From: Wilmer Ricciotti <ricciott@cs.unibo.it>
Date: Mon, 14 Jan 2008 07:43:37 +0000 (+0000)
Subject: Chebyshev's upper bound on prim
X-Git-Tag: make_still_working~5675
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=95f5a6fa9824a2dfef74e99379894b08357a16c6;p=helm.git

Chebyshev's upper bound on prim
---

diff --git a/helm/software/matita/library/nat/chebyshev_thm.ma b/helm/software/matita/library/nat/chebyshev_thm.ma
index bf616a17c..28e3ba5d2 100644
--- a/helm/software/matita/library/nat/chebyshev_thm.ma
+++ b/helm/software/matita/library/nat/chebyshev_thm.ma
@@ -249,49 +249,409 @@ intros;apply le_times_to_le_div;
                        |apply le_S_S;assumption]]]]]]]]
 qed.
 
-(*
-  
-theorem le_log_C2_stima : \forall n,b. S O < b \to
+theorem le_log_C2_stima : \forall n,b. S O < b \to b*b < n \to
 log b (C2 n) \leq 
-(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x))) (\lambda x.S O)) +
+(*(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x))) (\lambda x.S O)) +*)
+((S (((S (S (S (S O))))*(S (log b (pred n)))) + 
+              ((S (S (S (S O))))*n)))/(log b n)) +
 (((S (((S (S (S (S O))))*(S (log b (pred n)))) + 
               ((S (S (S (S O))))*n)))/(log b n)) + 
  (((sigma_p n (\lambda x.leb (S n) (x*x)) 
-              (\lambda i.((S (((S (S (S (S O))))*(S (log b (pred n)))) + 
-              ((S (S (S (S O))))*n)))/(log b n))* S (n!/i)))
-  *(S (log b 3)))/n!)).
-elim daemon.
-  
-theorem le_log_C2_sigma_p : \forall n,b. S O < b \to
-log b (C2 n) \leq 
-(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x))) (\lambda x.S O)) +
-(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x))) 
-   (\lambda p.(sigma_p (n+p) (\lambda i.leb p i) 
-       (\lambda i.S ((n+p)!/i*(S (log b 3)))))/(n+p)!)).
-intros;unfold C2;
-apply (trans_le ? (sigma_p (S n) (λx:nat.primeb x∧leb (S n) (x*x)) (λx:nat.1)
-+sigma_p (S n) (λx:nat.primeb x∧leb (S n) (x*x))
- (λi.log b (S (n/i)))))
-  [apply log_pi_p;assumption
+              (\lambda i.((S (((S (S (S (S O))))*(S (log b (pred i)))) + 
+              ((S (S (S (S O))))*i)))/(log b i))* S (n!/i)))
+  *(S (log b (S (S (S O))))))/n!)).intros.
+apply (trans_le ? ? ? (le_log_C2_sigma_p ? ? ?))
+  [assumption
   |apply le_plus
-     [apply le_n
-     |apply le_sigma_p;intros;cut (S (n/i) = (n+i)/i)
-        [rewrite > Hcut;apply le_log_div_sigma_p 
+     [apply (trans_le ? ? ? ? (le_prim_n_stima ? ? ? ?));
+      [unfold prim;apply le_sigma_p1;intros;
+       do 2 rewrite < times_n_SO;elim (primeb i)
+        [elim (leb (S n) (i*i));simplify [apply le_n|apply le_O_n]
+        |simplify;apply le_n]
+      |assumption
+      |apply (trans_le ? ? ? ? H1);apply le_S;apply le_times_n;
+       apply lt_to_le;assumption]
+   |apply le_plus
+     [apply le_prim_n_stima;
+       [assumption
+       |apply (trans_le ? (b*b))
+          [apply le_times_n;apply lt_to_le;assumption
+          |apply lt_to_le;assumption]]
+     |apply monotonic_div
+        [apply lt_O_fact
+        |apply le_times_l;apply le_sigma_p;intros;apply le_times_l;
+         apply le_prim_n_stima
            [assumption
-           |apply prime_to_lt_O;apply primeb_true_to_prime;
-            elim (and_true ? ? H2);assumption
-           |apply le_plus_n
-           |apply le_n]
-        |lapply (prime_to_lt_O i (primeb_true_to_prime ? (andb_true_true ? ? H2))) as H3;
-         apply (div_mod_spec_to_eq (n+i) i (S (n/i)) (n \mod i) ? ((n+i) \mod i))
-           [constructor 1
-              [apply lt_mod_m_m;assumption
-              |simplify;rewrite > assoc_plus;rewrite < div_mod;
-                 [apply sym_plus
+           |apply (le_exp_to_le1 ? ? (S (S O)));
+              [apply le_S;apply le_n
+              |do 2 rewrite > exp_SSO;apply (trans_le ? n)
+                 [apply lt_to_le;assumption
+                 |apply lt_to_le;apply leb_true_to_le;assumption]]]]]]]
+qed.
