From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Tue, 30 Apr 2013 15:39:19 +0000 (+0000)
Subject: This line, and those below, will be ignored--
X-Git-Tag: make_still_working~1170
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=d67b6a744a51aa8f0e33ea4bf5ff29ffce762ccb;p=helm.git

This line, and those below, will be ignored--

M    chebyshev/chebyshev_psi.ma
D    chebyshev/chebyshev_B.ma
---

diff --git a/matita/matita/lib/arithmetics/chebyshev/chebyshev_B.ma b/matita/matita/lib/arithmetics/chebyshev/chebyshev_B.ma
deleted file mode 100644
index 370195aec..000000000
--- a/matita/matita/lib/arithmetics/chebyshev/chebyshev_B.ma
+++ /dev/null
@@ -1,750 +0,0 @@
-(*
-    ||M||  This file is part of HELM, an Hypertextual, Electronic   
-    ||A||  Library of Mathematics, developed at the Computer Science 
-    ||T||  Department of the University of Bologna, Italy.           
-    ||I||                                                            
-    ||T||  
-    ||A||  
-    \   /  This file is distributed under the terms of the       
-     \ /   GNU General Public License Version 2   
-      V_____________________________________________________________*)
-
-include "arithmetics/chebyshev/chebyshev.ma".
-    
-definition B ≝ λn.
-∏_{p < S n | primeb p} 
-  (∏_{i < log p n} (exp p (mod (n /(exp p (S i))) 2))).
-  
-lemma Bdef : ∀n.B n = 
-  ∏_{p < S n | primeb p} 
-  (∏_{i < log p n} (exp p (mod (n /(exp p (S i))) 2))).
-// qed-.
-
-example B_SSSO: B 3 = 6. //
-qed.
-
-example B_SSSSO: B 4 = 6. //
-qed.
-
-example B_SSSSSO: B 5 = 30. //
-qed.
-
-example B_SSSSSSO: B 6 = 20. //
-qed.
-
-example B_SSSSSSSO: B 7 = 140. //
-qed.
-
-example B_SSSSSSSSO: B 8 = 70. //
-qed.
-
-theorem eq_fact_B:∀n. 1 < n →
-  (2*n)! = exp (n!) 2 * B(2*n).
-#n #lt1n >fact_pi_p3 @eq_f2
-  [@sym_eq >pi_p_primeb5 [@exp_fact_2|//] |// ]
-qed.
-
-theorem le_B_A: ∀n. B n ≤ A n.
-#n >eq_A_A' @le_pi #p #ltp #primep
-@le_pi #i #lti #_ >(exp_n_1 p) in ⊢ (??%); @le_exp
-  [@prime_to_lt_O @primeb_true_to_prime //
-  |@le_S_S_to_le @lt_mod_m_m @lt_O_S
-  ]
-qed.
-
-theorem le_B_A4: ∀n. O < n → 2 * B (4*n) ≤ A (4*n).
-#n #posn >eq_A_A'
-cut (2 < (S (4*n)))
-  [@le_S_S >(times_n_1 2) in ⊢ (?%?); @le_times //] #H2
-cut (O<log 2 (4*n))
-  [@lt_O_log [@le_S_S_to_le @H2 |@le_S_S_to_le @H2]] #Hlog
->Bdef >(bigop_diff ??? timesAC ? 2 ? H2) [2:%]
->Adef >(bigop_diff ??? timesAC ? 2 ? H2) in ⊢ (??%); [2:%]
-<associative_times @le_times
-  [>(bigop_diff ??? timesAC ? 0 ? Hlog) [2://]
-   >(bigop_diff ??? timesAC ? 0 ? Hlog) in ⊢ (??%); [2:%]
-   <associative_times >timesACdef @le_times 
-    [<exp_n_1 cut (4=2*2) [//] #H4 >H4 >associative_times
-     >commutative_times in ⊢ (?(??(??(?(?%?)?)))?);
-     >div_times [2://] >divides_to_mod_O
-      [@le_n |%{n} // |@lt_O_S]
-    |@le_pi #i #lti #H >(exp_n_1 2) in ⊢ (??%);
-     @le_exp [@lt_O_S |@le_S_S_to_le @lt_mod_m_m @lt_O_S]
-    ]
-  |@le_pi #p #ltp #Hp @le_pi #i #lti #H
-   >(exp_n_1 p) in ⊢ (??%); @le_exp
-    [@prime_to_lt_O @primeb_true_to_prime @(andb_true_r ?? Hp)
-    |@le_S_S_to_le @lt_mod_m_m @lt_O_S
-    ]
-  ]
-qed.
