From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Thu, 3 Jan 2013 16:49:01 +0000 (+0000)
Subject: refactoring
X-Git-Tag: make_still_working~1364
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=fd12cbee622c58bc45089d62c3e6f131c238beb5;p=helm.git

refactoring
---

diff --git a/matita/matita/lib/arithmetics/chebyshev/bertrand256.ma b/matita/matita/lib/arithmetics/chebyshev/bertrand256.ma
index 163125b81..c65e62381 100644
--- a/matita/matita/lib/arithmetics/chebyshev/bertrand256.ma
+++ b/matita/matita/lib/arithmetics/chebyshev/bertrand256.ma
@@ -236,6 +236,6 @@ theorem bertrand :
 ∀n. O < n → bertrand n.
 #n #posn elim (decidable_le n 256)
   [@bertrand_down //
-  |#len @le_to_bertrand2 @lt_to_le @not_le_to_lt @len]
+  |#len @bertrand_up @lt_to_le @not_le_to_lt @len]
 qed.
 
diff --git a/matita/matita/lib/arithmetics/chebyshev/chebyshev.ma b/matita/matita/lib/arithmetics/chebyshev/chebyshev.ma
deleted file mode 100644
index 50fb09fd7..000000000
--- a/matita/matita/lib/arithmetics/chebyshev/chebyshev.ma
+++ /dev/null
@@ -1,696 +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/log.ma".
-include "arithmetics/sigma_pi.ma". 
-include "arithmetics/ord.ma".
-
-(* (prim n) counts the number of prime numbers up to n included *)
-definition prim ≝ λn. ∑_{i < S n | primeb i} 1.
-
-lemma le_prim_n: ∀n. prim n ≤ n.
-#n elim n // -n #n #H
-whd in ⊢ (?%?); cases (primeb (S n)) whd in ⊢ (?%?);
-  [@le_S_S @H |@le_S @H]
-qed.
-
-lemma not_prime_times_2: ∀n. 1 < n → ¬prime (2*n).
-#n #ltn % * #H #H1 @(absurd (2 = 2*n))
-  [@H1 // %{n} //
-  |@lt_to_not_eq >(times_n_1 2) in ⊢ (?%?); @monotonic_lt_times_r //
-  ]
-qed.
-
-theorem eq_prim_prim_pred: ∀n. 1 < n → 
-  prim (2*n) = prim (pred (2*n)).
-#n #ltn 
-lapply (S_pred (2*n) ?) [>(times_n_1 0) in ⊢ (?%?); @lt_times //] #H2n
-lapply (not_prime_times_2 n ltn) #notp2n
-whd in ⊢ (??%?); >(not_prime_to_primeb_false … notp2n) whd in ⊢ (??%?);
-<H2n in ⊢ (??%?); % 
-qed.
-
-theorem le_prim_n1: ∀n. 4 ≤ n → 
-  prim (S(2*n)) ≤ n.
-#n #le4 elim le4 -le4
-  [@le_n
-  |#m #le4 cut (S (2*m) = pred (2*(S m))) [normalize //] #Heq >Heq
-   <eq_prim_prim_pred [2: @le_S_S @(transitive_le … le4) //] 
-   #H whd in ⊢ (?%?); cases (primeb (S (2*S m))) 
-    [@le_S_S @H |@le_S @H]
-  ]
-qed.
-    
-(* usefull to kill a successor in bertrand *) 
-theorem le_prim_n2: ∀n. 7 ≤ n → prim (S(2*n)) ≤ pred n.
-#n #le7 elim le7 -le7
-  [@le_n
-  |#m #le7 cut (S (2*m) = pred (2*(S m))) [normalize //] #Heq >Heq
-   <eq_prim_prim_pred [2: @le_S_S @(transitive_le … le7) //] 
-   #H whd in ⊢ (?%?); 
-   whd in ⊢ (??%); <(S_pred m) in ⊢ (??%); [2: @(transitive_le … le7) //]
-   cases (primeb (S (2*S m))) [@le_S_S @H |@le_S @H]
-  ]
-qed.
-
-lemma even_or_odd: ∀n.∃a.n=2*a ∨ n = S(2*a).
-#n elim n -n 
-  [%{0} %1 %
-  |#n * #a * #Hn [%{a} %2 @eq_f @Hn | %{(S a)} %1 >Hn normalize //
-  ]
-qed.
-
-(* la prova potrebbe essere migliorata *)
-theorem le_prim_n3: ∀n. 15 ≤ n →
-  prim n ≤ pred (n/2).
-#n #len cases (even_or_odd (pred n)) #a * #Hpredn
-  [cut (n = S (2*a)) [<Hpredn @sym_eq @S_pred @(transitive_le … len) //] #Hn
-   >Hn @(transitive_le ? (pred a))
-    [@le_prim_n2 @(le_times_to_le 2) [//|@le_S_S_to_le <Hn @len]
-    |@monotonic_pred @le_times_to_le_div //
-    ]
-  |cut (n = (2*S a)) 
-    [normalize normalize in Hpredn:(???%); <plus_n_Sm <Hpredn @sym_eq @S_pred
-     @(transitive_le … len) //] #Hn 
-   >Hn @(transitive_le ? (pred a)) 
-    [>eq_prim_prim_pred 
-      [2:@(lt_times_n_to_lt_r 2) <Hn @(transitive_le … len) //]
-     <Hn >Hpredn @le_prim_n2 @le_S_S_to_le @(lt_times_n_to_lt_r 2) <Hn @len
-    |@monotonic_pred @le_times_to_le_div //
-    ]
-  ]
-qed. 
