--- /dev/null
+(**************************************************************************)
+(* __ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+include "nat/iteration2.ma".
+include "nat/div_and_mod_diseq.ma".
+include "nat/binomial.ma".
+include "nat/log.ma".
+include "nat/chebyshev.ma".
+
+theorem sigma_p_div_exp: \forall n,m.
+sigma_p n (\lambda i.true) (\lambda i.m/(exp (S(S O)) i)) \le
+((S(S O))*m*(exp (S(S O)) n) - (S(S O))*m)/(exp (S(S O)) n).
+intros.
+elim n
+ [apply le_O_n.
+ |rewrite > true_to_sigma_p_Sn
+ [apply (trans_le ? (m/(S(S O))\sup(n1)+((S(S O))*m*(S(S O))\sup(n1)-(S(S O))*m)/(S(S O))\sup(n1)))
+ [apply le_plus_r.assumption
+ |rewrite > assoc_times in ⊢ (? ? (? (? % ?) ?)).
+ rewrite < distr_times_minus.
+ change in ⊢ (? ? (? ? %)) with ((S(S O))*(exp (S(S O)) n1)).
+ rewrite > sym_times in ⊢ (? ? (? % ?)).
+ rewrite > sym_times in ⊢ (? ? (? ? %)).
+ rewrite < lt_to_lt_to_eq_div_div_times_times
+ [apply (trans_le ? ((m+((S(S O))*m*((S(S O)))\sup(n1)-(S(S O))*m))/((S(S O)))\sup(n1)))
+ [apply le_plus_div.
+ apply lt_O_exp.
+ apply lt_O_S
+ |change in ⊢ (? (? (? ? (? ? %)) ?) ?) with (m + (m +O)).
+ rewrite < plus_n_O.
+ rewrite < eq_minus_minus_minus_plus.
+ rewrite > sym_plus.
+ rewrite > sym_times in ⊢ (? (? (? (? (? (? % ?) ?) ?) ?) ?) ?).
+ rewrite > assoc_times.
+ rewrite < plus_minus_m_m
+ [apply le_n
+ |apply le_plus_to_minus_r.
+ rewrite > plus_n_O in ⊢ (? (? ? %) ?).
+ change in ⊢ (? % ?) with ((S(S O))*m).
+ rewrite > sym_times.
+ apply le_times_r.
+ rewrite > times_n_SO in ⊢ (? % ?).
+ apply le_times_r.
+ apply lt_O_exp.
+ apply lt_O_S
+ ]
+ ]
+ |apply lt_O_S
+ |apply lt_O_exp.
+ apply lt_O_S
+ ]
+ ]
+ |reflexivity
+ ]
+ ]
+qed.
+
+theorem le_fact_exp: \forall i,m. i \le m \to m!≤ m \sup i*(m-i)!.
+intro.elim i
+ [rewrite < minus_n_O.
+ simplify.rewrite < plus_n_O.
+ apply le_n
+ |simplify.
+ apply (trans_le ? ((m)\sup(n)*(m-n)!))
+ [apply H.
+ apply lt_to_le.assumption
+ |rewrite > sym_times in ⊢ (? ? (? % ?)).
+ rewrite > assoc_times.
+ apply le_times_r.
+ generalize in match H1.
+ cases m;intro
+ [apply False_ind.
+ apply (lt_to_not_le ? ? H2).
+ apply le_O_n
+ |rewrite > minus_Sn_m.
+ simplify.
+ apply le_plus_r.
+ apply le_times_l.
+ apply le_minus_m.
+ apply le_S_S_to_le.
+ assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem le_fact_exp1: \forall n. S O < n \to (S(S O))*n!≤ n \sup n.
+intros.elim H
+ [apply le_n
+ |change with ((S(S O))*((S n1)*(fact n1)) \le (S n1)*(exp (S n1) n1)).
+ rewrite < assoc_times.
+ rewrite < sym_times in ⊢ (? (? % ?) ?).
+ rewrite > assoc_times.
+ apply le_times_r.
+ apply (trans_le ? (exp n1 n1))
+ [assumption
+ |apply monotonic_exp1.
+ apply le_n_Sn
+ ]
+ ]
+qed.
+
+theorem le_exp_sigma_p_exp: \forall n. (exp (S n) n) \le
+sigma_p (S n) (\lambda k.true) (\lambda k.(exp n n)/k!).
+intro.
+rewrite > exp_S_sigma_p.
+apply le_sigma_p.
+intros.unfold bc.
+apply (trans_le ? ((exp n (n-i))*((n \sup i)/i!)))
+ [rewrite > sym_times.
+ apply le_times_r.
+ rewrite > sym_times.
+ rewrite < eq_div_div_div_times
+ [apply monotonic_div
+ [apply lt_O_fact
+ |apply le_times_to_le_div2
+ [apply lt_O_fact
+ |apply le_fact_exp.
+ apply le_S_S_to_le.
+ assumption
+ ]
+ ]
+ |apply lt_O_fact
+ |apply lt_O_fact
+ ]
+ |rewrite > (plus_minus_m_m ? i) in ⊢ (? ? (? (? ? %) ?))
+ [rewrite > exp_plus_times.
+ apply le_times_div_div_times.
+ apply lt_O_fact
+ |apply le_S_S_to_le.
+ assumption
+ ]
+ ]
+qed.
+
+theorem lt_exp_sigma_p_exp: \forall n. S O < n \to
+(exp (S n) n) <
+sigma_p (S n) (\lambda k.true) (\lambda k.(exp n n)/k!).
+intros.
+rewrite > exp_S_sigma_p.
+apply lt_sigma_p
+ [intros.unfold bc.
+ apply (trans_le ? ((exp n (n-i))*((n \sup i)/i!)))
+ [rewrite > sym_times.
+ apply le_times_r.
+ rewrite > sym_times.
+ rewrite < eq_div_div_div_times
+ [apply monotonic_div
+ [apply lt_O_fact
+ |apply le_times_to_le_div2
+ [apply lt_O_fact
+ |apply le_fact_exp.
+ apply le_S_S_to_le.
+ assumption
+ ]
+ ]
+ |apply lt_O_fact
+ |apply lt_O_fact
+ ]
+ |rewrite > (plus_minus_m_m ? i) in ⊢ (? ? (? (? ? %) ?))
+ [rewrite > exp_plus_times.
+ apply le_times_div_div_times.
+ apply lt_O_fact
+ |apply le_S_S_to_le.
+ assumption
+ ]
+ ]
+ |apply (ex_intro ? ? n).
+ split
+ [split
+ [apply le_n
+ |reflexivity
+ ]
+ |rewrite < minus_n_n.
+ rewrite > bc_n_n.
+ simplify.unfold lt.
+ apply le_times_to_le_div
+ [apply lt_O_fact
+ |rewrite > sym_times.
+ apply le_fact_exp1.
+ assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem le_exp_SSO_fact:\forall n.
+exp (S(S O)) (pred n) \le n!.
+intro.elim n
+ [apply le_n
+ |change with ((S(S O))\sup n1 ≤(S n1)*n1!).
+ apply (nat_case1 n1);intros
+ [apply le_n
+ |change in ⊢ (? % ?) with ((S(S O))*exp (S(S O)) (pred (S m))).