-
-(* not usefull    
-theorem le_fact_A:\forall n.S O < n \to
-fact (2*n) \le exp (fact n) 2 * A (2*n).
-intros.
-rewrite > eq_fact_B
-  [apply le_times_r.
-   apply le_B_A
-  |assumption
-  ]
-qed. *)
-
-theorem lt_SO_to_le_B_exp: ∀n. 1 < n →
-  B (2*n) ≤ exp 2 (pred (2*n)).
-#n #lt1n @(le_times_to_le (exp (fact n) 2))
-  [@lt_O_exp //
-  |<(eq_fact_B … lt1n) <commutative_times in ⊢ (??%);
-   >exp_2 <associative_times @fact_to_exp 
-  ]
-qed.
-
-theorem le_B_exp: ∀n.
-  B (2*n) ≤ exp 2 (pred (2*n)).
-#n cases n
-  [@le_n|#n1 cases n1 [@le_n |#n2 @lt_SO_to_le_B_exp @le_S_S @lt_O_S]]
-qed.
-
-theorem lt_4_to_le_B_exp: ∀n.4 < n →
-  B (2*n) \le exp 2 ((2*n)-2).
-#n #lt4n @(le_times_to_le (exp (fact n) 2))
-  [@lt_O_exp //
-  |<eq_fact_B
-    [<commutative_times in ⊢ (??%); >exp_2 <associative_times
-     @lt_4_to_fact //
-    |@lt_to_le @lt_to_le @lt_to_le //
-    ]
-  ]
-qed.
-
-theorem lt_1_to_le_exp_B: ∀n. 1 < n →
-  exp 2 (2*n) ≤ 2 * n * B (2*n).
-#n #lt1n 
-@(le_times_to_le (exp (fact n) 2))
-  [@lt_O_exp @le_1_fact
-  |<associative_times in ⊢ (??%); >commutative_times in ⊢ (??(?%?));
-   >associative_times in ⊢ (??%); <(eq_fact_B … lt1n)
-   <commutative_times; @exp_to_fact2 @lt_to_le // 
-  ]
-qed.
-
-theorem le_exp_B: ∀n. O < n →
-  exp 2 (2*n) ≤ 2 * n * B (2*n).
-#n #posn cases posn
-  [@le_n |#m #lt1m @lt_1_to_le_exp_B @le_S_S // ]
-qed.
-
-let rec bool_to_nat b ≝ 
-  match b with [true ⇒ 1 | false ⇒ 0].
-  
-theorem eq_A_2_n: ∀n.O < n →
-A(2*n) =
- ∏_{p <S (2*n) | primeb p}
-  (∏_{i <log p (2*n)} (exp p (bool_to_nat (leb (S n) (exp p (S i)))))) *A n.
-#n #posn >eq_A_A' > eq_A_A' 
-cut (
- ∏_{p < S n | primeb p} (∏_{i <log p n} p)
- = ∏_{p < S (2*n) | primeb p}
-     (∏_{i <log p (2*n)} (p\sup(bool_to_nat (¬ (leb (S n) (exp p (S i))))))))
-  [2: #Hcut >Adef in ⊢ (???%); >Hcut
-   <times_pi >Adef @same_bigop
-    [//
-    |#p #lt1p #primep <times_pi @same_bigop
-      [//
-      |#i #lt1i #_ cases (true_or_false (leb (S n) (exp p (S i)))) #Hc >Hc
-        [normalize <times_n_1 >plus_n_O //
-        |normalize <plus_n_O <plus_n_O //
-        ]
-      ]
-    ]
-  |@(trans_eq ?? 
-    (∏_{p < S n | primeb p} 
-      (∏_{i < log p n} (p \sup(bool_to_nat (¬leb (S n) (exp p (S i))))))))
-    [@same_bigop
-      [//
-      |#p #lt1p #primep @same_bigop
-        [//
-        |#i #lti#_ >lt_to_leb_false
-          [normalize @plus_n_O
-          |@le_S_S @(transitive_le ? (exp p (log p n)))
-            [@le_exp [@prime_to_lt_O @primeb_true_to_prime //|//]
-            |@le_exp_log //
-            ]
-          ]
-        ]
-      ]
-    |@(trans_eq ?? 