-
-(* This is chebishev psi function *)
-definition Psi: nat → nat ≝
-  λn.∏_{p < S n | primeb p} (exp p (log p n)).
-  
-definition psi_def : ∀n. 
-  Psi n = ∏_{p < S n | primeb p} (exp p (log p n)).
-// qed.
-
-theorem le_Psil1: ∀n.
-  Psi n ≤ ∏_{p < S n | primeb p} n.
-#n cases n [@le_n |#m @le_pi #i #_ #_ @le_exp_log //]
-qed.
-
-theorem le_Psil: ∀n. Psi n ≤ exp n (prim n).
-#n <exp_sigma @le_Psil1
-qed.
-
-theorem lePsi_r2: ∀n.
-  exp n (prim n) ≤ Psi n * Psi n.
-#n elim (le_to_or_lt_eq ?? (le_O_n n)) #Hn
-  [<(exp_sigma (S n) n primeb) <times_pi
-   @le_pi #i #lti #primei 
-   cut (1<n) 
-     [@(lt_to_le_to_lt … (le_S_S_to_le … lti)) @prime_to_lt_SO 
-      @primeb_true_to_prime //] #lt1n
-   <exp_plus_times
-   @(transitive_le ? (exp i (S(log i n))))
-    [@lt_to_le @lt_exp_log @prime_to_lt_SO @primeb_true_to_prime //
-    |@le_exp 
-      [@prime_to_lt_O @primeb_true_to_prime //
-      |>(plus_n_O (log i n)) in ⊢ (?%?); >plus_n_Sm
-       @monotonic_le_plus_r @lt_O_log //
-       @le_S_S_to_le //
-      ]
-    ]
-  |<Hn @le_n
-  ]
-qed.
-
-(* an equivalent formulation for psi *)
-definition Psi': nat → nat ≝
-λn. ∏_{p < S n | primeb p} (∏_{i < log p n} p).
-
-lemma Psidef: ∀n. Psi' n = ∏_{p < S n | primeb p} (∏_{i < log p n} p).
-// qed-.
-
-theorem eq_Psi_Psi': ∀n.Psi n = Psi' n.
-#n @same_bigop // #i #lti #primebi
-@(trans_eq ? ? (exp i (∑_{x < log i n} 1)))
-  [@eq_f @sym_eq @sigma_const
-  |@sym_eq @exp_sigma
-  ]
-qed.
-
-theorem eq_pi_p_primeb_divides_b: ∀n,m.
-∏_{p<n | primeb p ∧ dividesb p m} (exp p (ord m p))
-  = ∏_{p<n | primeb p} (exp p (ord m p)).
-#n #m elim n // #n1 #Hind cases (true_or_false (primeb n1)) #Hprime
-  [>bigop_Strue in ⊢ (???%); //
-   cases (true_or_false (dividesb n1 m)) #Hdivides
-    [>bigop_Strue [@eq_f @Hind| @true_to_andb_true //]
-  |>bigop_Sfalse
-    [>not_divides_to_ord_O
-      [whd in ⊢ (???(?%?)); //
-      |@dividesb_false_to_not_divides //
-      |@primeb_true_to_prime // 
-      ]
-    |>Hprime >Hdivides % 
-    ]
-  ]
-|>bigop_Sfalse [>bigop_Sfalse // |>Hprime %]
-]
-qed.
-
-(* integrations to minimization *)
-theorem false_to_lt_max: ∀f,n,m.O < n →
-  f n = false → max m f ≤ n → max m f < n.
-#f #n #m #posn #Hfn #Hmax cases (le_to_or_lt_eq ?? Hmax) -Hmax #Hmax 
-  [//
-  |cases (exists_max_forall_false f m)
-    [* #_ #Hfmax @False_ind @(absurd ?? not_eq_true_false) //
-    |* //
-    ]
-  ]
-qed.
-
-(* integrations to minimization *)
-lemma lt_1_max_prime: ∀n. 1 <  n → 
-  1 < max (S n) (λi:nat.primeb i∧dividesb i n).
-#n #lt1n
-@(lt_to_le_to_lt ? (smallest_factor n))
-  [@lt_SO_smallest_factor //
-  |@true_to_le_max
-    [@le_S_S @le_smallest_factor_n
-    |@true_to_andb_true
-      [@prime_to_primeb_true @prime_smallest_factor_n //
-      |@divides_to_dividesb_true
-        [@lt_O_smallest_factor @lt_to_le //
-        |@divides_smallest_factor_n @lt_to_le //
-        ]
-      ]
-    ]
-  ]
-qed. 
-
-theorem lt_max_to_pi_p_primeb: ∀q,m. O<m → max (S m) (λi.primeb i ∧ dividesb i m)<q →
-  m = ∏_{p < q | primeb p ∧ dividesb p m} (exp p (ord m p)).
-#q elim q
-  [#m #posm #Hmax normalize @False_ind @(absurd … Hmax (not_le_Sn_O ?))