+ apply le_times
+ [apply le_S_S.apply le_S_S.apply le_O_n
+ |rewrite < H1.assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem lt_SO_to_lt_exp_Sn_n_SSSO: \forall n. S O < n \to
+(exp (S n) n) < (S(S(S O)))*(exp n n).
+intros.
+apply (lt_to_le_to_lt ? (sigma_p (S n) (\lambda k.true) (\lambda k.(exp n n)/k!)))
+ [apply lt_exp_sigma_p_exp.assumption
+ |apply (trans_le ? (sigma_p (S n) (\lambda i.true) (\lambda i.(exp n n)/(exp (S(S O)) (pred i)))))
+ [apply le_sigma_p.intros.
+ apply le_times_to_le_div
+ [apply lt_O_exp.
+ apply lt_O_S
+ |apply (trans_le ? ((S(S O))\sup (pred i)* n \sup n/i!))
+ [apply le_times_div_div_times.
+ apply lt_O_fact
+ |apply le_times_to_le_div2
+ [apply lt_O_fact
+ |rewrite < sym_times.
+ apply le_times_r.
+ apply le_exp_SSO_fact
+ ]
+ ]
+ ]
+ |rewrite > eq_sigma_p_pred
+ [rewrite > div_SO.
+ rewrite > sym_plus.
+ change in ⊢ (? ? %) with ((exp n n)+(S(S O)*(exp n n))).
+ apply le_plus_r.
+ apply (trans_le ? (((S(S O))*(exp n n)*(exp (S(S O)) n) - (S(S O))*(exp n n))/(exp (S(S O)) n)))
+ [apply sigma_p_div_exp
+ |apply le_times_to_le_div2
+ [apply lt_O_exp.
+ apply lt_O_S.
+ |apply le_minus_m
+ ]
+ ]
+ |reflexivity
+ ]
+ ]
+ ]
+qed.
+
+theorem lt_exp_Sn_n_SSSO: \forall n.
+(exp (S n) n) < (S(S(S O)))*(exp n n).
+intro.cases n;intros
+ [simplify.apply le_S.apply le_n
+ |cases n1;intros
+ [simplify.apply le_n
+ |apply lt_SO_to_lt_exp_Sn_n_SSSO.
+ apply le_S_S.apply le_S_S.apply le_O_n
+ ]
+ ]
+qed.
+
+theorem lt_exp_Sn_m_SSSO: \forall n,m. O < m \to
+divides n m \to
+(exp (S n) m) < (exp (S(S(S O))) (m/n)) *(exp n m).
+intros.
+elim H1.rewrite > H2.
+rewrite < exp_exp_times.
+rewrite < exp_exp_times.
+rewrite > sym_times in ⊢ (? ? (? (? ? (? % ?)) ?)).
+rewrite > lt_O_to_div_times
+ [rewrite > times_exp.
+ apply lt_exp1
+ [apply (O_lt_times_to_O_lt ? n).
+ rewrite > sym_times.
+ rewrite < H2.
+ assumption
+ |apply lt_exp_Sn_n_SSSO
+ ]
+ |apply (O_lt_times_to_O_lt ? n2).
+ rewrite < H2.
+ assumption
+ ]
+qed.
+
+theorem le_log_exp_Sn_log_exp_n: \forall n,m,p. O < m \to S O < p \to
+divides n m \to
+log p (exp (S n) m) \le S((m/n)*S(log p (S(S(S O))))) + log p (exp n m).
+intros.
+apply (trans_le ? (log p (((S(S(S O))))\sup(m/n)*((n)\sup(m)))))
+ [apply le_log
+ [assumption
+ |apply lt_to_le.
+ apply lt_exp_Sn_m_SSSO;assumption
+ ]
+ |apply (trans_le ? (S(log p (exp (S(S(S O))) (m/n)) + log p (exp n m))))
+ [apply log_times.
+ assumption
+ |change in ⊢ (? ? %) with (S (m/n*S (log p (S(S(S O))))+log p ((n)\sup(m)))).
+ apply le_S_S.
+ apply le_plus_l.
+ apply log_exp1.
+ assumption
+ ]
+ ]
+qed.
+
+theorem le_log_exp_fact_sigma_p: \forall a,b,n,p. S O < p \to
+O < a \to a \le b \to b \le n \to
+log p (exp b n!) - log p (exp a n!) \le
+sigma_p b (\lambda i.leb a i) (\lambda i.S((n!/i)*S(log p (S(S(S O)))))).
+intros 7.
+elim b
+ [simplify.
+ apply (lt_O_n_elim ? (lt_O_fact n)).intro.
+ apply le_n.
+ |apply (bool_elim ? (leb a n1));intro
+ [rewrite > true_to_sigma_p_Sn
+ [apply (trans_le ? (S (n!/n1*S (log p (S(S(S O)))))+(log p ((n1)\sup(n!))-log p ((a)\sup(n!)))))
+ [rewrite > sym_plus.
+ rewrite > plus_minus
+ [apply le_plus_to_minus_r.
+ rewrite < plus_minus_m_m
+ [rewrite > sym_plus.
+ apply le_log_exp_Sn_log_exp_n
+ [apply lt_O_fact
+ |assumption
+ |apply divides_fact;
+ [apply (trans_le ? ? ? H1);apply leb_true_to_le;assumption
+ |apply lt_to_le;assumption]]
+ |apply le_log
+ [assumption
+ |cut (O = exp O n!)
+ [apply monotonic_exp1;constructor 2;
+ apply leb_true_to_le;assumption
+ |elim n
+ [reflexivity
+ |simplify;rewrite > exp_plus_times;rewrite < H6;
+ rewrite > sym_times;rewrite < times_n_O;reflexivity]]]]
+ |apply le_log
+ [assumption
+ |apply monotonic_exp1;apply leb_true_to_le;assumption]]
+ |rewrite > sym_plus;rewrite > sym_plus in \vdash (? ? %);apply le_minus_to_plus;
+ rewrite < minus_plus_m_m;apply H3;apply lt_to_le;assumption]
+ |assumption]
+ |lapply (not_le_to_lt ? ? (leb_false_to_not_le ? ? H5));
+ rewrite > eq_minus_n_m_O
+ [apply le_O_n
+ |apply le_log
+ [assumption
+ |apply monotonic_exp1;assumption]]]]
+qed.
+
+theorem le_exp_div:\forall n,m,q. O < m \to
+exp (n/m) q \le (exp n q)/(exp m q).
+intros.
+apply le_times_to_le_div
+ [apply lt_O_exp.
+ assumption
+ |rewrite > times_exp.
+ rewrite < sym_times.
+ apply monotonic_exp1.
+ rewrite > (div_mod n m) in ⊢ (? ? %)
+ [apply le_plus_n_r
+ |assumption
+ ]
+ ]
+qed.
+
+theorem le_log_div_sigma_p: \forall a,b,n,p. S O < p \to
+O < a \to a \le b \to b \le n \to
+log p (b/a) \le
+(sigma_p b (\lambda i.leb a i) (\lambda i.S((n!/i)*S(log p (S(S(S O)))))))/n!.
+intros.