-      (∏_{p < S (2*n) | primeb p}
-       (∏_{i <log p n} (p \sup(bool_to_nat (¬leb (S n) (p \sup(S i))))))))
-      [@(pad_bigop_nil … timesA)
-        [@le_S_S //|#i #lti #upi %2 >lt_to_log_O //]
-      |@same_bigop 
-        [//
-        |#p #ltp #primep @(pad_bigop_nil … timesA)
-          [@le_log
-            [@prime_to_lt_SO @primeb_true_to_prime //|//]
-          |#i #lei #iup %2 >le_to_leb_true
-            [%
-            |@(lt_to_le_to_lt ? (exp p (S (log p n))))
-              [@lt_exp_log @prime_to_lt_SO @primeb_true_to_prime //
-              |@le_exp
-                [@prime_to_lt_O @primeb_true_to_prime //
-                |@le_S_S //
-                ]
-              ]
-            ]
-          ]
-        ]
-      ]
-    ]
-  ]
-qed.
-               
-theorem le_A_BA1: ∀n. O < n → 
-  A(2*n) ≤ B(2*n)*A n.
-#n #posn >(eq_A_2_n … posn) @le_times [2:@le_n]
->Bdef @le_pi #p #ltp #primep @le_pi #i #lti #_ @le_exp
-  [@prime_to_lt_O @primeb_true_to_prime //
-  |cases (true_or_false (leb (S n) (exp p (S i)))) #Hc >Hc
-    [whd in ⊢ (?%?);
-     cut (2*n/p\sup(S i) = 1) [2: #Hcut >Hcut @le_n]
-     @(div_mod_spec_to_eq (2*n) (exp p (S i)) 
-       ? (mod (2*n) (exp p (S i))) ? (minus (2*n) (exp p (S i))) )
-      [@div_mod_spec_div_mod @lt_O_exp @prime_to_lt_O @primeb_true_to_prime //
-      |cut (p\sup(S i)≤2*n)
-        [@(transitive_le ? (exp p (log p (2*n))))
-          [@le_exp [@prime_to_lt_O @primeb_true_to_prime // | //]
-          |@le_exp_log >(times_n_O O) in ⊢ (?%?); @lt_times // 
-          ]
-        ] #Hcut
-       @div_mod_spec_intro
-        [@lt_plus_to_minus
-          [// |normalize in ⊢ (? % ?); < plus_n_O @lt_plus @leb_true_to_le //]
-        |>commutative_plus >commutative_times in ⊢ (???(??%));
-         < times_n_1 @plus_minus_m_m //
-        ]
-      ]
-    |@le_O_n
-    ]
-  ]
-qed.
-
-theorem le_A_BA: ∀n. A(2*n) \le B(2*n)*A n.
-#n cases n [@le_n |#m @le_A_BA1 @lt_O_S]
-qed.
-
-theorem le_A_exp: ∀n. A(2*n) ≤ (exp 2 (pred (2*n)))*A n.
-#n @(transitive_le ? (B(2*n)*A n))
-  [@le_A_BA |@le_times [@le_B_exp |//]]
-qed.
-
-theorem lt_4_to_le_A_exp: ∀n. 4 < n →
-  A(2*n) ≤ exp 2 ((2*n)-2)*A n.
-#n #lt4n @(transitive_le ? (B(2*n)*A n))
-  [@le_A_BA|@le_times [@(lt_4_to_le_B_exp … lt4n) |@le_n]]
-qed.
-
-(* two technical lemmas *)
-lemma times_2_pred: ∀n. 2*(pred n) \le pred (2*n).
-#n cases n
-  [@le_n|#m @monotonic_le_plus_r @le_n_Sn]
-qed.
-
-lemma le_S_times_2: ∀n. O < n → S n ≤ 2*n.
-#n #posn elim posn
-  [@le_n
-  |#m #posm #Hind 
-   cut (2*(S m) = S(S(2*m))) [normalize <plus_n_Sm //] #Hcut >Hcut
-   @le_S_S @(transitive_le … Hind) @le_n_Sn
-  ]
-qed.
-
-theorem le_A_exp1: ∀n.
-  A(exp 2 n) ≤ exp 2 ((2*(exp 2 n)-(S(S n)))).
-#n elim n
-  [@le_n
-  |#n1 #Hind whd in ⊢ (?(?%)?); >commutative_times 
-   @(transitive_le ??? (le_A_exp ?)) 
-   @(transitive_le ? (2\sup(pred (2*2^n1))*2^(2*2^n1-(S(S n1)))))
-    [@monotonic_le_times_r // 
-    |<exp_plus_times @(le_exp … (lt_O_S ?))
