le.assumption
+  |apply not_le_to_lt.
+   apply leb_false_to_not_le.
+   apply (lt_max_to_false ? (S n1) (S (log p (S n1))))
+    [apply le_S_S.apply le_n
+    |apply lt_log_n_n
+      [assumption|apply lt_O_S]
+    ]
+  ]
+qed.
+
+theorem log_times1: \forall p,n,m. S O < p \to O < n \to O < m \to
+log p (n*m) \le S(log p n+log p m).
+intros.
+unfold in ⊢ (? (% ? ?) ?).
+apply f_false_to_le_max
+  [apply (ex_intro ? ? O).
+   split
+    [apply le_O_n
+    |apply le_to_leb_true.
+     simplify.
+     rewrite > times_n_SO.
+     apply le_times;assumption
+    ]
+  |intros.
+   apply lt_to_leb_false.
+   apply (lt_to_le_to_lt ? ((exp p (S(log p n)))*(exp p (S(log p m)))))
+    [apply lt_times;apply lt_exp_log;assumption
+    |rewrite < exp_plus_times.
+     apply le_exp
+      [apply lt_to_le.assumption
+      |simplify.
+       rewrite < plus_n_Sm.
+       assumption
+      ]
+    ]
+  ]
+qed.
+  
+theorem log_times: \forall p,n,m.S O < p \to log p (n*m) \le S(log p n+log p m).
+intros.
+cases n
+  [apply le_O_n
+  |cases m
+    [rewrite < times_n_O.
+     apply le_O_n
+    |apply log_times1
+      [assumption
+      |apply lt_O_S
+      |apply lt_O_S
+      ]
+    ]
+  ]
+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.
+unfold log in ⊢ (? ? (% ? ?) ?).
+apply max_spec_to_max.
+unfold max_spec.
+split
+  [split
+    [elim n
+      [simplify.
+       rewrite < plus_n_O.
+       apply le_log_n_n.
+       assumption
+      |simplify.
+       rewrite > assoc_times.
+       apply (trans_le ? ((S(S O))*(p\sup n1*m)))
+        [apply le_S_times_SSO
+          [rewrite > (times_n_O O) in ⊢ (? % ?).
+           apply lt_times
+            [apply lt_O_exp.
+             apply lt_to_le.
+             assumption
+            |assumption
+            ]
+          |assumption
+          ]
+        |apply le_times
+          [assumption
+          |apply le_n
+          ]
+        ]
+      ]
+    |simplify.
+     apply le_to_leb_true.
+     rewrite > exp_plus_times.
+     apply le_times_r.
+     apply le_exp_log.
+     assumption
+    ]
+  |intros.
+   simplify.
+   apply lt_to_leb_false.
+   apply (lt_to_le_to_lt ? ((exp p n)*(exp p (S(log p m)))))
+    [apply lt_times_r1
+      [apply lt_O_exp.apply lt_to_le.assumption
+      |apply lt_exp_log.assumption
+      ]
+    |rewrite < exp_plus_times.
+     apply le_exp
+      [apply lt_to_le.assumption
+      |rewrite < plus_n_Sm.
+       assumption
+      ]
+    ]
+  ]
+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
+  [simplify in ⊢ (? (? ? %) ?).
+   rewrite > log_SO
+    [apply le_O_n
+    |assumption
+    ]
+  |simplify.
+   apply (trans_le ? (S (log p n+log p (n\sup n1))))
+    [apply log_times.assumption
+    |apply le_S_S.
+     apply le_plus_r.
+     assumption
+    ]
+  ]
+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.
+
+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
+      |rewrite < times_n_SO.
+       assumption
+      ]
+    |assumption
+    |assumption
+    ]
+  |apply le_log
+    [assumption
+    |rewrite > (div_mod n m) in ⊢ (? ? %)
+      [apply le_plus_n_r
+      |assumption
+      ]
+    ]
+  ]
+qed.
+
+theorem log_n_n: \forall n. S O < n \to log n n = S O.
+intros.
+rewrite > exp_n_SO in ⊢ (? ? (? ? %) ?).
+rewrite > times_n_SO in ⊢ (? ? (? ? %) ?).
+rewrite > log_exp
+  [rewrite > log_SO
+    [reflexivity
+    |assumption
+    ]
+  |assumption
+  |apply le_n
+  ]
+qed.
+
+theorem log_i_SSOn: \forall n,i. S O < n \to n < i \to i \le ((S(S O))*n) \to
+log i ((S(S O))*n) = S O.
+intros.
+apply antisymmetric_le
+  [apply not_lt_to_le.intro.
+   apply (lt_to_not_le ((S(S O)) * n) (exp i (S(S O))))
+    [rewrite > exp_SSO.
+     apply lt_times
+      [apply (le_to_lt_to_lt ? n);assumption
+      |assumption
+      ]
+    |apply (trans_le ? (exp i (log i ((S(S O))*n))))
+      [apply le_exp
+        [apply (ltn_to_ltO ? ? H1)
+        |assumption
+        ]
+      |apply le_exp_log.
+       rewrite > (times_n_O O) in ⊢ (? % ?).
+       apply lt_times
+        [apply lt_O_S
+        |apply lt_to_le.assumption
+        ]
+      ]
+    ]
+  |apply (trans_le ? (log i i))
+    [rewrite < (log_n_n i) in ⊢ (? % ?)
+      [apply le_log
+        [apply (trans_lt ? n);assumption
+        |apply le_n
+        ]
+      |apply (trans_lt ? n);assumption
+      ]
+    |apply le_log
+      [apply (trans_lt ? n);assumption
+      |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