(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/log".
-
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).
]
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
-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.
[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
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