-     cut (S(S n1) ≤ 2*(exp 2 n1))
-      [elim n1
-        [@le_n
-        |#n2 >commutative_times in ⊢ (%→?); #Hind1 @(transitive_le ? (2*(S(S n2))))
-          [@le_S_times_2 @lt_O_S |@monotonic_le_times_r //] 
-        ]
-      ] #Hcut
-     @le_S_S_to_le cut(∀a,b. S a + b = S (a+b)) [//] #Hplus <Hplus >S_pred
-      [>eq_minus_S_pred in ⊢ (??%); >S_pred
-        [>plus_minus_commutative
-          [@monotonic_le_minus_l
-           cut (∀a. 2*a = a + a) [//] #times2 <times2 
-           @monotonic_le_times_r >commutative_times @le_n
-          |@Hcut
-          ]
-        |@lt_plus_to_minus_r whd in ⊢ (?%?);
-         @(lt_to_le_to_lt ? (2*(S(S n1))))
-          [>(times_n_1 (S(S n1))) in ⊢ (?%?); >commutative_times
-           @monotonic_lt_times_l [@lt_O_S |@le_n]
-          |@monotonic_le_times_r whd in ⊢ (??%); //
-          ]
-        ]
-      |whd >(times_n_1 1) in ⊢ (?%?); @le_times
-        [@le_n_Sn |@lt_O_exp @lt_O_S]
-      ]
-    ]
-  ]
-qed.
-
-theorem monotonic_A: monotonic nat le A.
-#n #m #lenm elim lenm
-  [@le_n
-  |#n1 #len #Hind @(transitive_le … Hind)
-   cut (∏_{p < S n1 | primeb p}(p^(log p n1))
-          ≤∏_{p < S n1 | primeb p}(p^(log p (S n1))))
-    [@le_pi #p #ltp #primep @le_exp
-      [@prime_to_lt_O @primeb_true_to_prime //
-      |@le_log [@prime_to_lt_SO @primeb_true_to_prime // | //]
-      ]
-    ] #Hcut
-   >psi_def in ⊢ (??%); cases (true_or_false (primeb (S n1))) #Hc
-    [>bigop_Strue in ⊢ (??%); [2://]
-     >(times_n_1 (A n1)) >commutative_times @le_times
-      [@lt_O_exp @lt_O_S |@Hcut]
-    |>bigop_Sfalse in ⊢ (??%); // 
-    ]
-  ]
-qed.
-
-(*
-theorem le_A_exp2: \forall n. O < n \to
-A(n) \le (exp (S(S O)) ((S(S O)) * (S(S O)) * n)).
-intros.
-apply (trans_le ? (A (exp (S(S O)) (S(log (S(S O)) n)))))
-  [apply monotonic_A.
-   apply lt_to_le.
-   apply lt_exp_log.
-   apply le_n
-  |apply (trans_le ? ((exp (S(S O)) ((S(S O))*(exp (S(S O)) (S(log (S(S O)) n)))))))
-    [apply le_A_exp1
-    |apply le_exp
-      [apply lt_O_S
-      |rewrite > assoc_times.apply le_times_r.
-       change with ((S(S O))*((S(S O))\sup(log (S(S O)) n))≤(S(S O))*n).
-       apply le_times_r.
-       apply le_exp_log.
-       assumption
-      ]
-    ]
-  ]
-qed.
-*)
-
-(* example *)
-example A_1: A 1 = 1. // qed.
-
-example A_2: A 2 = 2. // qed.
-
-example A_3: A 3 = 6. // qed.
-
-example A_4: A 4 = 12. // qed.
-
-(*
-(* a better result *)
-theorem le_A_exp3: \forall n. S O < n \to
-A(n) \le exp (pred n) (2*(exp 2 (2 * n)).
-intro.
-apply (nat_elim1 n).
-intros.
-elim (or_eq_eq_S m).
-elim H2
-  [elim (le_to_or_lt_eq (S O) a)
-    [rewrite > H3 in ⊢ (? % ?).
-     apply (trans_le ? ((exp (S(S O)) ((S(S O)*a)))*A a))
-      [apply le_A_exp
-      |apply (trans_le ? (((S(S O)))\sup((S(S O))*a)*
-         ((pred a)\sup((S(S O)))*((S(S O)))\sup((S(S O))*a))))
-        [apply le_times_r.
-         apply H
-          [rewrite > H3.
