(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/log".
-
include "datatypes/constructors.ma".
include "nat/minimization.ma".
include "nat/relevant_equations.ma".
]
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.
]
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.
[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
]
]