+
+lemma log_interval : \forall b,k,n. S O < b \to exp b k \leq n \to n < exp b (S k) \to
+                       log b n = k.
+intros 2.elim k
+  [simplify in H2;rewrite < times_n_SO in H2;apply lt_to_log_O;assumption
+  |cut (log b n1 < S (S n))
+     [cut (n < log b n1)
+        [apply antisymmetric_le
+           [apply le_S_S_to_le;assumption
+           |assumption]
+        |apply (trans_le ? (log b (exp b (S n))))
+           [rewrite > eq_log_exp
+              [apply le_n
+              |assumption]
+           |apply le_log;assumption]]
+     |apply le_S_S;apply (trans_le ? (log b (pred (exp b (S (S n))))))
+        [apply le_log
+           [assumption
+           |apply le_S_S_to_le;apply (trans_le ? ? ? H3);
+            rewrite > minus_n_O in \vdash (? ? (? (? %)));
+            rewrite < (eq_minus_S_pred (exp b (S (S n))) O);
+            rewrite > minus_n_O in \vdash (? % ?);
+            apply minus_le_S_minus_S]
+        |unfold log;apply f_false_to_le_max;
+           [apply (ex_intro ? ? (S n));split
+              [apply (trans_le ? (exp b (S n)));
+                 [apply lt_to_le;apply lt_m_exp_nm;assumption
+                 |rewrite > minus_n_O in ⊢ (? ? (? %));
+                  rewrite < eq_minus_S_pred;apply le_plus_to_minus_r;
+                  rewrite > sym_plus;
+                  change in \vdash (? % ?) with (S (O + exp b (S n)));
+                  apply lt_minus_to_plus;
+                  change in ⊢ (? ? (? % ?)) with (b * (exp b (S n)));
+                  rewrite > times_n_SO in \vdash (? ? (? ? %));
+                  rewrite > sym_times in \vdash (? ? (? % ?));
+                  rewrite < distributive_times_minus;unfold lt;
+                  rewrite > times_n_SO in \vdash (? % ?);apply le_times
+                    [apply lt_O_exp;apply (trans_le ? ? ? ? H1);
+                     apply le_S;apply le_n
+                    |apply le_plus_to_minus_r;simplify;assumption]]
+              |apply le_to_leb_true;
+               rewrite > minus_n_O in \vdash (? ? (? %));
+               rewrite < eq_minus_S_pred;apply le_plus_to_minus_r;
+               rewrite > sym_plus;change in \vdash (? % ?) with (S (exp b (S n)));
+               apply lt_exp;
+                 [assumption
+                 |apply le_n]]
+           |intros;apply lt_to_leb_false;unfold lt;
+            rewrite > minus_n_O in \vdash (? (? (? %)) ?);
+            rewrite < eq_minus_S_pred;rewrite < minus_Sn_m
+              [rewrite > minus_S_S;rewrite < minus_n_O;apply le_exp;
+                 [apply (trans_le ? ? ? ? H1);apply le_S;apply le_n
+                 |assumption]
+              |apply lt_O_exp;apply (trans_le ? ? ? ? H1);apply le_S;apply le_n]]]]]
+qed.              
+
+lemma log_strano : \forall b,i.S O < b \to S O < i \to
+                  ((S (S (S (S O)))) * log b (pred i)) + (S (S (S (S (S O))))) \leq
+                  (S (S (S O)))*i.
+alias num (instance 0) = "natural number".