-  |#n #Hind #m #posm #Hmax 
-   cases (true_or_false (primeb n∧dividesb n m)) #Hcase
-    [>bigop_Strue
-      [>(exp_ord n m … posm) in ⊢ (??%?);
-        [@eq_f >(Hind (ord_rem m n))
-          [@same_bigop
-            [#x #ltxn cases (true_or_false (primeb x)) #Hx >Hx
-              [cases (true_or_false (dividesb x (ord_rem m n))) #Hx1 >Hx1
-                [@sym_eq @divides_to_dividesb_true
-                  [@prime_to_lt_O @primeb_true_to_prime //
-                  |@(transitive_divides ? (ord_rem m n))
-                    [@dividesb_true_to_divides //
-                    |@(divides_ord_rem … posm) @(transitive_lt … ltxn) 
-                     @prime_to_lt_SO @primeb_true_to_prime //
-                    ]
-                  ]
-                |@sym_eq @not_divides_to_dividesb_false
-                  [@prime_to_lt_O @primeb_true_to_prime //
-                  |@(ord_O_to_not_divides … posm)
-                    [@primeb_true_to_prime //
-                    |<(ord_ord_rem n … posm … ltxn)
-                      [@not_divides_to_ord_O
-                        [@primeb_true_to_prime //
-                        |@dividesb_false_to_not_divides //
-                        ]
-                      |@primeb_true_to_prime //
-                      |@primeb_true_to_prime @(andb_true_l ?? Hcase)
-                      ]
-                    ]
-                  ]
-                ]
-              |//
-              ]
-            |#x #ltxn #Hx @eq_f @ord_ord_rem //
-              [@primeb_true_to_prime @(andb_true_l ? ? Hcase)
-              |@primeb_true_to_prime @(andb_true_l ? ? Hx)
-              ]
-            ]
-          |@not_eq_to_le_to_lt
-            [elim (exists_max_forall_false (λi:nat.primeb i∧dividesb i (ord_rem m n)) (S(ord_rem m n)))
-              [* #Hex #Hord_rem cases Hex #x * #H6 #H7 % #H 
-               >H in Hord_rem; #Hn @(absurd ?? (not_divides_ord_rem m n posm ?))
-                [@dividesb_true_to_divides @(andb_true_r ?? Hn)
-                |@prime_to_lt_SO @primeb_true_to_prime @(andb_true_l ?? Hn)
-                ]
-              |* #Hall #Hmax >Hmax @lt_to_not_eq @prime_to_lt_O
-               @primeb_true_to_prime @(andb_true_l ?? Hcase)
-              ]
-            |@(transitive_le ? (max (S m) (λi:nat.primeb i∧dividesb i (ord_rem m n))))
-              [@le_to_le_max @le_S_S @(divides_to_le … posm) @(divides_ord_rem … posm)
-               @prime_to_lt_SO @primeb_true_to_prime @(andb_true_l ?? Hcase)
-              |@(transitive_le ? (max (S m) (λi:nat.primeb i∧dividesb i m)))
-                [@le_max_f_max_g #i #ltim #Hi 
-                 cases (true_or_false (primeb i)) #Hprimei >Hprimei
-                  [@divides_to_dividesb_true
-                    [@prime_to_lt_O @primeb_true_to_prime //
-                    |@(transitive_divides ? (ord_rem m n))
-                      [@dividesb_true_to_divides @(andb_true_r ?? Hi)
-                      |@(divides_ord_rem … posm)
-                       @prime_to_lt_SO @primeb_true_to_prime
-                       @(andb_true_l ?? Hcase)
-                      ]
-                    ]
-                  |>Hprimei in Hi; #Hi @Hi 
-                  ]
-                |@le_S_S_to_le //
-                ]
-              ]
-            ]
-          |@(lt_O_ord_rem … posm) @prime_to_lt_SO
-           @primeb_true_to_prime @(andb_true_l ?? Hcase)
-          ]
-        |@prime_to_lt_SO @primeb_true_to_prime @(andb_true_l ?? Hcase)
-        ]
-      |//
-      ]
-    |cases (le_to_or_lt_eq ?? posm) #Hm
-      [>bigop_Sfalse
-        [@(Hind … posm) @false_to_lt_max
-          [@(lt_to_le_to_lt ? (max (S m) (λi:nat.primeb i∧dividesb i m)))
-            [@lt_to_le @lt_1_max_prime // 
-            |@le_S_S_to_le //
-            ]
-          |//
-          |@le_S_S_to_le //
-          ]
-        |@Hcase
-        ]
-      |<Hm 
-       <(bigop_false (S n) ? 1 times (λp:nat.p\sup(ord 1 p))) in ⊢ (??%?);
-       @same_bigop
-        [#i #lein cases (true_or_false (primeb i)) #primei >primei //
-         @sym_eq @not_divides_to_dividesb_false
-          [@prime_to_lt_O @primeb_true_to_prime //
-          |% #divi @(absurd ?? (le_to_not_lt i 1 ?))
-            [@prime_to_lt_SO @(primeb_true_to_prime ? primei)
-            |@divides_to_le // 
-            ]
-          ]
-        |// 
-        ]
-      ]
-    ]
-  ]
-qed.
-
-(* factorization of n into primes *)
-theorem pi_p_primeb_dividesb: ∀n. O < n → 
-  n = ∏_{ p < S n | primeb p ∧ dividesb p n} (exp p (ord n p)).
-#n #posn @lt_max_to_pi_p_primeb // @lt_max_n @le_S @posn
-qed.
-
-theorem pi_p_primeb: ∀n. O < n → 
-  n = ∏_{ p < (S n) | primeb p}(exp p (ord n p)).