-           rewrite > times_n_SO in ⊢ (? % ?).
-           rewrite > sym_times.
-           apply lt_times_l1
-            [apply lt_to_le.assumption
-            |apply le_n
-            ]
-          |assumption
-          ]
-        |rewrite > sym_times.
-         rewrite > assoc_times.
-         rewrite < exp_plus_times.
-         apply (trans_le ? 
-          (pred a\sup((S(S O)))*(S(S O))\sup(S(S O))*(S(S O))\sup((S(S O))*m)))
-          [rewrite > assoc_times.
-           apply le_times_r.
-           rewrite < exp_plus_times.
-           apply le_exp
-            [apply lt_O_S
-            |rewrite < H3.
-             simplify.
-             rewrite < plus_n_O.
-             apply le_S.apply le_S.
-             apply le_n
-            ]
-          |apply le_times_l.
-           rewrite > times_exp.
-           apply monotonic_exp1.
-           rewrite > H3.
-           rewrite > sym_times.
-           cases a
-            [apply le_n
-            |simplify.
-             rewrite < plus_n_Sm.
-             apply le_S.
-             apply le_n
-            ]
-          ]
-        ]
-      ]
-    |rewrite < H4 in H3.
-     simplify in H3.
-     rewrite > H3.
-     simplify.
-     apply le_S_S.apply le_S_S.apply le_O_n
-    |apply not_lt_to_le.intro.
-     apply (lt_to_not_le ? ? H1).
-     rewrite > H3.
-     apply (le_n_O_elim a)
-      [apply le_S_S_to_le.assumption
-      |apply le_O_n
-      ]
-    ]
-  |elim (le_to_or_lt_eq O a (le_O_n ?))
-    [apply (trans_le ? (A ((S(S O))*(S a))))
-      [apply monotonic_A.
-       rewrite > H3.
-       rewrite > times_SSO.
-       apply le_n_Sn
-      |apply (trans_le ? ((exp (S(S O)) ((S(S O)*(S a))))*A (S a)))
-        [apply le_A_exp
-        |apply (trans_le ? (((S(S O)))\sup((S(S O))*S a)*
-           (a\sup((S(S O)))*((S(S O)))\sup((S(S O))*(S a)))))
-          [apply le_times_r.
-           apply H
-            [rewrite > H3.
-             apply le_S_S.
-             simplify.
-             rewrite > plus_n_SO.
-             apply le_plus_r.
-             rewrite < plus_n_O.
-             assumption
-            |apply le_S_S.assumption
-            ]
-          |rewrite > sym_times.
-           rewrite > assoc_times.
-           rewrite < exp_plus_times.
-           apply (trans_le ? 
-            (a\sup((S(S O)))*(S(S O))\sup(S(S O))*(S(S O))\sup((S(S O))*m)))
-            [rewrite > assoc_times.
-             apply le_times_r.
-             rewrite < exp_plus_times.
-             apply le_exp
-              [apply lt_O_S
-              |rewrite > times_SSO.
-               rewrite < H3.
-               simplify.
-               rewrite < plus_n_Sm.
-               rewrite < plus_n_O.
-               apply le_n
-              ]
-            |apply le_times_l.
-             rewrite > times_exp.
-             apply monotonic_exp1.
-             rewrite > H3.
-             rewrite > sym_times.
-             apply le_n
-            ]
-          ]
-        ]
-      ]
-    |rewrite < H4 in H3.simplify in H3.
-     apply False_ind.
-     apply (lt_to_not_le ? ? H1).
-     rewrite > H3.
-     apply le_
-    ]
-  ]
-qed.
-*)
-
-theorem le_A_exp4: ∀n. 1 < n →
-  A(n) ≤ (pred n)*exp 2 ((2 * n) -3).