+cut (\forall b,i,k.S O < b \to S O < i \to
+            (exp b k) \leq i-1 \to i-1 < (exp b (S k)) \to
+                  ((S (S (S (S O)))) * log b (pred i)) + (S (S (S (S (S O))))) \leq
+                  (S (S (S O)))*i)
+  [intros;apply (Hcut ? ? (log b (i-1)) H H1);
+     [apply le_exp_log;rewrite > (minus_n_n 1) in \vdash (? % ?);
+      apply lt_plus_to_lt_minus;
+        [apply le_n
+        |rewrite < eq_minus_plus_plus_minus
+           [rewrite > sym_plus;rewrite > eq_minus_plus_plus_minus;
+              [rewrite < minus_n_n;rewrite < plus_n_O;assumption
+              |apply le_n]
+           |apply lt_to_le;assumption]]
+     |apply lt_exp_log;assumption]
+  |intros;rewrite > minus_n_O in ⊢ (? (? (? ? (? ? (? %))) ?) ?);
+   rewrite < eq_minus_S_pred;rewrite > (log_interval ? k)
+     [apply (trans_le ? (3*(exp b k) + 3))
+        [change in \vdash (? (? ? %) ?) with (2+3);
+         rewrite < assoc_plus;apply le_plus_l;
+         elim k
+           [simplify;apply le_S;apply le_n
+           |elim (decidable_eq_nat O n)
+              [rewrite < H5;apply (trans_le ? (3*(exp 2 1)));
+                 [simplify;apply le_n
+                 |apply le_times_r;apply monotonic_exp1;assumption]
+              |rewrite < times_n_Sm;apply (trans_le ? (3*(exp b n) + 4))
+                 [rewrite > assoc_plus;rewrite > sym_plus;apply le_plus_l;
+                  assumption
+                 |rewrite < sym_plus;change in \vdash (? % ?) with (S (3 + (3*(exp b n))));
+                  apply lt_minus_to_plus;
+                  change in ⊢ (? ? (? (? ? %) ?)) with (b*(exp b n));
+                  rewrite > sym_times in \vdash (? ? (? (? ? %) ?));
+                  rewrite < assoc_times;
+                  rewrite > times_n_SO in ⊢ (? ? (? ? (? ? %)));
+                  rewrite < assoc_times;rewrite < distr_times_minus;
+                  apply (trans_le ? (3*2*1))
+                    [simplify;apply le_S;apply le_S;apply le_n
+                    |apply le_times
+                       [apply le_times_r;apply (trans_le ? (exp 2 n))
+                          [rewrite > exp_n_SO in \vdash (? % ?);apply le_exp
+                             [apply le_S;apply le_n
+                             |generalize in match H5;cases n
+                                [intro;elim H6;reflexivity
+                                |intro;apply le_S_S;apply le_O_n]]
+                          |apply monotonic_exp1;assumption]
+                       |apply le_S_S_to_le;rewrite < minus_Sn_m;
+                          [simplify;rewrite < minus_n_O;assumption
+                          |apply lt_to_le;assumption]]]]]]
+        |rewrite > times_n_SO in \vdash (? (? ? %) ?);
+         rewrite < distr_times_plus;apply le_times_r;
+         rewrite < plus_n_SO;apply (trans_le ? (S (i-1)))
+           [apply le_S_S;assumption
+           |rewrite < minus_Sn_m
+              [simplify;rewrite < minus_n_O;apply le_n
+              |apply lt_to_le;assumption]]]
+     |assumption
+     |assumption
+     |assumption]]
+qed.
+                  
+lemma eq_div_div_times : \forall x,y,z.O < z \to O < y \to x/y = (z*x)/(z*y).
+intros.rewrite > (div_mod x y) in \vdash (? ? ? %);
+  [rewrite > distr_times_plus;rewrite > sym_times;rewrite > assoc_times;
+   rewrite > sym_times in ⊢ (? ? ? (? (? (? ? %) ?) ?));
+   rewrite > div_plus_times
+     [reflexivity
+     |generalize in match H;cases z;intros
+        [elim (not_le_Sn_O ? H2)
+        |apply lt_times_r;apply lt_mod_m_m;assumption]]
+  |assumption]
+qed.
+                  
+alias num (instance 0) = "natural number".