-#n #posn <eq_pi_p_primeb_divides_b @pi_p_primeb_dividesb //
-qed.
-
-(* more result on order functions *)
-theorem le_ord_log: ∀n,p. O < n → 1 < p →
-  ord n p ≤ log p n.
-#n #p #posn #lt1p >(exp_ord p ? lt1p posn) in ⊢ (??(??%)); 
->log_exp // @lt_O_ord_rem //
-qed.
-
-theorem sigma_p_dividesb:
-∀m,n,p. O < n → prime p → p ∤ n →
-m = ∑_{ i < m | dividesb (p\sup (S i)) ((exp p m)*n)} 1.
-#m elim m // -m #m #Hind #n #p #posn #primep #ndivp
->bigop_Strue
-  [>commutative_plus <plus_n_Sm @eq_f <plus_n_O
-   >(Hind n p posn primep ndivp) in ⊢ (? ? % ?); 
-   @same_bigop
-    [#i #ltim cases (true_or_false (dividesb (p\sup(S i)) (p\sup m*n))) #Hc >Hc
-      [@sym_eq @divides_to_dividesb_true
-        [@lt_O_exp @prime_to_lt_O //
-        |%{((exp p (m - i))*n)} <associative_times
-         <exp_plus_times @eq_f2 // @eq_f normalize @eq_f >commutative_plus 
-         @plus_minus_m_m @lt_to_le //
-        ]
-      |@False_ind @(absurd ?? (dividesb_false_to_not_divides ? ? Hc))
-       %{((exp p (m - S i))*n)} <associative_times <exp_plus_times @eq_f2
-        [@eq_f >commutative_plus @plus_minus_m_m //
-           assumption
-        |%
-        ]
-      ]
-    |// 
-    ]
-  |@divides_to_dividesb_true
-    [@lt_O_exp @prime_to_lt_O // | %{n} //]
-  ]
-qed.
-  
-theorem sigma_p_dividesb1:
-∀m,n,p,k. O < n → prime p → p ∤ n → m ≤ k →
-  m = ∑_{i < k | dividesb (p\sup (S i)) ((exp p m)*n)} 1.
-#m #n #p #k #posn #primep #ndivp #lemk
-lapply (prime_to_lt_SO ? primep) #lt1p
-lapply (prime_to_lt_O ? primep) #posp
->(sigma_p_dividesb m n p posn primep ndivp) in ⊢ (??%?);
->(pad_bigop k m) // @same_bigop
-  [#i #ltik cases (true_or_false (leb m i)) #Him > Him
-    [whd in ⊢(??%?); @sym_eq 
-     @not_divides_to_dividesb_false
-      [@lt_O_exp //
-      |lapply (leb_true_to_le … Him) -Him #Him
-       % #Hdiv @(absurd ?? (le_to_not_lt ?? Him))
-       (* <(ord_exp p m lt1p) *) >(plus_n_O m)
-       <(not_divides_to_ord_O ?? primep ndivp)
-       <(ord_exp p m lt1p)
-       <ord_times //
-        [whd <(ord_exp p (S i) lt1p)
-         @divides_to_le_ord //
-          [@lt_O_exp // 
-          |>(times_n_O O) @lt_times // @lt_O_exp //
-          ]
-        |@lt_O_exp //
-        ]
-      ]
-    |%
-    ]
-  |//
-  ]
-qed.
-
-theorem eq_ord_sigma_p:
-∀n,m,x. O < n → prime x → 
-exp x m ≤ n → n < exp x (S m) →
-ord n x= ∑_{i < m | dividesb (x\sup (S i)) n} 1.
-#n #m #x #posn #primex #Hexp #ltn
-lapply (prime_to_lt_SO ? primex) #lt1x 
->(exp_ord x n) in ⊢ (???%); // @sigma_p_dividesb1 
-  [@lt_O_ord_rem // 
-  |//
-  |@not_divides_ord_rem // 
-  |@le_S_S_to_le @(le_to_lt_to_lt ? (log x n))
-    [@le_ord_log // 
-    |@(lt_exp_to_lt x)
-      [@lt_to_le //
-      |@(le_to_lt_to_lt ? n ? ? ltn) @le_exp_log //
-      ]
-    ]
-  ]
-qed.
-    
-theorem pi_p_primeb1: ∀n. O < n → 
-n = ∏_{p < S n| primeb p} 
-  (∏_{i < log p n| dividesb (exp p (S i)) n} p).
-#n #posn >(pi_p_primeb n posn) in ⊢ (??%?);
-@same_bigop
-  [// 
-  |#p #ltp #primep >exp_sigma @eq_f @eq_ord_sigma_p 
-    [//
-    |@primeb_true_to_prime //
-    |@le_exp_log // 
-    |@lt_exp_log @prime_to_lt_SO @primeb_true_to_prime //
-    ]
-  ]
-qed.
-
-(* the factorial function *)
-theorem eq_fact_pi_p:∀n.
-  fact n = ∏_{i < S n | leb 1 i} i.
-#n elim n // #n1 #Hind whd in ⊢ (??%?); >commutative_times >bigop_Strue 
-  [@eq_f // | % ]
-qed.
-   
-theorem eq_sigma_p_div: ∀n,q.O < q →
-  ∑_{ m < S n | leb (S O) m ∧ dividesb q m} 1 =n/q.