-#n @(nat_elim1 n)
-#m #Hind cases (even_or_odd m)
-#a * #Hm >Hm #Hlt
-  [cut (0<a) 
-    [cases a in Hlt; 
-      [whd in ⊢ (??%→?); #lt10 @False_ind @(absurd ? lt10 (not_le_Sn_O 1))
-    |#b #_ //]
-    ] #Hcut 
-   cases (le_to_or_lt_eq … Hcut) #Ha
-    [@(transitive_le ? (exp 2 (pred(2*a))*A a))
-      [@le_A_exp
-      |@(transitive_le ? (2\sup(pred(2*a))*((pred a)*2\sup((2*a)-3))))
-        [@monotonic_le_times_r @(Hind ?? Ha)
-         >Hm >(times_n_1 a) in ⊢ (?%?); >commutative_times
-         @monotonic_lt_times_l [@lt_to_le // |@le_n]
-        |<Hm <associative_times >commutative_times in ⊢ (?(?%?)?);
-         >associative_times; @le_times
-          [>Hm cases a[@le_n|#b normalize @le_plus_n_r]
-          |<exp_plus_times @le_exp
-            [@lt_O_S
-            |@(transitive_le ? (m+(m-3)))
-              [@monotonic_le_plus_l // 
-              |normalize <plus_n_O >plus_minus_commutative
-                [@le_n
-                |>Hm @(transitive_le ? (2*2) ? (le_n_Sn 3))
-                 @monotonic_le_times_r //
-                ]
-              ]
-            ]
-          ]
-        ]
-      ]
-    |<Ha normalize @le_n
-    ]
-  |cases (le_to_or_lt_eq O a (le_O_n ?)) #Ha
-    [@(transitive_le ? (A (2*(S a))))
-      [@monotonic_A >Hm normalize <plus_n_Sm @le_n_Sn
-      |@(transitive_le … (le_A_exp ?) ) 
-       @(transitive_le ? ((2\sup(pred (2*S a)))*(a*(exp 2 ((2*(S a))-3)))))
-        [@monotonic_le_times_r @Hind
-          [>Hm @le_S_S >(times_n_1 a) in ⊢ (?%?); >commutative_times
-           @monotonic_lt_times_l //
-          |@le_S_S //
-          ]
-        |cut (pred (S (2*a)) = 2*a) [//] #Spred >Spred
-         cut (pred (2*(S a)) = S (2 * a)) [normalize //] #Spred1 >Spred1
-         cut (2*(S a) = S(S(2*a))) [normalize <plus_n_Sm //] #times2 
-         cut (exp 2 (2*S a -3) = 2*(exp 2 (S(2*a) -3))) 
-          [>(commutative_times 2) in ⊢ (???%); >times2 >minus_Sn_m [%]
-           @le_S_S >(times_n_1 2) in ⊢ (?%?); @monotonic_le_times_r @Ha
-          ] #Hcut >Hcut
-         <associative_times in ⊢ (? (? ? %) ?); <associative_times
-         >commutative_times in ⊢ (? (? % ?) ?);
-         >commutative_times in ⊢ (? (? (? % ?) ?) ?);
-         >associative_times @monotonic_le_times_r
-         <exp_plus_times @(le_exp … (lt_O_S ?))
-         >plus_minus_commutative
-          [normalize >(plus_n_O (a+(a+0))) in ⊢ (?(?(??%)?)?); @le_n
-          |@le_S_S >(times_n_1 2) in ⊢ (?%?); @monotonic_le_times_r @Ha
-          ]
-        ]
-      ]
-    |@False_ind <Ha in Hlt; normalize #Hfalse @(absurd ? Hfalse) //
-    ]
-  ]
-qed.
-
-theorem le_n_8_to_le_A_exp: ∀n. n ≤ 8 → 
-  A(n) ≤ exp 2 ((2 * n) -3).
-#n cases n
-  [#_ @le_n
-  |#n1 cases n1
-    [#_ @le_n
-    |#n2 cases n2
-      [#_ @le_n
-      |#n3 cases n3
-        [#_ @leb_true_to_le //
-        |#n4 cases n4
-          [#_ @leb_true_to_le //
-          |#n5 cases n5
-            [#_ @leb_true_to_le //
-            |#n6 cases n6
-              [#_ @leb_true_to_le //
-              |#n7 cases n7
-                [#_ @leb_true_to_le //
-                |#n8 cases n8
-                  [#_ @leb_true_to_le //
-                  |#n9 #H @False_ind @(absurd ?? (lt_to_not_le ?? H))
-                   @leb_true_to_le //
-                  ]
-                ]
-              ]
-            ]
-          ]
-        ]
-      ]
-    ]
-  ]
-qed.
-           
-theorem le_A_exp5: ∀n. A(n) ≤ exp 2 ((2 * n) -3).
-#n @(nat_elim1 n) #m #Hind
-cases (decidable_le 9 m)
-  [#lem cases (even_or_odd m) #a * #Hm
-    [>Hm in ⊢ (?%?); @(transitive_le … (le_A_exp ?))