+apply le_times_to_le_div
+ [apply lt_O_fact
+ |apply (trans_le ? (log p (exp (b/a) n!)))
+ [apply log_exp2
+ [assumption
+ |apply le_times_to_le_div
+ [assumption
+ |rewrite < times_n_SO.
+ assumption
+ ]
+ ]
+ |apply (trans_le ? (log p ((exp b n!)/(exp a n!))))
+ [apply le_log
+ [assumption
+ |apply le_exp_div.assumption
+ ]
+ |apply (trans_le ? (log p (exp b n!) - log p (exp a n!)))
+ [apply log_div
+ [assumption
+ |apply lt_O_exp.
+ assumption
+ |apply monotonic_exp1.
+ assumption
+ ]
+ |apply le_log_exp_fact_sigma_p;assumption
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem sigma_p_log_div1: \forall n,b. S O < b \to
+sigma_p (S n) (\lambda p.(primeb p \land (leb (S n) (p*p)))) (\lambda p.(log b (n/p)))
+\le sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)*S(log b (S(S(S O)))))))/n!
+).
+intros.
+apply le_sigma_p.intros.
+apply le_log_div_sigma_p
+ [assumption
+ |apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ apply (andb_true_true ? ? H2)
+ |apply le_S_S_to_le.
+ assumption
+ |apply le_n
+ ]
+qed.
+
+theorem sigma_p_log_div2: \forall n,b. S O < b \to
+sigma_p (S n) (\lambda p.(primeb p \land (leb (S n) (p*p)))) (\lambda p.(log b (n/p)))
+\le
+(sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)))))*S(log b (S(S(S O))))/n!).
+intros.
+apply (trans_le ? (sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)*S(log b (S(S(S O)))))))/n!
+)))
+ [apply sigma_p_log_div1.assumption
+ |apply le_times_to_le_div
+ [apply lt_O_fact
+ |rewrite > distributive_times_plus_sigma_p.
+ rewrite < sym_times in ⊢ (? ? %).
+ rewrite > distributive_times_plus_sigma_p.
+ apply le_sigma_p.intros.
+ apply (trans_le ? ((n!*(sigma_p n (λj:nat.leb i j) (λi:nat.S (n!/i*S (log b (S(S(S O)))))))/n!)))
+ [apply le_times_div_div_times.
+ apply lt_O_fact
+ |rewrite > sym_times.
+ rewrite > lt_O_to_div_times
+ [rewrite > distributive_times_plus_sigma_p.
+ apply le_sigma_p.intros.
+ rewrite < times_n_Sm in ⊢ (? ? %).
+ rewrite > plus_n_SO.
+ rewrite > sym_plus.
+ apply le_plus
+ [apply le_S_S.apply le_O_n
+ |rewrite < sym_times.
+ apply le_n
+ ]
+ |apply lt_O_fact
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem sigma_p_log_div: \forall n,b. S O < b \to
+sigma_p (S n) (\lambda p.(primeb p \land (leb (S n) (p*p)))) (\lambda p.(log b (n/p)))
+\le (sigma_p n (\lambda i.leb (S n) (i*i)) (\lambda i.(prim i)*S(n!/i)))*S(log b (S(S(S O))))/n!
+.
+intros.
+apply (trans_le ? (sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)))))*S(log b (S(S(S O))))/n!))
+ [apply sigma_p_log_div2.assumption
+ |apply monotonic_div
+ [apply lt_O_fact
+ |apply le_times_l.
+ unfold prim.
+ cut
+ (sigma_p (S n) (λp:nat.primeb p∧leb (S n) (p*p))
+ (λp:nat.sigma_p n (λi:nat.leb p i) (λi:nat.S (n!/i)))
+ = sigma_p n (λi:nat.leb (S n) (i*i))
+ (λi:nat.sigma_p (S n) (\lambda p.primeb p \land leb (S n) (p*p) \land leb p i) (λp:nat.S (n!/i))))
+ [rewrite > Hcut.
+ apply le_sigma_p.intros.
+ rewrite < sym_times.
+ rewrite > distributive_times_plus_sigma_p.
+ rewrite < times_n_SO.
+ cut
+ (sigma_p (S n) (λp:nat.primeb p∧leb (S n) (p*p) \land leb p i) (λp:nat.S (n!/i))
+ = sigma_p (S i) (\lambda p.primeb p \land leb (S n) (p*p) \land leb p i) (λp:nat.S (n!/i)))
+ [rewrite > Hcut1.
+ apply le_sigma_p1.intros.
+ rewrite < andb_sym.
+ rewrite < andb_sym in ⊢ (? (? (? (? ? %)) ?) ?).
+ apply (bool_elim ? (leb i1 i));intros
+ [apply (bool_elim ? (leb (S n) (i1*i1)));intros
+ [apply le_n
+ |apply le_O_n
+ ]
+ |apply le_O_n
+ ]
+ |apply or_false_to_eq_sigma_p
+ [apply le_S.assumption
+ |intros.
+ left.rewrite > (lt_to_leb_false i1 i)
+ [rewrite > andb_sym.reflexivity
+ |assumption
+ ]
+ ]
+ ]
+ |apply sigma_p_sigma_p.intros.
+ apply (bool_elim ? (leb x y));intros
+ [apply (bool_elim ? (leb (S n) (x*x)));intros
+ [rewrite > le_to_leb_true in ⊢ (? ? ? (? % ?))
+ [reflexivity
+ |apply (trans_le ? (x*x))
+ [apply leb_true_to_le.assumption
+ |apply le_times;apply leb_true_to_le;assumption
+ ]
+ ]
+ |rewrite < andb_sym in ⊢ (? ? (? % ?) ?).
+ rewrite < andb_sym in ⊢ (? ? ? (? ? (? % ?))).
+ rewrite < andb_sym in ⊢ (? ? ? %).
+ reflexivity
+ ]
+ |rewrite < andb_sym.
+ rewrite > andb_assoc in ⊢ (? ? ? %).
+ rewrite < andb_sym in ⊢ (? ? ? (? % ?)).
+ reflexivity
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem le_sigma_p_div_log_div_pred_log : \forall n,b,m. S O < b \to b*b \leq n \to
+sigma_p (S n) (\lambda i.leb (S n) (i*i)) (\lambda i.m/(log b i))
+\leq ((S (S O)) * n * m)/(pred (log b n)).
+intros.