-#n #q #posq
-@(div_mod_spec_to_eq n q ? (n \mod q) ? (n \mod q))
-  [@div_mod_spec_intro
-    [@lt_mod_m_m // 
-    |elim n
-      [normalize cases q // 
-      |#n1 #Hind cases (or_div_mod1 n1 q posq)
-        [* #divq #eqn1  >divides_to_mod_O //
-         <plus_n_O >bigop_Strue
-          [>eqn1 in ⊢ (??%?); @eq_f2
-            [<commutative_plus <plus_n_Sm <plus_n_O @eq_f
-             @(div_mod_spec_to_eq n1 q (div n1 q) (mod n1 q) ? (mod n1 q))
-              [@div_mod_spec_div_mod //
-              |@div_mod_spec_intro [@lt_mod_m_m // | //]
-              ]
-            |%
-            ]
-          |@true_to_andb_true [//|@divides_to_dividesb_true //]
-          ]
-        |* #ndiv #eqn1 >bigop_Sfalse
-          [>(mod_S … posq) 
-            [< plus_n_Sm @eq_f //
-            |cases (le_to_or_lt_eq (S (mod n1 q)) q ?)
-              [//
-              |#eqq @False_ind cases ndiv #Hdiv @Hdiv
-               %{(S(div n1 q))} <times_n_Sm <commutative_plus //
-              |@lt_mod_m_m //
-              ]
-            ]
-          |>not_divides_to_dividesb_false //
-          ]
-        ]
-      ]
-    ]
-  |@div_mod_spec_div_mod //
-  ]
-qed.       
-
-lemma timesACdef: ∀n,m. timesAC n m = n * m.
-// qed-.
-
-(* still another characterization of the factorial *)
-theorem fact_pi_p: ∀n. 
-fact n = ∏_{ p < S n | primeb p} 
-           (∏_{i < log p n} (exp p (n /(exp p (S i))))).
-#n >eq_fact_pi_p
-@(trans_eq ?? 
-  (∏_{m < S n | leb 1 m}
-   (∏_{p < S m | primeb p}
-    (∏_{i < log p m | dividesb (exp p (S i)) m} p))))
-  [@same_bigop [// |#x #Hx1 #Hx2  @pi_p_primeb1 @leb_true_to_le //]
-  |@(trans_eq ?? 
-    (∏_{m < S n | leb 1 m}
-      (∏_{p < S m | primeb p ∧ leb p m}
-       (∏_{ i < log p m | dividesb ((p)\sup(S i)) m} p))))
-    [@same_bigop
-      [//
-      |#x #Hx1 #Hx2 @same_bigop
-        [#p #ltp >(le_to_leb_true … (le_S_S_to_le …ltp))
-         cases (primeb p) //
-        |//
-        ]
-      ]
-    |@(trans_eq ?? 
-      (∏_{m < S n | leb 1 m}
-       (∏_{p < S n | primeb p ∧ leb p m}
-         (∏_{i < log p m |dividesb ((p)\sup(S i)) m} p))))
-      [@same_bigop
-        [//
-        |#p #Hp1 #Hp2 @pad_bigop1 
-          [@Hp1
-          |#i #lti #upi >lt_to_leb_false
-            [cases (primeb i) //|@lti]
-          ] 
-        ]
-      |(* make a general theorem *)
-       @(trans_eq ?? 
-        (∏_{p < S n | primeb p}
-          (∏_{m < S n | leb p m}
-           (∏_{i < log p m | dividesb (p^(S i)) m} p))))
-        [@(bigop_commute … timesAC … (lt_O_S n) (lt_O_S n))
-         #i #j #lti #ltj
-         cases (true_or_false (primeb j ∧ leb j i)) #Hc >Hc
-          [>(le_to_leb_true 1 i)
-            [//
-            |@(transitive_le ? j)
-              [@prime_to_lt_O @primeb_true_to_prime @(andb_true_l ? ? Hc)
-              |@leb_true_to_le @(andb_true_r ?? Hc)
-              ]
-            ]
-          |cases(leb 1 i) // 
-          ]
-        |@same_bigop
-          [//
-          |#p #Hp1 #Hp2
-           @(trans_eq ?? 
-            (∏_{m < S n | leb p m}
-             (∏_{i < log p n | dividesb (p\sup(S i)) m} p)))
-            [@same_bigop
-              [//
-              |#x #Hx1 #Hx2 @sym_eq 
-               @sym_eq @pad_bigop1
-                [@le_log
-                  [@prime_to_lt_SO @primeb_true_to_prime //
-                  |@le_S_S_to_le //
-                  ]
-                |#i #Hi1 #Hi2 @not_divides_to_dividesb_false
-                  [@lt_O_exp @prime_to_lt_O @primeb_true_to_prime //
-                  |@(not_to_not … (lt_to_not_le x (exp p (S i)) ?)) 
-                    [#H @divides_to_le // @(lt_to_le_to_lt ? p)
-                      [@prime_to_lt_O @primeb_true_to_prime //
-                      |@leb_true_to_le //
-                      ]
-                    |@(lt_to_le_to_lt ? (exp p (S(log p x))))
-                      [@lt_exp_log @prime_to_lt_SO @primeb_true_to_prime //
-                      |@le_exp
-                        [@ prime_to_lt_O @primeb_true_to_prime //
-                        |@le_S_S //
-                        ]
-                      ]
-                    ]
-                  ]
-                ]
-              ]
-            |@
-             (trans_eq ? ? 