+lemma le_sigma_p_lemma1 : \forall n,b. S O < b \to b*b < n \to
+            (sigma_p n (\lambda x.leb (S n) (x*x)) 
+              (\lambda i.((S (((S (S (S (S O))))*(S (log b (pred i)))) + 
+              ((S (S (S (S O))))*i)))/(log b i))* S (n!/i)))
+            \leq ((28 * n * n!)/(pred (log b n))).
+intros.apply (trans_le ? (sigma_p n (\lambda x.leb (S n) (x*x)) (\lambda i. ((7*i)/(log b i))*(S (n!/i)))))
+  [apply le_sigma_p;intros;cut (b \leq i)
+     [cut (1 < i) [|apply (trans_le ? ? ? H Hcut)]
+      apply le_times_l;apply monotonic_div
+        [apply lt_O_log
+           [generalize in match H3;cases i
+              [simplify;intros;destruct H4
+              |intro;apply le_S_S;apply le_O_n]
+           |assumption]
+        |rewrite > sym_times;simplify in ⊢ (? (? (? % ?)) ?);
+         change in ⊢ (? % ?) with (1 + ((4 + (log b (pred i) * 4)) + 4*i));
+         rewrite > assoc_plus;rewrite < assoc_plus;
+         simplify in \vdash (? (? % ?) ?);rewrite < assoc_plus;
+         apply (trans_le ? (3*i + 4*i))
+           [apply le_minus_to_plus;rewrite > eq_minus_plus_plus_minus
+              [rewrite < minus_n_n;rewrite < plus_n_O;
+               rewrite > sym_plus;rewrite > sym_times;apply log_strano
+                 [assumption
+                 |lapply (leb_true_to_le ? ? H3);apply (trans_le ? ? ? H);
+                  apply (le_exp_to_le1 ? ? 2);
+                    [apply le_S_S;apply le_O_n
+                    |apply lt_to_le;do 2 rewrite > exp_SSO;apply (trans_le ? ? ? H1);
+                     apply lt_to_le;assumption]]
+              |apply le_n]
+           |rewrite > sym_times in \vdash (? (? % ?) ?);
+            rewrite > sym_times in \vdash (? (? ? %) ?);
+            rewrite < distr_times_plus;rewrite > sym_times;apply le_n]]
+     |apply (le_exp_to_le1 ? ? 2);
+        [apply le_S_S;apply le_O_n
+        |apply lt_to_le;do 2 rewrite > exp_SSO;apply (trans_le ? ? ? H1);
+         apply (trans_le ? ? ? ? (leb_true_to_le ? ? H3));apply le_S;
+         apply le_n]]
+  |apply (trans_le ? (sigma_p n (λx:nat.leb (S n) (x*x)) (λi:nat.7*i/log b i*((2*(n!))/i))))
+     [apply le_sigma_p;intros;apply le_times_r;apply (trans_le ? (n!/i + n!/i))
+        [change in \vdash (? % ?) with (1 + n!/i);apply le_plus_l;
+         apply le_times_to_le_div
+           [generalize in match H3;cases i;simplify;intro
+              [destruct H4
+              |apply le_S_S;apply le_O_n]
+           |generalize in match H2;cases n;intro
+              [elim (not_le_Sn_O ? H4)
+              |change in \vdash (? ? %) with ((S n1)*(n1!));apply le_times
+                 [apply lt_to_le;assumption
+                 |apply lt_O_fact]]]
+        |rewrite > plus_n_O in \vdash (? (? ? %) ?);
+         change in \vdash (? % ?) with (2 * (n!/i));
+         apply le_times_div_div_times;
+         generalize in match H3;cases i;simplify;intro
+           [destruct H4
+           |apply le_S_S;apply le_O_n]]
+     |apply (trans_le ? (sigma_p n (\lambda x:nat.leb (S n) (x*x)) (\lambda i.((14*(n!))/log b i))))
+        [apply le_sigma_p;intros;
+         cut (b \leq i)
+           [|apply (le_exp_to_le1 ? ? 2);
+              [apply le_S_S;apply le_O_n
+              |apply lt_to_le;do 2 rewrite > exp_SSO;apply (trans_le ? ? ? H1);
+               apply (trans_le ? ? ? ? (leb_true_to_le ? ? H3));apply le_S;
+               apply le_n]]
+         cut (1 < i)
+           [|apply (trans_le ? ? ? H Hcut)]
+         change in ⊢ (? ? (? % ?)) with ((7*2)*(n!));
+         rewrite > assoc_times in \vdash (? ? (? % ?));
+         apply (trans_le ? ? ? (le_times_div_div_times ? ? ? ?));
+           [apply lt_to_le;assumption
+           |rewrite > (eq_div_div_times ? ? 7)
+              [rewrite > sym_times in ⊢ (? (? (? ? %) ?) ?);
+               rewrite < assoc_times in \vdash (? (? % ?) ?);
+               apply (trans_le ? ? ? (le_div_times_m ? ? ? ? ?))