+apply (trans_le ? (sigma_p (S n)
+ (\lambda i.leb (S n) (i*i)) (\lambda i.(S (S O))*m/(pred (log b n)))))
+ [apply le_sigma_p;intros;apply le_times_to_le_div
+ [rewrite > minus_n_O in ⊢ (? ? (? %));rewrite < eq_minus_S_pred;
+ apply le_plus_to_minus_r;simplify;
+ rewrite < (eq_log_exp b ? H);
+ apply le_log;
+ [assumption
+ |simplify;rewrite < times_n_SO;assumption]
+ |apply (trans_le ? ((pred (log b n) * m)/log b i))
+ [apply le_times_div_div_times;apply lt_O_log
+ [elim (le_to_or_lt_eq ? ? (le_O_n i))
+ [assumption
+ |apply False_ind;apply not_eq_true_false;rewrite < H3;rewrite < H4;
+ reflexivity]
+ |apply (le_exp_to_le1 ? ? (S (S O)))
+ [apply lt_O_S;
+ |apply (trans_le ? (S n))
+ [apply le_S;simplify;rewrite < times_n_SO;assumption
+ |rewrite > exp_SSO;apply leb_true_to_le;assumption]]]
+ |apply le_times_to_le_div2
+ [apply lt_O_log
+ [elim (le_to_or_lt_eq ? ? (le_O_n i))
+ [assumption
+ |apply False_ind;apply not_eq_true_false;rewrite < H3;rewrite < H4;
+ reflexivity]
+ |apply (le_exp_to_le1 ? ? (S (S O)))
+ [apply lt_O_S;
+ |apply (trans_le ? (S n))
+ [apply le_S;simplify;rewrite < times_n_SO;assumption
+ |rewrite > exp_SSO;apply leb_true_to_le;assumption]]]
+ |rewrite > sym_times in \vdash (? ? %);rewrite < assoc_times;
+ apply le_times_l;rewrite > sym_times;
+ rewrite > minus_n_O in \vdash (? (? %) ?);
+ rewrite < eq_minus_S_pred;apply le_plus_to_minus;
+ simplify;rewrite < plus_n_O;apply (trans_le ? (log b (i*i)))
+ [apply le_log
+ [assumption
+ |apply lt_to_le;apply leb_true_to_le;assumption]
+ |rewrite > sym_plus;simplify;apply log_times;assumption]]]]
+ |rewrite > times_n_SO in \vdash (? (? ? ? (\lambda i:?.%)) ?);
+ rewrite < distributive_times_plus_sigma_p;
+ apply (trans_le ? ((((S (S O))*m)/(pred (log b n)))*n))
+ [apply le_times_r;apply (trans_le ? (sigma_p (S n) (\lambda i:nat.leb (S O) (i*i)) (\lambda Hbeta1:nat.S O)))
+ [apply le_sigma_p1;intros;do 2 rewrite < times_n_SO;
+ apply (bool_elim ? (leb (S n) (i*i)))
+ [intro;cut (leb (S O) (i*i) = true)
+ [rewrite > Hcut;apply le_n
+ |apply le_to_leb_true;apply (trans_le ? (S n))
+ [apply le_S_S;apply le_O_n
+ |apply leb_true_to_le;assumption]]
+ |intro;simplify in \vdash (? % ?);apply le_O_n]
+ |elim n
+ [simplify;apply le_n
+ |apply (bool_elim ? (leb (S O) ((S n1)*(S n1))));intro
+ [rewrite > true_to_sigma_p_Sn
+ [change in \vdash (? % ?) with (S (sigma_p (S n1) (\lambda i:nat.leb (S O) (i*i)) (\lambda Hbeta1:nat.S O)));
+ apply le_S_S;assumption
+ |assumption]
+ |rewrite > false_to_sigma_p_Sn
+ [apply le_S;assumption
+ |assumption]]]]
+ |rewrite > sym_times in \vdash (? % ?);
+ rewrite > sym_times in \vdash (? ? (? (? % ?) ?));
+ rewrite > assoc_times;
+ apply le_times_div_div_times;
+ rewrite > minus_n_O in ⊢ (? ? (? %));rewrite < eq_minus_S_pred;
+ apply le_plus_to_minus_r;simplify;
+ rewrite < (eq_log_exp b ? H);
+ apply le_log;
+ [assumption
+ |simplify;rewrite < times_n_SO;assumption]]]
+qed.
+
+lemma neper_sigma_p1 : \forall n,a.n \divides a \to
+exp (a * S n) n =
+sigma_p (S n) (\lambda x.true) (\lambda k.(bc n k)*(exp n (n-k))*(exp a n)).
+intros;rewrite < times_exp;rewrite > exp_S_sigma_p;
+rewrite > distributive_times_plus_sigma_p;
+apply eq_sigma_p;intros;
+ [reflexivity
+ |rewrite > sym_times;reflexivity;]
+qed.
+
+lemma eq_exp_pi_p : \forall a,n.(exp a n) = pi_p n (\lambda x.true) (\lambda x.a).
+intros;elim n
+ [simplify;reflexivity
+ |change in \vdash (? ? % ?) with (a*exp a n1);rewrite > true_to_pi_p_Sn
+ [apply eq_f2
+ [reflexivity
+ |assumption]
+ |reflexivity]]
+qed.
+
+lemma eq_fact_pi_p : \forall n.n! = pi_p n (\lambda x.true) (\lambda x.S x).
+intros;elim n
+ [simplify;reflexivity
+ |rewrite > true_to_pi_p_Sn
+ [change in \vdash (? ? % ?) with (S n1*n1!);apply eq_f2
+ [reflexivity
+ |assumption]
+ |reflexivity]]
+qed.
+
+lemma divides_pi_p : \forall m,n,p,f.m \leq n \to pi_p m p f \divides pi_p n p f.
+intros;elim H
+ [apply divides_n_n
+ |apply (bool_elim ? (p n1));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > sym_times;rewrite > times_n_SO;apply divides_times
+ [assumption
+ |apply divides_SO_n]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]
+qed.
+
+lemma divides_fact_fact : \forall m,n.m \leq n \to m! \divides n!.
+intros;do 2 rewrite > eq_fact_pi_p;apply divides_pi_p;assumption.
+qed.
+
+lemma divides_times_to_eq : \forall a,b,c.O < c \to c \divides a \to a*b/c = a/c*b.
+intros;elim H1;rewrite > H2;cases H;rewrite > assoc_times;do 2 rewrite > div_times;
+reflexivity;
+qed.
+
+lemma divides_pi_p_to_eq : \forall k,p,f,g.(\forall x.p x = true \to O < g x \land (g x \divides f x)) \to
+pi_p k p f/pi_p k p g = pi_p k p (\lambda x.(f x)/(g x)).