-              (∏_{i < log p n}
-                (∏_{m < S n | leb p m ∧ dividesb (p\sup(S i)) m} p)))
-              [@(bigop_commute ?????? nat 1 timesAC (λm,i.p) ???) //
-               cut (p ≤ n) [@le_S_S_to_le //] #lepn 
-               @(lt_O_log … lepn) @(lt_to_le_to_lt … lepn) @prime_to_lt_SO 
-               @primeb_true_to_prime //
-              |@same_bigop
-                [//
-                |#m #ltm #_ >exp_sigma @eq_f
-                 @(trans_eq ?? 
-                  (∑_{i < S n |leb 1 i∧dividesb (p\sup(S m)) i} 1))
-                  [@same_bigop
-                    [#i #lti
-                     cases (true_or_false (dividesb (p\sup(S m)) i)) #Hc >Hc 
-                      [cases (true_or_false (leb p i)) #Hp3 >Hp3 
-                        [>le_to_leb_true 
-                          [//
-                          |@(transitive_le ? p)
-                            [@prime_to_lt_O @primeb_true_to_prime //
-                            |@leb_true_to_le //
-                            ]
-                          ]
-                        |>lt_to_leb_false
-                          [//
-                          |@not_le_to_lt
-                           @(not_to_not ??? (leb_false_to_not_le ?? Hp3)) #posi
-                           @(transitive_le ? (exp p (S m)))
-                            [>(exp_n_1 p) in ⊢ (?%?);
-                             @le_exp
-                              [@prime_to_lt_O @primeb_true_to_prime //
-                              |@le_S_S @le_O_n
-                              ]
-                            |@(divides_to_le … posi)
-                             @dividesb_true_to_divides //
-                            ]
-                          ]
-                        ]
-                      |cases (leb p i) cases (leb 1 i) //
-                      ]
-                    |// 
-                    ]
-                  |@eq_sigma_p_div @lt_O_exp
-                   @prime_to_lt_O @primeb_true_to_prime //
-                  ]
-                ]
-              ]
-            ]
-          ]
-        ]
-      ]
-    ]
-  ]
-qed.
-
-theorem fact_pi_p2: ∀n. fact (2*n) =
-∏_{p < S (2*n) | primeb p} 
-  (∏_{i < log p (2*n)}
-    (exp p (2*(n /(exp p (S i))))*(exp p (mod (2*n /(exp p (S i))) 2)))).
-#n >fact_pi_p @same_bigop
-  [//
-  |#p #ltp #primep @same_bigop
-    [//
-    |#i #lti #_ <exp_plus_times @eq_f
-     >commutative_times in ⊢ (???(?%?));
-     cut (0 < p ^ (S i)) 
-       [@lt_O_exp @prime_to_lt_O @primeb_true_to_prime //]
-     generalize in match (p ^(S i)); #a #posa
-     >(div_times_times n a 2) // >(commutative_times n 2)
-     <eq_div_div_div_times //
-    ]
-  ]
-qed.
-
-theorem fact_pi_p3: ∀n. fact (2*n) =
-∏_{p < S (2*n) | primeb p}
-  (∏_{i < log p (2*n)}(exp p (2*(n /(exp p (S i)))))) *
-∏_{p < S (2*n) | primeb p}
-  (∏_{i < log p (2*n)}(exp p (mod (2*n /(exp p (S i))) 2))).
-#n <times_pi >fact_pi_p2 @same_bigop
-  [// 
-  |#p #ltp #primep @times_pi
-  ]
-qed.
-
-theorem pi_p_primeb4: ∀n. 1 < n →
-∏_{p < S (2*n) | primeb p} 
-  (∏_{i < log p (2*n)}(exp p (2*(n /(exp p (S i))))))
-= 
-∏_{p < S n | primeb p}
-  (∏_{i < log p (2*n)}(exp p (2*(n /(exp p (S i)))))).
-#n #lt1n
-@sym_eq @(pad_bigop_nil … timesAC)
-  [@le_S_S /2 by /
-  |#i #ltn #lti %2
-   >log_i_2n //
-    [>bigop_Strue // whd in ⊢ (??(??%)?); <times_n_1
-     <exp_n_1 >eq_div_O //
-    |@le_S_S_to_le // 
-    ]
-  ]
-qed.
-   
-theorem pi_p_primeb5: ∀n. 1 < n →
-∏_{p < S (2*n) | primeb p}
-  (∏_{i < log p (2*n)} (exp p (2*(n /(exp p (S i))))))
-= 
-∏_{p < S n | primeb p} 
-  (∏_{i < log p n} (exp p (2*(n /(exp p (S i)))))).
-#n #lt1n >(pi_p_primeb4 ? lt1n) @same_bigop
-  [//
-  |#p #lepn #primebp @sym_eq @(pad_bigop_nil … timesAC) 
-    [@le_log
-      [@prime_to_lt_SO @primeb_true_to_prime //
-      |// 
-      ]
-    |#i #lelog #lti %2 >eq_div_O //
-     @(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 exp_fact_2: ∀n.
-exp (fact n) 2 = 
-  ∏_{p < S n| primeb p}
-    (∏_{i < log p n} (exp p (2*(n /(exp p (S i)))))).