+                 [apply lt_O_log
+                    [apply lt_to_le;assumption
+                    |assumption]
+                 |unfold lt;rewrite > times_n_SO in \vdash (? % ?);
+                  apply le_times;
+                    [apply le_S_S;apply le_O_n
+                    |apply lt_to_le;assumption]
+                 |apply le_n]
+              |apply le_S_S;apply le_O_n
+              |apply lt_to_le;assumption]]
+        |apply (trans_le ? (sigma_p (S n) (\lambda x.leb (S n) (x*x))
+                      (\lambda i.14*n!/log b i)))
+           [apply (bool_elim ? (leb (S n) (n*n)));intro
+              [rewrite > true_to_sigma_p_Sn
+                 [rewrite > sym_plus;rewrite > plus_n_O in \vdash (? % ?);
+                  apply le_plus_r;apply le_O_n
+                 |assumption]
+              |rewrite > false_to_sigma_p_Sn
+                 [apply le_n
                  |assumption]]
-           |apply div_mod_spec_div_mod;assumption]]]]
+           |apply (trans_le ? ? ? (le_sigma_p_div_log_div_pred_log ? ? ? ? ?));
+              [assumption
+              |apply lt_to_le;assumption
+              |rewrite < assoc_times;
+               rewrite > sym_times in ⊢ (? (? (? % ?) ?) ?);
+               rewrite < assoc_times;apply le_n]]]]]
+qed.
+
+theorem le_log_C2_stima2 : \forall n,b. S O < b \to b*b < n \to
+log b (C2 n) \leq 
+(14*n/(log b n)) + 
+ ((((28*n*n!)/pred (log b n))
+  *(S (log b (S (S (S O))))))/n!).intros.
+apply (trans_le ? ? ? (le_log_C2_stima ? ? H H1));
+rewrite < assoc_plus in \vdash (? % ?);apply le_plus
+  [rewrite > times_SSO_n in \vdash (? % ?);
+   apply (trans_le ? ? ? (le_times_div_div_times ? ? ? ?))
+     [apply lt_O_log
+       [apply (trans_le ? (b*b))
+          [rewrite > times_n_SO;apply le_times;apply lt_to_le;assumption
+          |apply lt_to_le;assumption]
+       |apply (trans_le ? (b*b))
+          [rewrite > times_n_SO in \vdash (? % ?);apply le_times
+             [apply le_n|apply lt_to_le;assumption]
+          |apply lt_to_le;assumption]]
+     |change in \vdash (? ? (? (? % ?) ?)) with (2*7);
+      apply monotonic_div;
+        [apply lt_O_log
+          [apply (trans_le ? (b*b))
+             [rewrite > times_n_SO;apply le_times;apply lt_to_le;assumption
+             |apply lt_to_le;assumption]
+          |apply (trans_le ? (b*b))
+             [rewrite > times_n_SO in \vdash (? % ?);apply le_times
+                [apply le_n|apply lt_to_le;assumption]
+             |apply lt_to_le;assumption]]
+        |rewrite > assoc_times;apply le_times_r;
+         change in \vdash (? (? (? (? ? %) ?)) ?) with (1 + (log b (pred n)));
+         rewrite > distr_times_plus;
+         change in \vdash (? % ?) with (1 + (4*1+4*log b (pred n)+4*n));
+         do 2 rewrite < assoc_plus;simplify in ⊢ (? (? (? % ?) ?) ?);
+         change in ⊢ (? ? %) with ((3+4)*n);rewrite > sym_times in \vdash (? ? %);
+         rewrite > distr_times_plus;
+         rewrite > sym_times in \vdash (? ? (? % ?));
+         rewrite > sym_times in \vdash (? ? (? ? %));
+         apply le_plus_l;rewrite > sym_plus;apply log_strano
+           [assumption
+           |apply (trans_le ? ? ? H);apply (trans_le ? (b*b))
+              [rewrite > times_n_SO in \vdash (? % ?);
+               apply le_times_r;apply lt_to_le;assumption
+              |apply lt_to_le;assumption]]]]
+  |apply monotonic_div
+     [apply lt_O_fact
+     |apply le_times_l;apply (le_sigma_p_lemma1 ? ? H H1)]] 
+qed.