+intros;
+cut (\forall k1.(pi_p k1 p g \divides pi_p k1 p f))
+ [|intro;elim k1
+ [simplify;apply divides_n_n
+ |apply (bool_elim ? (p n));intro;
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_pi_p_Sn
+ [elim (H n)
+ [elim H4;elim H1;rewrite > H5;rewrite > H6;
+ rewrite < assoc_times;rewrite > assoc_times in ⊢ (? ? (? % ?));
+ rewrite > sym_times in ⊢ (? ? (? (? ? %) ?));
+ rewrite > assoc_times;rewrite > assoc_times;
+ apply divides_times
+ [apply divides_n_n
+ |rewrite > times_n_SO in \vdash (? % ?);apply divides_times
+ [apply divides_n_n
+ |apply divides_SO_n]]
+ |assumption]
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_pi_p_Sn
+ [assumption
+ |assumption]
+ |assumption]]]]
+elim k
+ [simplify;reflexivity
+ |apply (bool_elim ? (p n))
+ [intro;rewrite > true_to_pi_p_Sn;
+ [rewrite > true_to_pi_p_Sn;
+ [rewrite > true_to_pi_p_Sn;
+ [elim (H n);
+ [elim H4;rewrite > H5;rewrite < eq_div_div_div_times;
+ [cases H3
+ [rewrite > assoc_times;do 2 rewrite > div_times;
+ elim (Hcut n);rewrite > H6;rewrite < assoc_times;
+ rewrite < sym_times in \vdash (? ? (? (? % ?) ?) ?);
+ cut (O < pi_p n p g)
+ [rewrite < H1;rewrite > H6;cases Hcut1;
+ rewrite > assoc_times;do 2 rewrite > div_times;reflexivity
+ |elim n
+ [simplify;apply le_n
+ |apply (bool_elim ? (p n3));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > (times_n_O O);apply lt_times
+ [elim (H n3);assumption
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]]
+ |rewrite > assoc_times;do 2 rewrite > div_times;
+ elim (Hcut n);rewrite > H7;rewrite < assoc_times;
+ rewrite < sym_times in \vdash (? ? (? (? % ?) ?) ?);
+ cut (O < pi_p n p g)
+ [rewrite < H1;rewrite > H7;cases Hcut1;
+ rewrite > assoc_times;do 2 rewrite > div_times;reflexivity
+ |elim n
+ [simplify;apply le_n
+ |apply (bool_elim ? (p n3));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > (times_n_O O);apply lt_times
+ [elim (H n3);assumption
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]]]
+ |assumption
+ |(*già usata 2 volte: fattorizzare*)
+ elim n
+ [simplify;apply le_n
+ |apply (bool_elim ? (p n1));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > (times_n_O O);apply lt_times
+ [elim (H n1);assumption
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]]
+ |assumption]
+ |assumption]
+ |assumption]
+ |assumption]
+ |intro;rewrite > (false_to_pi_p_Sn ? ? ? H2);
+ rewrite > (false_to_pi_p_Sn ? ? ? H2);rewrite > (false_to_pi_p_Sn ? ? ? H2);
+ assumption]]
+qed.
+
+lemma divides_times_to_divides_div : \forall a,b,c.O < b \to
+ a*b \divides c \to a \divides c/b.
+intros;elim H1;rewrite > H2;rewrite > sym_times in \vdash (? ? (? (? % ?) ?));
+rewrite > assoc_times;cases H;rewrite > div_times;rewrite > times_n_SO in \vdash (? % ?);
+apply divides_times
+ [1,3:apply divides_n_n
+ |*:apply divides_SO_n]
+qed.
+
+lemma neper_sigma_p2 : \forall n,a.O < n \to n \divides a \to
+sigma_p (S n) (\lambda x.true) (\lambda k.((bc n k)*(exp a n)*(exp n (n-k)))/(exp n n))
+= sigma_p (S n) (\lambda x.true)
+(\lambda k.(exp a (n-k))*(pi_p k (\lambda y.true) (\lambda i.a - (a*i/n)))/k!).
+intros;apply eq_sigma_p;intros;
+ [reflexivity
+ |transitivity (bc n x*exp a n/exp n x)
+ [rewrite > minus_n_O in ⊢ (? ? (? ? (? ? %)) ?);
+ rewrite > (minus_n_n x);
+ rewrite < (eq_plus_minus_minus_minus n x x);
+ [rewrite > exp_plus_times;
+ rewrite > sym_times;rewrite > sym_times in \vdash (? ? (? ? %) ?);
+ rewrite < sym_times;
+ rewrite < sym_times in ⊢ (? ? (? ? %) ?);
+ rewrite < lt_to_lt_to_eq_div_div_times_times;
+ [reflexivity
+ |*:apply lt_O_exp;assumption]
+ |apply le_n
+ |apply le_S_S_to_le;assumption]
+ |rewrite > minus_n_O in ⊢ (? ? (? (? ? (? ? %)) ?) ?);
+ rewrite > (minus_n_n x);
+ rewrite < (eq_plus_minus_minus_minus n x x);
+ [rewrite > exp_plus_times;
+ unfold bc;
+ elim (bc2 n x)
+ [rewrite > H3;cut (x!*n2 = pi_p x (\lambda i.true) (\lambda i.(n - i)))
+ [rewrite > sym_times in ⊢ (? ? (? (? (? (? % ?) ?) ?) ?) ?);
+ rewrite > assoc_times;
+ rewrite < sym_times in ⊢ (? ? (? (? (? % ?) ?) ?) ?);
+ rewrite < lt_to_lt_to_eq_div_div_times_times;
+ [rewrite > Hcut;rewrite < assoc_times;
+ cut (pi_p x (λi:nat.true) (λi:nat.n-i)/x!*(a)\sup(x)
+ = pi_p x (λi:nat.true) (λi:nat.(n-i))*pi_p x (\lambda i.true) (\lambda i.a)/x!)
+ [rewrite > Hcut1;rewrite < times_pi_p;
+ rewrite > divides_times_to_eq in \vdash (? ? % ?);
+ [rewrite > eq_div_div_div_times;
+ [rewrite > sym_times in ⊢ (? ? (? (? ? %) ?) ?);
+ rewrite < eq_div_div_div_times;
+ [cut (exp n x = pi_p x (\lambda i.true) (\lambda i.n))
+ [rewrite > Hcut2;rewrite > divides_pi_p_to_eq
+ [rewrite > sym_times in \vdash (? ? ? %);
+ rewrite > divides_times_to_eq in \vdash (? ? ? %);
+ [apply eq_f2
+ [apply eq_f2
+ [apply eq_pi_p;intros
+ [reflexivity
+ |rewrite > sym_times;
+ rewrite > distr_times_minus;elim H1;
+ rewrite > H5;(* in ⊢ (? ? (? (? ? (? % ?)) ?) ?);*)
+ rewrite > sym_times;rewrite > assoc_times;
+ rewrite < distr_times_minus;
+ generalize in match H;cases n;intros
+ [elim (not_le_Sn_O ? H6)
+ |do 2 rewrite > div_times;reflexivity]]
+ |reflexivity]
+ |reflexivity]
+ |apply lt_O_fact
+ |cut (pi_p x (λy:nat.true) (λi:nat.a-a*i/n) = (exp a x)/(exp n x)*(n!/(n-x)!))