-     @(transitive_le ? (2\sup(pred(2*a))*(2\sup((2*a)-3))))
-      [@monotonic_le_times_r @Hind >Hm >(times_n_1 a) in ⊢ (?%?); 
-       >commutative_times @(monotonic_lt_times_l  … (le_n ?))
-       @(transitive_lt ? 3)
-        [@lt_O_S |@(le_times_to_le 2) [@lt_O_S |<Hm @lt_to_le @lem]]
-      |<Hm <exp_plus_times @(le_exp … (lt_O_S ?))
-       whd in match (times 2 m); >commutative_times <times_n_1
-       <plus_minus_commutative
-        [@monotonic_le_plus_l @le_pred_n
-        |@(transitive_le … lem) @leb_true_to_le //
-        ]
-      ]
-    |@(transitive_le ? (A (2*(S a))))
-      [@monotonic_A >Hm normalize <plus_n_Sm @le_n_Sn
-      |@(transitive_le ? ((exp 2 ((2*(S a))-2))*A (S a)))
-        [@lt_4_to_le_A_exp @le_S_S
-         @(le_times_to_le 2)[@le_n_Sn|@le_S_S_to_le <Hm @lem]
-        |@(transitive_le ? ((2\sup((2*S a)-2))*(exp 2 ((2*(S a))-3))))
-          [@monotonic_le_times_r @Hind >Hm @le_S_S
-           >(times_n_1 a) in ⊢ (?%?); 
-           >commutative_times @(monotonic_lt_times_l  … (le_n ?))
-           @(transitive_lt ? 3)
-            [@lt_O_S |@(le_times_to_le 2) [@lt_O_S |@le_S_S_to_le <Hm @lem]]
-          |cut (∀a. 2*(S a) = S(S(2*a))) [normalize #a <plus_n_Sm //] #times2
-           >times2 <Hm <exp_plus_times @(le_exp … (lt_O_S ?))
-           cases m
-            [@le_n
-            |#n1 cases n1
-              [@le_n
-              |#n2 normalize <minus_n_O <plus_n_O <plus_n_Sm
-               normalize <minus_n_O <plus_n_Sm @le_n
-              ]
-            ]
-          ]
-        ]
-      ]
-    ]
-  |#H @le_n_8_to_le_A_exp @le_S_S_to_le @not_le_to_lt //
-  ]
-qed.       
-
-theorem le_exp_Al:∀n. O < n → exp 2 n ≤ A (2 * n).
-#n #posn @(transitive_le ? ((exp 2 (2*n))/(2*n)))
-  [@le_times_to_le_div
-    [>(times_n_O O) in ⊢ (?%?); @lt_times [@lt_O_S|//]
-    |normalize in ⊢ (??(??%)); < plus_n_O >exp_plus_times
-     @le_times [2://] elim posn [//]
-     #m #le1m #Hind whd in ⊢ (??%); >commutative_times in ⊢ (??%);
-     @monotonic_le_times_r @(transitive_le … Hind) 
-     >(times_n_1 m) in ⊢ (?%?); >commutative_times 
-     @(monotonic_lt_times_l  … (le_n ?)) @le1m
-    ]
-  |@le_times_to_le_div2
-    [>(times_n_O O) in ⊢ (?%?); @lt_times [@lt_O_S|//]
-    |@(transitive_le ? ((B (2*n)*(2*n))))
-      [<commutative_times in ⊢ (??%); @le_exp_B //
-      |@le_times [@le_B_A|@le_n]
-      ]
-    ]
-  ]
-qed.
-
-theorem le_exp_A2:∀n. 1 < n → exp 2 (n / 2) \le A n.
-#n #lt1n @(transitive_le ? (A(n/2*2)))
-  [>commutative_times @le_exp_Al
-   cases (le_to_or_lt_eq ? ? (le_O_n (n/2))) [//]
-   #Heq @False_ind @(absurd ?? (lt_to_not_le ?? lt1n))
-   >(div_mod n 2) <Heq whd in ⊢ (?%?);
-   @le_S_S_to_le @(lt_mod_m_m n 2) @lt_O_S
-  |@monotonic_A >(div_mod n 2) in ⊢ (??%); @le_plus_n_r
-  ]
-qed.
-
-theorem eq_sigma_pi_SO_n: ∀n.∑_{i < n} 1 = n.
-#n elim n //
-qed.
-
-theorem leA_prim: ∀n.
-  exp n (prim n) \le A n * ∏_{p < S n | primeb p} p.