+
+theorem le_log_C2_stima3 : \forall n,b. S O < b \to b*b < n \to
+log b (C2 n) \leq 
+(14*n/(log b n)) + 
+ ((28*n)*(S (log b (S (S (S O)))))/pred (log b n)).intros.
+apply (trans_le ? ? ? (le_log_C2_stima2 ? ? H H1));apply le_plus_r;
+(*apply (trans_le ? ((28*n)*(28*n*n!/pred (log b n)*S (log b 3)/(28*n*n!))))
+  [*)
+rewrite > (eq_div_div_times ? ? (28*n)) in \vdash (? % ?)
+  [rewrite > sym_times in ⊢ (? (? (? ? %) ?) ?);
+   rewrite < assoc_times;
+   apply le_div_times_m;
+     [apply (trans_le ? (pred (log b (b*b))))
+        [rewrite < exp_SSO;rewrite > eq_log_exp;
+           [apply le_n
+           |assumption]
+        |rewrite < exp_SSO;rewrite > eq_log_exp;
+           [simplify;rewrite > minus_n_O in \vdash (? ? (? %));
+            rewrite < eq_minus_S_pred;
+            apply le_plus_to_minus_r;simplify;
+            rewrite < (eq_log_exp b 2);
+              [apply le_log
+                 [assumption
+                 |rewrite > exp_SSO;apply lt_to_le;assumption]
+              |assumption]
+           |assumption]]
+     |unfold lt;rewrite > times_n_SO in \vdash (? % ?);apply le_times
+        [rewrite > times_n_SO in \vdash (? % ?);apply le_times
+           [apply le_S_S;apply le_O_n
+           |apply (trans_le ? ? ? ? H1);apply le_S_S;
+            rewrite > times_n_SO;apply le_times
+              [apply le_O_n
+              |apply lt_to_le;assumption]]
+        |apply lt_O_fact]]
+  |unfold lt;rewrite > times_n_SO in \vdash (? % ?);apply le_times
+     [apply le_S_S;apply le_O_n
+     |apply (trans_le ? ? ? ? H1);apply le_S_S;
+      rewrite > times_n_SO;apply le_times
+        [apply le_O_n
+        |apply lt_to_le;assumption]]
+  |apply lt_O_fact]
+qed.
+
+lemma le_prim_log1: \forall n,b. S O < b \to O < n \to 
+                     (prim n)*(log b n) \leq
+                     log b (A n) + log b (C1 n) + log b (C2 n) + 2.
+intros.change in \vdash (? ? (? ? %)) with (1+1).
+rewrite < assoc_plus;rewrite > assoc_plus in ⊢ (? ? (? (? % ?) ?)).
+rewrite > assoc_plus in ⊢ (? ? (? % ?));
+apply (trans_le ? (log b (A n) + (log b (C1 n * C2 n)) + 1));
+  [apply (trans_le ? (log b (A n * (C1 n * C2 n))))
+     [apply (trans_le ? (log b (exp n (prim n))))
+        [apply log_exp2;assumption
+        |apply le_log
+           [assumption
+           |rewrite < jj;apply asdasd]]
+     |rewrite > sym_plus;simplify;apply log_times;assumption]
+  |apply le_plus_l;apply le_plus_r;rewrite > sym_plus;simplify;apply log_times;
+   assumption]
+qed.
+
+lemma le_log_A1 : \forall n,b. S O < b \to S O < n \to
+                  log b (A n) \leq 2*(S (log b (pred n))) + (2*n)*(S (log b 2)) + 1.