+ [rewrite > Hcut3;rewrite > times_n_SO;
+ rewrite > sym_times;apply divides_times
+ [apply divides_SO_n;
+ |apply divides_times_to_divides_div;
+ [apply lt_O_fact
+ |apply bc2;apply le_S_S_to_le;assumption]]
+ |cut (pi_p x (\lambda y.true) (\lambda i. a - a*i/n) =
+ pi_p x (\lambda y.true) (\lambda i. a*(n-i)/n))
+ [rewrite > Hcut3;
+ rewrite < (divides_pi_p_to_eq ? ? (\lambda i.(a*(n-i))) (\lambda i.n))
+ [rewrite > (times_pi_p ? ? (\lambda i.a) (\lambda i.(n-i)));
+ rewrite > divides_times_to_eq;
+ [apply eq_f2
+ [apply eq_f2;rewrite < eq_exp_pi_p;reflexivity
+ |rewrite < Hcut;rewrite > H3;
+ rewrite < sym_times in ⊢ (? ? ? (? (? % ?) ?));
+ rewrite > (S_pred ((n-x)!));
+ [rewrite > assoc_times;
+ rewrite > div_times;reflexivity
+ |apply lt_O_fact]]
+ |unfold lt;cut (1 = pi_p x (\lambda y.true) (\lambda y.1))
+ [rewrite > Hcut4;apply le_pi_p;intros;assumption
+ |elim x
+ [simplify;reflexivity
+ |rewrite > true_to_pi_p_Sn;
+ [rewrite < H4;reflexivity
+ |reflexivity]]]
+ |elim x
+ [simplify;apply divides_SO_n
+ |rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_pi_p_Sn
+ [apply divides_times;assumption
+ |reflexivity]
+ |reflexivity]]]
+ |intros;split
+ [assumption
+ |rewrite > times_n_SO;apply divides_times
+ [assumption
+ |apply divides_SO_n]]]
+ |apply eq_pi_p;intros;
+ [reflexivity
+ |elim H1;rewrite > H5;rewrite > (S_pred n);
+ [rewrite > assoc_times;
+ rewrite > assoc_times;
+ rewrite > div_times;
+ rewrite > div_times;
+ rewrite > distr_times_minus;
+ rewrite > sym_times;
+ reflexivity
+ |assumption]]]]]
+ |intros;split
+ [assumption
+ |rewrite > sym_times;rewrite > times_n_SO;
+ apply divides_times
+ [assumption
+ |apply divides_SO_n]]]
+ |rewrite < eq_exp_pi_p;reflexivity]
+ |apply lt_O_exp;assumption
+ |apply lt_O_fact]
+ |apply lt_O_fact
+ |apply lt_O_exp;assumption]
+ |apply lt_O_exp;assumption
+ |rewrite > (times_pi_p ? ? (\lambda x.(n-x)) (\lambda x.a));
+ rewrite > divides_times_to_eq
+ [rewrite > times_n_SO;rewrite > sym_times;apply divides_times
+ [apply divides_SO_n
+ |elim x
+ [simplify;apply divides_SO_n
+ |change in \vdash (? % ?) with (n*(exp n n1));
+ rewrite > true_to_pi_p_Sn
+ [apply divides_times;assumption
+ |reflexivity]]]
+ |apply lt_O_fact
+ |apply (witness ? ? n2);symmetry;assumption]]
+ |rewrite > divides_times_to_eq;
+ [apply eq_f2
+ [reflexivity
+ |elim x
+ [simplify;reflexivity
+ |change in \vdash (? ? % ?) with (a*(exp a n1));
+ rewrite > true_to_pi_p_Sn
+ [apply eq_f2
+ [reflexivity
+ |assumption]
+ |reflexivity]]]
+ |apply lt_O_fact
+ |apply (witness ? ? n2);symmetry;assumption]]
+ |apply lt_O_fact
+ |apply lt_O_fact]
+ |apply (inj_times_r (pred ((n-x)!)));
+ rewrite < (S_pred ((n-x)!))
+ [rewrite < assoc_times;rewrite < sym_times in \vdash (? ? (? % ?) ?);
+ rewrite < H3;generalize in match H2;elim x
+ [rewrite < minus_n_O;simplify;rewrite < times_n_SO;reflexivity
+ |rewrite < fact_minus in H4;
+ [rewrite > true_to_pi_p_Sn
+ [rewrite < assoc_times;rewrite > sym_times in \vdash (? ? ? (? % ?));
+ apply H4;apply lt_to_le;assumption
+ |reflexivity]
+ |apply le_S_S_to_le;assumption]]
+ |apply lt_O_fact]]
+ |apply le_S_S_to_le;assumption]
+ |apply le_n
+ |apply le_S_S_to_le;assumption]]]
+qed.
+
+lemma divides_sigma_p_to_eq : \forall k,p,f,b.O < b \to
+ (\forall x.p x = true \to b \divides f x) \to
+ (sigma_p k p f)/b = sigma_p k p (\lambda x. (f x)/b).
+intros;cut (\forall k1.b \divides sigma_p k1 p f)
+ [|intro;elim k1
+ [simplify;apply (witness ? ? O);rewrite < times_n_O;reflexivity
+ |apply (bool_elim ? (p n));intro
+ [rewrite > true_to_sigma_p_Sn;
+ [elim (H1 n);
+ [elim H2;rewrite > H4;rewrite > H5;rewrite < distr_times_plus;
+ rewrite > times_n_SO in \vdash (? % ?);apply divides_times
+ [apply divides_n_n
+ |apply divides_SO_n]
+ |assumption]
+ |assumption]
+ |rewrite > false_to_sigma_p_Sn;assumption]]]
+elim k
+ [cases H;simplify;reflexivity
+ |apply (bool_elim ? (p n));intro
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn
+ [elim (H1 n);
+ [elim (Hcut n);rewrite > H4;rewrite > H5;rewrite < distr_times_plus;
+ rewrite < H2;rewrite > H5;cases H;do 3 rewrite > div_times;
+ reflexivity
+ |assumption]
+ |assumption]
+ |assumption]
+ |do 2 try rewrite > false_to_sigma_p_Sn;assumption]]
+qed.
+
+lemma neper_sigma_p3 : \forall a,n.O < a \to O < n \to n \divides a \to (exp (S n) n)/(exp n n) =
+sigma_p (S n) (\lambda x.true)
+(\lambda k.(exp a (n-k))*(pi_p k (\lambda y.true) (\lambda i.a - (a*i/n)))/k!)/(exp a n).
+intros;transitivity ((exp a n)*(exp (S n) n)/(exp n n)/(exp a n))
+ [rewrite > eq_div_div_div_times
+ [rewrite < sym_times; rewrite < lt_to_lt_to_eq_div_div_times_times;
+ [reflexivity
+ |apply lt_O_exp;assumption
+ |apply lt_O_exp;assumption]
+ |apply lt_O_exp;assumption
+ |apply lt_O_exp;assumption]
+ |apply eq_f2;
+ [rewrite > times_exp;rewrite > neper_sigma_p1
+ [transitivity (sigma_p (S n) (λx:nat.true) (λk:nat.bc n k*(a)\sup(n)*(exp n (n-k))/(exp n n)))
+ [elim H2;rewrite > H3;rewrite < times_exp;rewrite > sym_times in ⊢ (? ? (? (? ? ? (λ_:?.? ? %)) ?) ?);
+ rewrite > sym_times in ⊢ (? ? ? (? ? ? (λ_:?.? (? (? ? %) ?) ?)));
+ transitivity (sigma_p (S n) (λx:nat.true)
+(λk:nat.(exp n n)*(bc n k*(n)\sup(n-k)*(n2)\sup(n)))/exp n n)
+ [apply eq_f2
+ [apply eq_sigma_p;intros;
+ [reflexivity
+ |rewrite < assoc_times;rewrite > sym_times;reflexivity]
+ |reflexivity]
+ |rewrite < (distributive_times_plus_sigma_p ? ? ? (\lambda k.bc n k*(exp n (n-k))*(exp n2 n)));
+ transitivity (sigma_p (S n) (λx:nat.true)
+ (λk:nat.bc n k*(n2)\sup(n)*(n)\sup(n-k)))
+ [rewrite > (S_pred (exp n n))
+ [rewrite > div_times;apply eq_sigma_p;intros
+ [reflexivity
+ |rewrite > sym_times;rewrite > sym_times in ⊢ (? ? ? (? % ?));
+ rewrite > assoc_times in \vdash (? ? ? %);reflexivity]
+ |apply lt_O_exp;assumption]
+ |apply eq_sigma_p;intros
+ [reflexivity
+ |rewrite < assoc_times;rewrite > assoc_times in \vdash (? ? ? %);
+ rewrite > sym_times in \vdash (? ? ? (? (? ? %) ?));
+ rewrite < assoc_times;rewrite > sym_times in \vdash (? ? ? %);
+ rewrite > (S_pred (exp n n))
+ [rewrite > div_times;reflexivity
+ |apply lt_O_exp;assumption]]]]
+ |rewrite > neper_sigma_p2;
+ [reflexivity
+ |assumption
+ |assumption]]
+ |assumption]
+ |reflexivity]]
+qed.