-#n <(exp_sigma (S n) n primeb) <times_pi @le_pi
-#p #ltp #primep @lt_to_le @lt_exp_log
-@prime_to_lt_SO @primeb_true_to_prime //
-qed.
-
-theorem le_prim_log : ∀n,b. 1 < b →
-  log b (A n) ≤ prim n * (S (log b n)).
-#n #b #lt1b @(transitive_le … (log_exp1 …)) [@le_log // | //]
-qed.
-
-(*********************** the inequalities ***********************)
-lemma exp_Sn: ∀b,n. exp b (S n) = b * (exp b n).
-normalize // 
-qed.
-
-theorem le_exp_priml: ∀n. O < n →
-  exp 2 (2*n) ≤ exp (2*n) (S(prim (2*n))).
-#n #posn @(transitive_le ? (((2*n*(B (2*n))))))
-  [@le_exp_B // 
-  |>exp_Sn @monotonic_le_times_r @(transitive_le ? (A (2*n)))
-    [@le_B_A |@le_Al]
-  ]
-qed.
-
-theorem le_exp_prim4l: ∀n. O < n →
-  exp 2 (S(4*n)) ≤ exp (4*n) (S(prim (4*n))).
-#n #posn @(transitive_le ? (2*(4*n*(B (4*n)))))
-  [>exp_Sn @monotonic_le_times_r
-   cut (4*n = 2*(2*n)) [<associative_times //] #Hcut
-   >Hcut @le_exp_B @lt_to_le whd >(times_n_1 2) in ⊢ (?%?);
-   @monotonic_le_times_r //
-  |>exp_Sn <associative_times >commutative_times in ⊢ (?(?%?)?);
-   >associative_times @monotonic_le_times_r @(transitive_le ? (A (4*n)))
-    [@le_B_A4 // |@le_Al]
-  ]
-qed.
-
-theorem le_priml: ∀n. O < n →
-  2*n ≤ (S (log 2 (2*n)))*S(prim (2*n)).
-#n #posn <(eq_log_exp 2 (2*n) ?) in ⊢ (?%?);
-  [@(transitive_le ? ((log 2) (exp (2*n) (S(prim (2*n))))))
-    [@le_log [@le_n |@le_exp_priml //]
-    |>commutative_times in ⊢ (??%); @log_exp1 @le_n
-    ]
-  |@le_n
-  ]
-qed.
-
-theorem le_exp_primr: ∀n.
-  exp n (prim n) ≤ exp 2 (2*(2*n-3)).
-#n @(transitive_le ? (exp (A n) 2))
-  [>exp_Sn >exp_Sn whd in match (exp ? 0); <times_n_1 @leA_r2
-  |>commutative_times <exp_exp_times @le_exp1 [@lt_O_S |@le_A_exp5]
-  ]
-qed.
-
-(* bounds *)
-theorem le_primr: ∀n. 1 < n → prim n \le 2*(2*n-3)/log 2 n.
-#n #lt1n @le_times_to_le_div
-  [@lt_O_log // 
-  |@(transitive_le ? (log 2 (exp n (prim n))))
-    [>commutative_times @log_exp2
-      [@le_n |@lt_to_le //]
-    |<(eq_log_exp 2 (2*(2*n-3))) in ⊢ (??%);
-      [@le_log [@le_n |@le_exp_primr]
-      |@le_n
-      ]
-    ]
-  ]
-qed.
-     
-theorem le_priml1: ∀n. O < n →
-  2*n/((log 2 n)+2) - 1 ≤ prim (2*n).
-#n #posn @le_plus_to_minus @le_times_to_le_div2
-  [>commutative_plus @lt_O_S
-  |>commutative_times in ⊢ (??%); <plus_n_Sm <plus_n_Sm in ⊢ (??(??%));
-   <plus_n_O <commutative_plus <log_exp
-    [@le_priml // | //| @le_n]
-  ]
-qed.
-
-
-
-
diff --git a/matita/matita/lib/arithmetics/chebyshev/chebyshev_psi.ma b/matita/matita/lib/arithmetics/chebyshev/chebyshev_psi.ma
index 5c4e93bba..26409e9ce 100644
--- a/matita/matita/lib/arithmetics/chebyshev/chebyshev_psi.ma
+++ b/matita/matita/lib/arithmetics/chebyshev/chebyshev_psi.ma
@@ -122,6 +122,7 @@ theorem lePsi_r2: ∀n.
        @monotonic_le_plus_r @lt_O_log //
        @le_S_S_to_le //
       ]
+
     ]
   |<Hn @le_n
   ]