]
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).
+intros.
+apply (le_times_to_le (exp (fact n) (S(S O))))
+ [apply lt_O_exp.
+ apply lt_O_fact
+ |rewrite < assoc_times in ⊢ (? ? %).
+ rewrite > sym_times in ⊢ (? ? (? % ?)).
+ rewrite > assoc_times in ⊢ (? ? %).
+ rewrite < eq_fact_B
+ [rewrite < sym_times.
+ apply fact3.
+ apply lt_to_le.assumption
+ |assumption
+ ]
+ ]
+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).
+intros.
+elim H
+ [apply le_n
+ |apply lt_SO_to_le_exp_B.
+ apply le_S_S.assumption
+ ]
+qed.
+
theorem eq_A_SSO_n: \forall n.O < n \to
A((S(S O))*n) =
pi_p (S ((S(S O))*n)) primeb
]
qed.
-theorem times_exp:
-\forall n,m,p. exp n p * exp m p = exp (n*m) p.
-intros.elim p
- [simplify.reflexivity
- |simplify.
- rewrite > assoc_times.
- rewrite < assoc_times in ⊢ (? ? (? ? %) ?).
- rewrite < sym_times in ⊢ (? ? (? ? (? % ?)) ?).
- rewrite > assoc_times in ⊢ (? ? (? ? %) ?).
- rewrite < assoc_times.
- rewrite < H.
- reflexivity
- ]
-qed.
-
-theorem monotonic_exp1: \forall n.
-monotonic nat le (\lambda x.(exp x n)).
-unfold monotonic. intros.
-simplify.elim n
- [apply le_n
- |simplify.
- apply le_times;assumption
- ]
-qed.
-
(* a better result *)
theorem le_A_exp3: \forall n. S O < n \to
A(n) \le exp (pred n) (S(S O))*(exp (S(S O)) ((S(S O)) * n)).
]
qed.
+theorem eq_sigma_pi_SO_n: \forall n.
+sigma_p n (\lambda i.true) (\lambda i.S O) = n.
+intro.elim n
+ [reflexivity
+ |rewrite > true_to_sigma_p_Sn
+ [rewrite > H.reflexivity
+ |reflexivity
+ ]
+ ]
+qed.
+
+theorem leA_prim: \forall n.
+exp n (prim n) \le A n * pi_p (S n) primeb (λp:nat.p).
+intro.
+unfold prim.
+rewrite < (exp_sigma_p (S n) n primeb).
+unfold A.
+rewrite < times_pi_p.
+apply le_pi_p.
+intros.
+rewrite > sym_times.
+change in ⊢ (? ? %) with (exp i (S (log i n))).
+apply lt_to_le.
+apply lt_exp_log.
+apply prime_to_lt_SO.
+apply primeb_true_to_prime.
+assumption.
+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))).
+intros.
+apply (trans_le ? ((((S(S O))*n*(B ((S(S O))*n))))))
+ [apply le_exp_B.assumption
+ |change in ⊢ (? ? %) with ((((S(S O))*n))*(((S(S O))*n))\sup (prim ((S(S O))*n))).
+ apply le_times_r.
+ apply (trans_le ? (A ((S(S O))*n)))
+ [apply le_B_A
+ |apply le_Al
+ ]
+ ]
+qed.
+
+theorem le_priml: \forall n. O < n \to
+(S(S O))*n \le (S (log (S(S O)) ((S(S O))*n)))*S(prim ((S(S O))*n)).
+intros.
+rewrite < (eq_log_exp (S(S O))) in ⊢ (? % ?)
+ [apply (trans_le ? ((log (S(S O)) (exp ((S(S O))*n) (S(prim ((S(S O))*n)))))))
+ [apply le_log
+ [apply le_n
+ |apply lt_O_exp.apply lt_O_S
+ |apply le_exp_priml.assumption
+ ]
+ |rewrite > sym_times in ⊢ (? ? %).
+ apply log_exp1.
+ apply le_n
+ ]
+ |apply le_n
+ ]
+qed.
+
+theorem le_exp_primr: \forall n. S O < n \to
+exp n (prim n) \le exp (pred n) ((S(S O))*(S(S O)))*(exp (S(S O)) ((S(S O))*(S(S O)) * n)).
+intros.
+apply (trans_le ? (exp (A n) (S(S O))))
+ [change in ⊢ (? ? %) with ((A n)*((A n)*(S O))).
+ rewrite < times_n_SO.
+ apply leA_r2
+ |apply (trans_le ? (exp (exp (pred n) (S(S O))*(exp (S(S O)) ((S(S O)) * n))) (S(S O))))
+ [apply monotonic_exp1.
+ apply le_A_exp3.
+ assumption
+ |rewrite < times_exp.
+ rewrite > exp_exp_times.
+ rewrite > exp_exp_times.
+ rewrite > sym_times in ⊢ (? (? ? (? ? %)) ?).
+ rewrite < assoc_times in ⊢ (? (? ? (? ? %)) ?).
+ apply le_n
+ ]
+ ]
+qed.
(* da spostare *)