+
+theorem neper_monotonic : \forall n. O < n \to
+(exp (S n) n)/(exp n n) \leq (exp (S (S n)) (S n))/(exp (S n) (S n)).
+intros;rewrite > (neper_sigma_p3 (n*S n) n)
+ [rewrite > (neper_sigma_p3 (n*S n) (S n))
+ [change in ⊢ (? ? (? ? %)) with ((n * S n)*(exp (n * S n) n));
+ rewrite < eq_div_div_div_times
+ [apply monotonic_div;
+ [apply lt_O_exp;rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]
+ |apply le_times_to_le_div
+ [rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]
+ |rewrite > distributive_times_plus_sigma_p;
+ apply (trans_le ? (sigma_p (S n) (λx:nat.true)
+ (λk:nat.((n*S n))\sup(S n-k)*pi_p k (λy:nat.true) (λi:nat.n*S n-n*S n*i/S n)/k!)))
+ [apply le_sigma_p;intros;rewrite > sym_times in ⊢ (? (? ? %) ?);
+ rewrite > sym_times in \vdash (? ? (? % ?));
+ rewrite > divides_times_to_eq in \vdash (? ? %)
+ [rewrite > divides_times_to_eq in \vdash (? % ?)
+ [rewrite > sym_times in \vdash (? (? ? %) ?);
+ rewrite < assoc_times;
+ rewrite > sym_times in \vdash (? ? %);
+ rewrite > minus_Sn_m;
+ [apply le_times_r;apply monotonic_div
+ [apply lt_O_fact
+ |apply le_pi_p;intros;apply monotonic_le_minus_r;
+ rewrite > assoc_times in \vdash (? % ?);
+ rewrite > sym_times in \vdash (? % ?);
+ rewrite > assoc_times in \vdash (? % ?);
+ rewrite > div_times;
+ rewrite > (S_pred n) in \vdash (? ? %);
+ [rewrite > assoc_times;rewrite > div_times;
+ rewrite < S_pred
+ [rewrite > sym_times;apply le_times_l;apply le_S;apply le_n
+ |assumption]
+ |assumption]]
+ |apply le_S_S_to_le;assumption]
+ |apply lt_O_fact
+ |cut (pi_p i (λy:nat.true) (λi:nat.n*S n-n*S n*i/n) =
+ pi_p i (\lambda y.true) (\lambda i.S n) *
+ pi_p i (\lambda y.true) (\lambda i.(n-i)))
+ [rewrite > Hcut;rewrite > times_n_SO;
+ rewrite > sym_times;apply divides_times
+ [apply divides_SO_n
+ |elim (bc2 n i);
+ [apply (witness ? ? n2);
+ cut (pi_p i (\lambda y.true) (\lambda i.n-i) = (n!/(n-i)!))
+ [rewrite > Hcut1;rewrite > H3;rewrite > sym_times in ⊢ (? ? (? (? % ?) ?) ?);
+ rewrite > (S_pred ((n-i)!))
+ [rewrite > assoc_times;rewrite > div_times;
+ reflexivity
+ |apply lt_O_fact]
+ |generalize in match H1;elim i
+ [rewrite < minus_n_O;rewrite > div_n_n;
+ [reflexivity
+ |apply lt_O_fact]
+ |rewrite > true_to_pi_p_Sn
+ [rewrite > H4
+ [rewrite > sym_times;rewrite < divides_times_to_eq
+ [rewrite < fact_minus
+ [rewrite > sym_times in ⊢ (? ? (? ? %) ?);
+ rewrite < lt_to_lt_to_eq_div_div_times_times;
+ [reflexivity
+ |apply lt_to_lt_O_minus;apply le_S_S_to_le;
+ assumption
+ |apply lt_O_fact;]
+ |apply le_S_S_to_le;assumption]
+ |apply lt_O_fact
+ |apply divides_fact_fact;
+ apply le_plus_to_minus;
+ rewrite > plus_n_O in \vdash (? % ?);
+ apply le_plus_r;apply le_O_n]
+ |apply lt_to_le;assumption]
+ |reflexivity]]]
+ |apply le_S_S_to_le;assumption]]
+ |rewrite < times_pi_p;apply eq_pi_p;intros;
+ [reflexivity
+ |rewrite > distr_times_minus;rewrite > assoc_times;
+ rewrite > (S_pred n) in \vdash (? ? (? ? (? (? % ?) %)) ?)
+ [rewrite > div_times;rewrite > sym_times;reflexivity
+ |assumption]]]]
+ |apply lt_O_fact
+ |cut (pi_p i (λy:nat.true) (λi:nat.n*S n-n*S n*i/S n) =
+ pi_p i (\lambda y.true) (\lambda i.n) *
+ pi_p i (\lambda y.true) (\lambda i.(S n-i)))
+ [rewrite > Hcut;rewrite > times_n_SO;rewrite > sym_times;
+ apply divides_times
+ [apply divides_SO_n
+ |elim (bc2 (S n) i);
+ [apply (witness ? ? n2);
+ cut (pi_p i (\lambda y.true) (\lambda i.S n-i) = ((S n)!/(S n-i)!))
+ [rewrite > Hcut1;rewrite > H3;rewrite > sym_times in ⊢ (? ? (? (? % ?) ?) ?);
+ rewrite > (S_pred ((S n-i)!))