+intros.apply (trans_le ? (log b ((exp (pred n) 2)*(exp 2 (2*n)))))
+  [apply le_log
+     [assumption
+     |simplify in ⊢ (? ? (? (? % ?) ?));apply le_A_exp3;assumption]
+  |rewrite < sym_plus;apply (trans_le ? (1 + ((log b (exp (pred n) 2)) + (log b (exp 2 (2*n))))));
+     [change in \vdash (? ? %) with (S (log b ((pred n)\sup(2))+log b ((2)\sup(2*n))));
+      apply log_times;assumption
+     |apply le_plus_r;apply le_plus;apply log_exp1;assumption]]
+qed.
+
+lemma le_theta_pi_A : \forall n.theta_pi n \leq A n.
+intro.unfold theta_pi.unfold A.apply le_pi_p.intros.
+rewrite > exp_n_SO in \vdash (? % ?).
+cut (O < i)
+  [apply le_exp
+     [assumption
+     |apply lt_O_log
+        [apply (trans_le ? ? ? Hcut);apply le_S_S_to_le;assumption
+        |apply le_S_S_to_le;assumption]]
+  |apply prime_to_lt_O;apply primeb_true_to_prime;assumption]
 qed.
-*)
 
 definition sqrt \def
   \lambda n.max n (\lambda x.leb (x*x) n).
@@ -304,6 +664,17 @@ intros;apply (trans_le ? (sqrt m * sqrt m))
      [apply le_O_n
      |simplify;reflexivity]]
 qed.
+ 
+theorem lt_sqrt_to_le_times_l : \forall m,n.n < sqrt m \to n*n < m.
+intros;apply (trans_le ? (sqrt m * sqrt m))
+  [apply (trans_le ? (S n * S n))
+     [simplify in \vdash (? ? %);apply le_S_S;apply (trans_le ? (n * S n))
+        [apply le_times_r;apply le_S;apply le_n
+        |rewrite > sym_plus;rewrite > plus_n_O in \vdash (? % ?);
+         apply le_plus_r;apply le_O_n]  
+     |apply le_times;assumption]
+  |apply le_sqrt_to_le_times_l;apply le_n]
+qed.
 
 theorem le_sqrt_to_le_times_r : \forall m,n.sqrt m < n \to m < n*n.
 intros;apply not_le_to_lt;intro;
@@ -343,3 +714,106 @@ intro;unfold theta_pi;unfold C1;rewrite > (false_to_eq_pi_p (S (sqrt n)) (S n))
         |apply le_sqrt_to_le_times_r;assumption]
      |reflexivity]]
 qed.
+
+lemma le_sqrt_n_n : \forall n.sqrt n \leq n.
+intro.unfold sqrt.apply le_max_n.
+qed.
+
+lemma le_prim_log_stima: \forall n,b. S O < b \to b < sqrt n \to 
+                     (prim n)*(log b n) \leq
+                      2*S (log b (pred n))+2*n*S (log b 2)
+                      +(2*S (log b (pred (sqrt n)))+2*sqrt n*S (log b 2))
+                      +(14*n/log b n+28*n*S (log b 3)/pred (log b n))
+                      +4.
+intros.cut (1 < n)
+  [apply (trans_le ? ((2*(S (log b (pred n))) + (2*n)*(S (log b 2)) + 1) + 
+                     (2*(S (log b (pred (sqrt n)))) + (2*(sqrt n))*(S (log b 2)) + 1) +
+                     ((14*n/(log b n)) + ((28*n)*(S (log b (S (S (S O)))))/pred (log b n))) + 2))
+    [apply (trans_le ? ? ? (le_prim_log1 ? ? H ?))