-#n >fact_pi_p <exp_pi @same_bigop
-  [//
-  |#p #ltp #primebp @sym_eq 
-   @(trans_eq ?? (∏_{x < log p n} (exp (exp p (n/(exp p (S x)))) 2)))
-  [@same_bigop
-    [//
-    |#x #ltx #_ @sym_eq >commutative_times @exp_exp_times
-    ]
-  |@exp_pi
-  ]
-qed.
-
diff --git a/matita/matita/lib/arithmetics/chebyshev/chebyshev_teta.ma b/matita/matita/lib/arithmetics/chebyshev/chebyshev_teta.ma
deleted file mode 100644
index d2fb6bbd4..000000000
--- a/matita/matita/lib/arithmetics/chebyshev/chebyshev_teta.ma
+++ /dev/null
@@ -1,214 +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/binomial.ma".
-include "arithmetics/gcd.ma".
-include "arithmetics/chebyshev/chebyshev_psi.ma". 
-
-(* This is chebishev theta function *)
-
-definition theta: nat → nat ≝ λn. 
-  ∏_{p < S n| primeb p} p.
-  
-lemma theta_def: ∀n.theta n = ∏_{p < S n| primeb p} p.
-// qed.
-
-theorem lt_O_theta: ∀n. O < theta n.
-#n elim n
-  [@le_n
-  |#n1 #Hind cases (true_or_false (primeb (S n1))) #Hc 
-    [>theta_def >bigop_Strue
-      [>(times_n_O O) @lt_times // | //]
-    |>theta_def >bigop_Sfalse // 
-    ]
-  ]
-qed.
-
-theorem divides_fact_to_divides: ∀p,n. prime p → divides p n! → 
-  ∃m.O < m ∧ m \le n ∧ divides p m.  
-#p #n #primep elim n
-  [normalize in ⊢ (%→?); #H @False_ind @(absurd (p ≤1))
-    [@divides_to_le //|@lt_to_not_le @prime_to_lt_SO @primep]
-  |#n1 #Hind >factS #Hdiv
-   cases (divides_times_to_divides ??? primep Hdiv) #Hcase
-    [%{(S n1)} %[ % [@lt_O_S |@le_n] |@Hcase]
-    |cases(Hind Hcase) #a * * #posa #lea #diva
-     %{a} % [% [// |@le_S //] |//]
-    ]
-  ]
-qed.
-      
-theorem divides_fact_to_le: ∀p,n. prime p → divides p n! → p ≤ n.  
-#p #n #primep #divp cases (divides_fact_to_divides p n primep divp)
-#a * * #posa #lea #diva @(transitive_le ? a) [@divides_to_le // | //]
-qed.
-     
-theorem prime_to_divides_M: ∀m,p. 
-  prime p → S m < p → p ≤ S(2*m) → divides p (M m).      
-#m #p #primep #lemp #lep >Mdef >bceq
-cases (bc2 (S(2*m)) m ?)
-  [#q #Hq >Hq >commutative_times >div_times
-    [cases (divides_times_to_divides p (m!*(S (2*m)-m)!) q primep ?)
-      [#Hdiv @False_ind
-       cases (divides_times_to_divides p (m!) (S (2*m)-m)! primep ?)
-        [-Hdiv #Hdiv @(absurd (p ≤ m))
-          [@divides_fact_to_le //
-          |@(lt_to_not_le ?? (lt_to_le ?? lemp))
-          ]
-        |-Hdiv #Hdiv @(absurd (p ≤S m))
-          [@(divides_fact_to_le … primep)
-           cut (S m = S(2*m)-m) 
-            [normalize in ⊢ (???(?%?)); <plus_n_O 
-             >plus_n_Sm >commutative_plus @minus_plus_m_m
-            ] #HSm
-           >HSm //
-          |@lt_to_not_le //
-          ]
-        |//  
-        ]
-      |//
-      |<Hq @divides_fact [@prime_to_lt_O // |//]
-      ]
-    |>(times_n_O O) in ⊢ (?%?); @lt_times //
-    ]
-  |normalize in ⊢ (??(?%)); <plus_n_O //
-  ]
-qed.
-             
-theorem divides_pi_p_M1: ∀m.∀i. i ≤ S(S(2*m)) → 
-  ∏_{p < i | leb (S(S m)) p ∧ primeb p} p ∣ M m .
-#m #i elim i
-  [#_ @(quotient ?? (M m)) >commutative_times @times_n_1
-  |#n #Hind #len
-   cases (true_or_false (leb (S (S m)) n ∧ primeb n)) #Hc 
-    [>bigop_Strue
-      [@divides_to_divides_times
-        [@primeb_true_to_prime @(andb_true_r ?? Hc)
-        |cut (∀p.prime p → n ≤ p → ¬p∣∏_{p < n | leb (S (S m)) p∧primeb p} p)
-          [2: #Hcut @(Hcut … (le_n ?)) @primeb_true_to_prime @(andb_true_r ?? Hc)]
-         #p #primep elim n
-          [#_ normalize @(not_to_not ? (p ≤ 1)) 
-            [@divides_to_le @lt_O_S|@lt_to_not_le @prime_to_lt_SO //]
-          |#n1 #Hind1 #Hn1 cases (true_or_false (leb (S (S m)) n1∧primeb n1)) #Hc1
-            [>bigop_Strue 
-              [% #Habs cases(divides_times_to_divides ??? primep Habs)
-                [-Habs #Habs @(absurd … Hn1) @le_to_not_lt
-                 @(divides_to_le … Habs) @prime_to_lt_O
-                 @primeb_true_to_prime @(andb_true_r ?? Hc1)
-                |-Habs #Habs @(absurd … Habs) @Hind1 @lt_to_le //
-                ]
-              |@Hc1
-              ]
-            |>bigop_Sfalse // @Hind1 @lt_to_le //
-            ]
-          ]
-        |@prime_to_divides_M
-          [@primeb_true_to_prime @(andb_true_r ?? Hc)
-          |@leb_true_to_le @(andb_true_l ?? Hc)
-          |@le_S_S_to_le //
-          ]
-        |@Hind @lt_to_le //
-        ]
-      |@Hc
-      ]
-    |>bigop_Sfalse // @Hind @lt_to_le @len
-    ]
-  ]
-qed.