+ [rewrite > assoc_times;rewrite > div_times;
+ reflexivity
+ |apply lt_O_fact]
+ |generalize in match H1;elim i
+ [rewrite < minus_n_O;rewrite > div_n_n;
+ [reflexivity
+ |apply lt_O_fact]
+ |rewrite > true_to_pi_p_Sn
+ [rewrite > H4
+ [rewrite > sym_times;rewrite < divides_times_to_eq
+ [rewrite < fact_minus
+ [rewrite > sym_times in ⊢ (? ? (? ? %) ?);
+ rewrite < lt_to_lt_to_eq_div_div_times_times;
+ [reflexivity
+ |apply lt_to_lt_O_minus;apply lt_to_le;
+ assumption
+ |apply lt_O_fact]
+ |apply lt_to_le;assumption]
+ |apply lt_O_fact
+ |apply divides_fact_fact;
+ apply le_plus_to_minus;
+ rewrite > plus_n_O in \vdash (? % ?);
+ apply le_plus_r;apply le_O_n]
+ |apply lt_to_le;assumption]
+ |reflexivity]]]
+ |apply lt_to_le;assumption]]
+ |rewrite < times_pi_p;apply eq_pi_p;intros;
+ [reflexivity
+ |rewrite > distr_times_minus;rewrite > sym_times in \vdash (? ? (? ? (? (? % ?) ?)) ?);
+ rewrite > assoc_times;rewrite > div_times;reflexivity]]]
+ |rewrite > true_to_sigma_p_Sn in \vdash (? ? %)
+ [rewrite > sym_plus;rewrite > plus_n_O in \vdash (? % ?);
+ apply le_plus_r;apply le_O_n
+ |reflexivity]]]]
+ |rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]
+ |apply lt_O_exp;rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]]
+ |rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]
+ |apply lt_O_S
+ |apply (witness ? ? n);apply sym_times]
+ |rewrite > (times_n_O O);apply lt_times
+ [assumption
+ |apply lt_O_S]
+ |assumption
+ |apply (witness ? ? (S n));reflexivity]
+qed.
+
+theorem le_SSO_neper : \forall n.O < n \to (2 \leq (exp (S n) n)/(exp n n)).
+intros;elim H
+ [simplify;apply le_n
+ |apply (trans_le ? ? ? H2);apply neper_monotonic;assumption]
+qed.
+
+theorem le_SSO_exp_neper : \forall n.O < n \to 2*(exp n n) \leq exp (S n) n.
+intros;apply (trans_le ? ((exp (S n) n)/(exp n n)*(exp n n)))
+ [apply le_times_l;apply le_SSO_neper;assumption
+ |rewrite > sym_times;apply (trans_le ? ? ? (le_times_div_div_times ? ? ? ?))
+ [apply lt_O_exp;assumption
+ |cases (lt_O_exp ? n H);rewrite > div_times;apply le_n]]
+qed.
+
+(* theorem le_log_exp_Sn_log_exp_n: \forall n,m,a,p. O < m \to S O < p \to
+divides n m \to
+log p (exp n m) - log p (exp a m) \le
+sigma_p (S n) (\lambda i.leb (S a) i) (\lambda i.S((m/i)*S(log p (S(S(S O)))))).
+intros.
+elim n
+ [rewrite > false_to_sigma_p_Sn.
+ simplify.
+ apply (lt_O_n_elim ? H).intro.
+ simplify.apply le_O_n
+ |apply (bool_elim ? (leb a n1));intro
+ [rewrite > true_to_sigma_p_Sn
+ [apply (trans_le ? (S (m/S n1*S (log p (S(S(S O)))))+(log p ((n1)\sup(m))-log p ((a)\sup(m)))))
+ [rewrite > sym_plus.
+ rewrite > plus_minus
+ [apply le_plus_to_minus_r.
+ rewrite < plus_minus_m_m
+ [rewrite > sym_plus.
+ apply le_log_exp_Sn_log_exp_n.
+
+
+* a generalization
+theorem le_exp_sigma_p_exp_m: \forall m,n. (exp (S m) n) \le
+sigma_p (S n) (\lambda k.true) (\lambda k.((exp m (n-k))*(exp n k))/(k!)).
+intros.
+rewrite > exp_S_sigma_p.
+apply le_sigma_p.
+intros.unfold bc.
+apply (trans_le ? ((exp m (n-i))*((n \sup i)/i!)))
+ [rewrite > sym_times.
+ apply le_times_r.
+ rewrite > sym_times.
+ rewrite < eq_div_div_div_times
+ [apply monotonic_div
+ [apply lt_O_fact
+ |apply le_times_to_le_div2
+ [apply lt_O_fact
+ |apply le_fact_exp.
+ apply le_S_S_to_le.
+ assumption
+ ]
+ ]
+ |apply lt_O_fact
+ |apply lt_O_fact
+ ]
+ |apply le_times_div_div_times.
+ apply lt_O_fact
+ ]
+qed.
+
+theorem lt_exp_sigma_p_exp_m: \forall m,n. S O < n \to
+(exp (S m) n) <
+sigma_p (S n) (\lambda k.true) (\lambda k.((exp m (n-k))*(exp n k))/(k!)).
+intros.
+rewrite > exp_S_sigma_p.
+apply lt_sigma_p
+ [intros.unfold bc.
+ apply (trans_le ? ((exp m (n-i))*((n \sup i)/i!)))
+ [rewrite > sym_times.
+ apply le_times_r.
+ rewrite > sym_times.
+ rewrite < eq_div_div_div_times
+ [apply monotonic_div
+ [apply lt_O_fact
+ |apply le_times_to_le_div2
+ [apply lt_O_fact
+ |apply le_fact_exp.
+ apply le_S_S_to_le.
+ assumption
+ ]
+ ]
+ |apply lt_O_fact
+ |apply lt_O_fact
+ ]
+ |apply le_times_div_div_times.
+ apply lt_O_fact
+ ]
+ |apply (ex_intro ? ? n).
+ split
+ [split
+ [apply le_n
+ |reflexivity
+ ]
+ |rewrite < minus_n_n.
+ rewrite > bc_n_n.
+ simplify.unfold lt.
+ apply le_times_to_le_div
+ [apply lt_O_fact
+ |rewrite > sym_times.
+ rewrite < plus_n_O.
+ apply le_fact_exp1.
+ assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem lt_SO_to_lt_exp_Sn_n_SSSO_bof: \forall m,n. S O < n \to
+(exp (S m) n) < (S(S(S O)))*(exp m n).
+intros.
+apply (lt_to_le_to_lt ? (sigma_p (S n) (\lambda k.true) (\lambda k.((exp m (n-k))*(exp n k))/(k!))))
+ [apply lt_exp_sigma_p_exp_m.assumption
+ |apply (trans_le ? (sigma_p (S n) (\lambda i.true) (\lambda i.(exp n n)/(exp (S(S O)) (pred i)))))
+ [apply le_sigma_p.intros.
+ apply le_times_to_le_div
+ [apply lt_O_exp.
+ apply lt_O_S
+ |apply (trans_le ? ((S(S O))\sup (pred i)* n \sup n/i!))
+ [apply le_times_div_div_times.
+ apply lt_O_fact
+ |apply le_times_to_le_div2
+ [apply lt_O_fact
+ |rewrite < sym_times.
+ apply le_times_r.
+ apply le_exp_SSO_fact
+ ]
+ ]
+ ]
+ |rewrite > eq_sigma_p_pred
+ [rewrite > div_SO.
+ rewrite > sym_plus.
+ change in ⊢ (? ? %) with ((exp n n)+(S(S O)*(exp n n))).
+ apply le_plus_r.
+ apply (trans_le ? (((S(S O))*(exp n n)*(exp (S(S O)) n) - (S(S O))*(exp n n))/(exp (S(S O)) n)))
+ [apply sigma_p_div_exp
+ |apply le_times_to_le_div2
+ [apply lt_O_exp.
+ apply lt_O_S.
+ |apply le_minus_m
+ ]
+ ]
+ |reflexivity
+ ]
+ ]
+ ]
+qed.
+*)
projects / helm.git / blobdiff