+       [apply lt_to_le;assumption
+       |apply le_plus_l;apply le_plus
+          [apply le_plus
+             [apply le_log_A1;assumption
+             |rewrite < eq_theta_pi_sqrt_C1;apply (trans_le ? (log b (A (sqrt n))))
+                [apply le_log
+                   [assumption
+                   |apply le_theta_pi_A]
+                |apply le_log_A1
+                   [assumption
+                   |apply (trans_le ? ? ? H);apply lt_to_le;assumption]]]
+          |apply le_log_C2_stima3;
+             [assumption
+             |apply lt_sqrt_to_le_times_l;assumption]]]
+    |rewrite > assoc_plus in ⊢ (? (? % ?) ?);
+     rewrite > sym_plus in ⊢ (? (? (? ? %) ?) ?);
+     rewrite > assoc_plus in \vdash (? % ?);
+     rewrite > assoc_plus in ⊢ (? (? ? %) ?);
+     rewrite > assoc_plus in ⊢ (? (? % ?) ?);
+     rewrite > assoc_plus in \vdash (? % ?);
+     rewrite < assoc_plus in ⊢ (? (? ? %) ?);
+     rewrite > assoc_plus in ⊢ (? (? ? (? % ?)) ?);
+     rewrite > sym_plus in ⊢ (? (? ? (? (? ? %) ?)) ?);
+     rewrite < assoc_plus in ⊢ (? (? ? (? % ?)) ?);
+     rewrite < assoc_plus in \vdash (? % ?);
+     rewrite < assoc_plus in ⊢ (? (? % ?) ?);
+     rewrite > assoc_plus in \vdash (? % ?);
+     rewrite > sym_plus in ⊢ (? (? ? %) ?);
+     rewrite > assoc_plus in ⊢ (? (? ? %) ?);
+     rewrite > assoc_plus in ⊢ (? (? ? (? % ?)) ?);
+     rewrite > assoc_plus in ⊢ (? (? ? %) ?);     
+     rewrite > assoc_plus in ⊢ (? (? ? (? ? %)) ?);
+     simplify in ⊢ (? (? ? (? ? (? ? %))) ?);
+     rewrite < assoc_plus in ⊢ (? (? ? %) ?);
+     rewrite < assoc_plus in ⊢ (? % ?);apply le_plus_l;
+     rewrite > assoc_plus in \vdash (? % ?);
+     rewrite > assoc_plus in ⊢ (? (? ? %) ?);
+     rewrite > sym_plus in ⊢ (? (? ? (? ? %)) ?);
+     rewrite < assoc_plus in ⊢ (? (? ? %) ?);
+     rewrite < assoc_plus in \vdash (? % ?);apply le_plus_l;
+     rewrite > assoc_plus in \vdash (? ? %);apply le_n]
+  |apply (trans_le ? ? ? H);apply lt_to_le;apply (trans_le ? ? ? H1);
+   apply le_sqrt_n_n]
+qed.
+      
+(*intros;apply lt_to_le;lapply (lt_div_S (((S (S (S (S O))))* log b (pred i)) + (S (S (S (S (S O)))))) i)
+   [apply (trans_le ? ? ? Hletin);apply le_times_l;apply le_S_S;
+    elim H1
+      [rewrite > log_SO;
+         [simplify;apply le_n
+         |assumption]
+      |
+    apply le_times_to_le_div2;
+      |*)
+       
+(*   
+theorem le_log_C2_sigma_p : \forall n,b. S O < b \to
+log b (C2 n) \leq 
+(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x))) (\lambda x.S O)) +
+(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x))) 
+   (\lambda p.(sigma_p (n+p) (\lambda i.leb p i) 
+       (\lambda i.S ((n+p)!/i*(S (log b 3)))))/(n+p)!)).
+intros;unfold C2;
+apply (trans_le ? (sigma_p (S n) (λx:nat.primeb x∧leb (S n) (x*x)) (λx:nat.1)
++sigma_p (S n) (λx:nat.primeb x∧leb (S n) (x*x))
+ (λi.log b (S (n/i)))))
+  [apply log_pi_p;assumption
+  |apply le_plus
+     [apply le_n
+     |apply le_sigma_p;intros;cut (S (n/i) = (n+i)/i)
+        [rewrite > Hcut;apply le_log_div_sigma_p 
+           [assumption
+           |apply prime_to_lt_O;apply primeb_true_to_prime;
+            elim (and_true ? ? H2);assumption
+           |apply le_plus_n
+           |apply le_n]
+        |lapply (prime_to_lt_O i (primeb_true_to_prime ? (andb_true_true ? ? H2))) as H3;
+         apply (div_mod_spec_to_eq (n+i) i (S (n/i)) (n \mod i) ? ((n+i) \mod i))
+           [constructor 1
+              [apply lt_mod_m_m;assumption
+              |simplify;rewrite > assoc_plus;rewrite < div_mod;
+                 [apply sym_plus
+                 |assumption]]
+           |apply div_mod_spec_div_mod;assumption]]]]
+qed.
+*)
+