Cleanup.

diff --git a/helm/software/matita/library/nat/chebyshev.ma b/helm/software/matita/library/nat/chebyshev.ma
index ff7db61cc241eaeaebcd241f06ceceafecb08ceb..6ef8624e4983922e9d8276633f72a6d0163d42fc 100644 (file)
--- a/helm/software/matita/library/nat/chebyshev.ma
+++ b/helm/software/matita/library/nat/chebyshev.ma
@@ -18,7 +18,7 @@ include "nat/factorization.ma".
 include "nat/factorial2.ma".
 
 definition prim: nat \to nat \def
-\lambda n. sigma_p (S n) primeb (\lambda p.(S O)).
+\lambda n. sigma_p (S n) primeb (\lambda p.1).
 
 theorem le_prim_n: \forall n. prim n \le n.
 intros.unfold prim. elim n
@@ -152,7 +152,7 @@ apply nat_elim2;intros
   ]
 qed.
 
-(* si dovrebbe poter migliorare *)
+(* la prova potrebbe essere migliorata *)
 theorem le_prim_n3: \forall n. 15 \le n \to
 prim n \le pred (n/2).
 intros.
@@ -331,7 +331,7 @@ elim n
     ]
   ]
 qed.
-    
+
 theorem lt_max_to_pi_p_primeb: \forall q,m. O < m \to max m (\lambda i.primeb i \land divides_b i m) < q \to
 m = pi_p q (\lambda i.primeb i \land divides_b i m) (\lambda p.exp p (ord m p)).
 intro.elim q
@@ -544,7 +544,7 @@ apply pi_p_primeb_divides_b.
 assumption.
 qed.
 
-theorem le_ord_log: \forall n,p. O < n \to S O < p \to
+theorem le_ord_log: \forall n,p. O < n \to 1 < p \to
 ord n p \le log p n.
 intros.
 rewrite > (exp_ord p) in ⊢ (? ? (? ? %))
@@ -655,7 +655,7 @@ qed.
 theorem eq_ord_sigma_p:
 \forall n,m,x. O < n \to prime x \to 
 exp x m \le n \to n < exp x (S m) \to
-ord n x=sigma_p m (λi:nat.divides_b (x\sup (S i)) n) (λx:nat.S O).
+ord n x=sigma_p m (λi:nat.divides_b (x\sup (S i)) n) (λx:nat.1).
 intros.
 lapply (prime_to_lt_SO ? H1).
 rewrite > (exp_ord x n) in ⊢ (? ? ? (? ? (λi:?.? ? %) ?))
@@ -797,20 +797,20 @@ pi_p (S n) primeb
 intros.
 rewrite > eq_fact_pi_p.
 apply (trans_eq ? ? 
-  (pi_p (S n) (λi:nat.leb (S O) i) 
-   (λn.pi_p (S n) primeb 
-    (\lambda p.(pi_p (log p n) 
-     (\lambda i.divides_b (exp p (S i)) n) (\lambda i.p))))))
+  (pi_p (S n) (λm:nat.leb (S O) m) 
+   (λm.pi_p (S m) primeb 
+    (\lambda p.(pi_p (log p m) 
+     (\lambda i.divides_b (exp p (S i)) m) (\lambda i.p))))))
   [apply eq_pi_p1;intros
     [reflexivity
     |apply pi_p_primeb1.
      apply leb_true_to_le.assumption
     ]
   |apply (trans_eq ? ? 
-    (pi_p (S n) (λi:nat.leb (S O) i)
-     (λn:nat
-      .pi_p (S n) (\lambda p.primeb p\land leb p n)
-       (λp:nat.pi_p (log p n) (λi:nat.divides_b ((p)\sup(S i)) n) (λi:nat.p)))))
+    (pi_p (S n) (λm:nat.leb (S O) m)
+     (λm:nat
+      .pi_p (S m) (\lambda p.primeb p\land leb p m)
+       (λp:nat.pi_p (log p m) (λi:nat.divides_b ((p)\sup(S i)) m) (λi:nat.p)))))
     [apply eq_pi_p1
       [intros.reflexivity
       |intros.apply eq_pi_p1
@@ -825,7 +825,7 @@ apply (trans_eq ? ?
         ]
       ]
     |apply (trans_eq ? ? 
-      (pi_p (S n) (λi:nat.leb (S O) i)
+      (pi_p (S n) (λm:nat.leb (S O) m)
        (λm:nat
         .pi_p (S n) (λp:nat.primeb p∧leb p m)
          (λp:nat.pi_p (log p m) (λi:nat.divides_b ((p)\sup(S i)) m) (λi:nat.p)))))
@@ -1009,10 +1009,10 @@ intro.elim n
   |
 *)
 
-theorem fact_pi_p2: \forall n. fact ((S(S O))*n) =
-pi_p (S ((S(S O))*n)) primeb 
-  (\lambda p.(pi_p (log p ((S(S O))*n)) 
-    (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i))))*(exp p (mod ((S(S O))*n /(exp p (S i))) (S(S O)))))))).
+theorem fact_pi_p2: \forall n. fact (2*n) =
+pi_p (S (2*n)) primeb 
+  (\lambda p.(pi_p (log p (2*n)) 
+    (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i))))*(exp p (mod (2*n /(exp p (S i))) 2)))))).
 intros.rewrite > fact_pi_p.
 apply eq_pi_p1
   [intros.reflexivity
@@ -1031,13 +1031,13 @@ apply eq_pi_p1
   ]
 qed.
 