-
-theorem divides_pi_p_M:∀m.
-  ∏_{p < S(S(2*m)) | leb (S(S m)) p ∧ primeb p} p ∣ (M m).
-#m  @divides_pi_p_M1 @le_n
-qed.  
-
-theorem theta_pi_p_theta: ∀m. theta (S (2*m))
-= (∏_{p < S (S (2*m)) | leb (S (S m)) p∧primeb p} p)*theta (S m).
-#m >theta_def >theta_def 
-<(bigop_I ???? timesA)
->(bigop_sumI 0 (S(S m)) (S(S(2*m))) (λp.primeb p) … timesA (λp.p))
-  [2:@le_S_S @le_S_S // |3:@le_O_n]
-@eq_f2
-  [>bigop_I_gen // |@(bigop_I … timesA)]
-qed.
-
-theorem div_theta_theta: ∀m. 
-  theta (S(2*m))/theta (S m) = 
-    ∏_{p < S(S(2*m)) | leb (S(S m)) p ∧ primeb p} p.
-#m @(div_mod_spec_to_eq ????? 0 (div_mod_spec_div_mod …))
-  [@lt_O_theta
-  |@div_mod_spec_intro [@lt_O_theta |<plus_n_O @theta_pi_p_theta]
-  ]
-qed.
-                     
-theorem le_theta_M_theta: ∀m. 
-  theta (S(2*m)) ≤ (M m)*theta (S m).  
-#m >theta_pi_p_theta @le_times [2://] @divides_to_le
-  [@lt_O_bc @lt_to_le @le_S_S // | @divides_pi_p_M
-  ]
-qed.
-
-theorem lt_O_to_le_theta_exp_theta: ∀m. O < m→
-  theta (S(2*m)) < exp 2 (2*m)*theta (S m). 
-#m #posm @(le_to_lt_to_lt ? (M m*theta (S m)))
-  [@le_theta_M_theta
-  |@monotonic_lt_times_l [@lt_O_theta|@lt_M //]
-  ]
-qed.
-
-theorem theta_pred: ∀n. 1 < n → theta (2*n) = theta (pred (2*n)).
-#n #lt1n >theta_def >theta_def >bigop_Sfalse
-  [>S_pred
-    [//
-    |>(times_n_O 2) in ⊢ (?%?); @monotonic_lt_times_r @lt_to_le //
-    ]
-  |@not_prime_to_primeb_false % * #lt2n #Hprime
-   @(absurd (2=2*n))
-    [@(Hprime … (le_n ?)) %{n} // 
-    |@lt_to_not_eq >(times_n_1 2) in ⊢ (?%?); @monotonic_lt_times_r //
-    ]
-  ]
-qed.
-  
-theorem le_theta: ∀m.theta m ≤ exp 2 (2*m).
-#m @(nat_elim1 m) #m1 #Hind
-cut (∀a. 2 *a = a+a) [//] #times2
-cases (even_or_odd m1) #a * #Ha >Ha
-  [lapply Ha cases a
-    [#_ @le_n
-    |#n cases n 
-      [#_ @leb_true_to_le //
-      |#n1 #Hn1 >theta_pred
-        [cut (pred (2*S(S n1)) = (S (2*S n1)))
-          [normalize >plus_n_Sm in ⊢ (???%); //] #Hcut >Hcut
-         @(transitive_le ? (exp 2 (2*(S n1))*theta (S (S n1))))
-          [@lt_to_le @lt_O_to_le_theta_exp_theta @lt_O_S
-          |>times2 in ⊢ (??%);>exp_plus_times in ⊢ (??%); @le_times
-            [@le_exp [@lt_O_S |@monotonic_le_times_r @le_n_Sn]
-            |@Hind >Hn1 >times2 //
-            ]
-          ]
-        |@le_S_S @lt_O_S
-        ]
-      ]
-    ]
-  |lapply Ha cases a
-    [#_ @leb_true_to_le // 
-    |#n #Hn @(transitive_le ? (exp 2 (2*(S n))*theta (S (S n))))
-      [@lt_to_le @lt_O_to_le_theta_exp_theta @lt_O_S
-      |>times2 in ⊢ (??%); <plus_n_Sm <commutative_plus >plus_n_Sm
-       >exp_plus_times in ⊢ (??%); @monotonic_le_times_r
-       cut (∀a. 2*(S a) = S(S(2*a))) [#a normalize <plus_n_Sm //] #timesSS
-       <timesSS @Hind >Hn @le_S_S >times2 //
-      ]
-    ]
-  ]
-qed.
-