-theorem fact_pi_p3: \forall n. fact ((S(S O))*n) =
-pi_p (S ((S(S O))*n)) primeb 
-  (\lambda p.(pi_p (log p ((S(S O))*n)) 
-    (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i))))))))*
-pi_p (S ((S(S O))*n)) primeb 
-  (\lambda p.(pi_p (log p ((S(S O))*n))   
-    (\lambda i.true) (\lambda i.(exp p (mod ((S(S O))*n /(exp p (S i))) (S(S O))))))).
+theorem fact_pi_p3: \forall n. fact (2*n) =
+pi_p (S (2*n)) primeb 
+  (\lambda p.(pi_p (log p (2*n)) 
+    (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i))))))))*
+pi_p (S (2*n)) primeb 
+  (\lambda p.(pi_p (log p (2*n))   
+    (\lambda i.true) (\lambda i.(exp p (mod (2*n /(exp p (S i))) 2))))).
 intros.
 rewrite < times_pi_p.
 rewrite > fact_pi_p2.
@@ -1047,14 +1047,14 @@ apply eq_pi_p;intros
   ]
 qed.
 
-theorem pi_p_primeb4: \forall n. S O < n \to
-pi_p (S ((S(S O))*n)) primeb 
-  (\lambda p.(pi_p (log p ((S(S O))*n)) 
-    (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i))))))))
+theorem pi_p_primeb4: \forall n. 1 < n \to
+pi_p (S (2*n)) primeb 
+  (\lambda p.(pi_p (log p (2*n)) 
+    (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i))))))))
 = 
 pi_p (S n) primeb 
-  (\lambda p.(pi_p (log p (S(S O)*n)) 
-    (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i)))))))).
+  (\lambda p.(pi_p (log p (2*n)) 
+    (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i)))))))).
 intros.
 apply or_false_eq_SO_to_eq_pi_p
   [apply le_S_S.
@@ -1076,14 +1076,14 @@ apply or_false_eq_SO_to_eq_pi_p
   ]
 qed.
    
-theorem pi_p_primeb5: \forall n. S O < n \to
-pi_p (S ((S(S O))*n)) primeb 
+theorem pi_p_primeb5: \forall n. 1 < n \to
+pi_p (S (2*n)) primeb 
   (\lambda p.(pi_p (log p ((S(S O))*n)) 
-    (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i))))))))
+    (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i))))))))
 = 
 pi_p (S n) primeb 
   (\lambda p.(pi_p (log p n) 
-    (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i)))))))).
+    (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i)))))))).
 intros.
 rewrite > (pi_p_primeb4 ? H).
 apply eq_pi_p1;intros
@@ -1118,11 +1118,11 @@ apply eq_pi_p1;intros
 qed.
 
 theorem exp_fact_SSO: \forall n.
-exp (fact n) (S(S O))
+exp (fact n) 2
 = 
 pi_p (S n) primeb 
   (\lambda p.(pi_p (log p n) 
-    (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i)))))))).
+    (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i)))))))).
 intros.
 rewrite > fact_pi_p.
 rewrite < exp_pi_p.
@@ -1164,7 +1164,7 @@ reflexivity.
 qed.
 
 theorem eq_fact_B:\forall n.S O < n \to
-fact ((S(S O))*n) = exp (fact n) (S(S O)) * B((S(S O))*n).
+fact (2*n) = exp (fact n) 2 * B(2*n).
 intros. unfold B.
 rewrite > fact_pi_p3.
 apply eq_f2
@@ -1275,7 +1275,8 @@ cut ((S(S O)) < (S ((S(S(S(S O))))*n)))
     ]
   ]
 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.
@@ -1284,7 +1285,7 @@ rewrite > eq_fact_B
    apply le_B_A
   |assumption
   ]
-qed.
+qed. *)
 
 theorem lt_SO_to_le_B_exp: \forall n.S O < n \to
 B (2*n) \le exp 2 (pred (2*n)).
@@ -1331,8 +1332,8 @@ apply (le_times_to_le (exp (fact n) (S(S O))))
   ]
 qed.
 
-theorem lt_SO_to_le_exp_B: \forall n. S O < n \to
-exp (S(S O)) ((S(S O))*n) \le (S(S O)) * n * B ((S(S O))*n).
+theorem lt_SO_to_le_exp_B: \forall n. 1 < n \to
+exp 2 (2*n) \le 2 * n * B (2*n).
 intros.
 apply (le_times_to_le (exp (fact n) (S(S O))))
   [apply lt_O_exp.
@@ -1350,7 +1351,7 @@ apply (le_times_to_le (exp (fact n) (S(S O))))
 qed.
 
 theorem le_exp_B: \forall n. O < n \to
-exp (S(S O)) ((S(S O))*n) \le (S(S O)) * n * B ((S(S O))*n).
+exp 2 (2*n) \le 2 * n * B (2*n).
 intros.
 elim H
   [apply le_n
@@ -1360,9 +1361,9 @@ elim H
 qed.
 
 theorem eq_A_SSO_n: \forall n.O < n \to
-A((S(S O))*n) =
- pi_p (S ((S(S O))*n)) primeb 
-  (\lambda p.(pi_p (log p ((S(S O))*n) )   
+A(2*n) =
+ pi_p (S (2*n)) primeb 
+  (\lambda p.(pi_p (log p (2*n) )   
     (\lambda i.true) (\lambda i.(exp p (bool_to_nat (leb (S n) (exp p (S i))))))))
  *A n.
 intro.
@@ -1458,7 +1459,7 @@ cut (
 qed.
                
 theorem le_A_BA1: \forall n. O < n \to 
-A((S(S O))*n) \le B((S(S O))*n)*A n.
+A(2*n) \le B(2*n)*A n.
 intros.
 rewrite > eq_A_SSO_n
   [apply le_times_l.
@@ -2084,6 +2085,64 @@ elim (decidable_le 9 m)
   ]
 qed.       
 
+theorem le_exp_Al:\forall n. O < n \to exp 2 n \le A (2 * n).
+intros.
+apply (trans_le ? ((exp 2 (2*n))/(2*n)))
+  [apply le_times_to_le_div
+    [rewrite > (times_n_O O) in ⊢ (? % ?).
+     apply lt_times
+      [apply lt_O_S
+      |assumption
+      ]
+    |simplify in ⊢ (? ? (? ? %)).
+     rewrite < plus_n_O.
+     rewrite > exp_plus_times.
+     apply le_times_l.
+     alias id "le_times_SSO_n_exp_SSO_n" = "cic:/matita/nat/o/le_times_SSO_n_exp_SSO_n.con".
+     apply le_times_SSO_n_exp_SSO_n
+    ]
+  |apply le_times_to_le_div2
+    [rewrite > (times_n_O O) in ⊢ (? % ?).
+     apply lt_times
+      [apply lt_O_S
+      |assumption
+      ]
+    |apply (trans_le ? ((B (2*n)*(2*n))))
+      [rewrite < sym_times in ⊢ (? ? %).
+       apply le_exp_B.
+       assumption
+      |apply le_times_l.
+       apply le_B_A
+      ]
+    ]
+  ]
+qed.
+
+theorem le_exp_A2:\forall n. 1 < n \to exp 2 (n / 2) \le A n.
+intros.
+apply (trans_le ? (A(n/2*2)))
+  [rewrite > sym_times.
+   apply le_exp_Al.
+   elim (le_to_or_lt_eq ? ? (le_O_n (n/2)))
+    [assumption
+    |apply False_ind.
+     apply (lt_to_not_le ? ? H).
+     rewrite > (div_mod n 2)
+      [rewrite < H1.
+       change in ⊢ (? % ?) with (n\mod 2).
+       apply le_S_S_to_le.
+       apply lt_mod_m_m.
+       apply lt_O_S
+      |apply lt_O_S
+      ]
+    ]
+  |apply monotonic_A.
+   rewrite > (div_mod n 2) in ⊢ (? ? %).
+   apply le_plus_n_r.
+   apply lt_O_S
+  ]
+qed.
+
 theorem eq_sigma_pi_SO_n: \forall n.
 sigma_p n (\lambda i.true) (\lambda i.S O) = n.
 intro.elim n
@@ -2126,13 +2185,13 @@ qed.
 
 (* the inequalities *)
 theorem le_exp_priml: \forall n. O < n \to
-exp (S(S O)) ((S(S O))*n) \le exp ((S(S O))*n) (S(prim ((S(S O))*n))).
+exp 2 (2*n) \le exp (2*n) (S(prim (2*n))).
 intros.
-apply (trans_le ? ((((S(S O))*n*(B ((S(S O))*n))))))
+apply (trans_le ? (((2*n*(B (2*n))))))
   [apply le_exp_B.assumption
-  |change in ⊢ (? ? %) with ((((S(S O))*n))*(((S(S O))*n))\sup (prim ((S(S O))*n))).
+  |change in ⊢ (? ? %) with (((2*n))*((2*n))\sup (prim (2*n))).
    apply le_times_r.
-   apply (trans_le ? (A ((S(S O))*n)))
+   apply (trans_le ? (A (2*n)))
     [apply le_B_A
     |apply le_Al
     ]