--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / Matita is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/nat/chebishev".
+
+include "nat/log.ma".
+include "nat/pi_p.ma".
+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)).
+
+definition A: nat \to nat \def
+\lambda n. pi_p (S n) primeb (\lambda p.exp p (log p n)).
+
+theorem le_Al1: \forall n.
+A n \le pi_p (S n) primeb (\lambda p.n).
+intro.unfold A.
+cases n
+ [simplify.apply le_n
+ |apply le_pi_p.
+ intros.
+ apply le_exp_log.
+ apply lt_O_S
+ ]
+qed.
+
+theorem le_Al: \forall n.
+A n \le exp n (prim n).
+intro.unfold prim.
+rewrite < exp_sigma_p.
+apply le_Al1.
+qed.
+
+theorem leA_r2: \forall n.
+exp n (prim n) \le A n * A n.
+intro.unfold prim.
+elim (le_to_or_lt_eq ? ? (le_O_n n))
+ [rewrite < (exp_sigma_p (S n) n primeb).
+ unfold A.
+ rewrite < times_pi_p.
+ apply le_pi_p.
+ intros.
+ rewrite < exp_plus_times.
+ apply (trans_le ? (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
+ |apply le_exp
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |rewrite > plus_n_O.
+ simplify.
+ rewrite > plus_n_Sm.
+ apply le_plus_r.
+ apply lt_O_log
+ [assumption
+ |apply le_S_S_to_le.
+ apply H1
+ ]
+ ]
+ ]
+ |rewrite < H. apply le_n
+ ]
+qed.
+
+definition A': nat \to nat \def
+\lambda n. pi_p (S n) primeb
+ (\lambda p.(pi_p (log p n) (\lambda i.true) (\lambda i.p))).
+
+theorem eq_A_A': \forall n.A n = A' n.
+intro.unfold A.unfold A'.
+apply eq_pi_p
+ [intros.reflexivity
+ |intros.
+ apply (trans_eq ? ? (exp x (sigma_p (log x n) (λi:nat.true) (λi:nat.(S O)))))
+ [apply eq_f.apply sym_eq.apply sigma_p_true
+ |apply sym_eq.apply exp_sigma_p
+ ]
+ ]
+qed.
+
+theorem eq_pi_p_primeb_divides_b: \forall n,m.
+pi_p n (\lambda p.primeb p \land divides_b p m) (\lambda p.exp p (ord m p))
+= pi_p n primeb (\lambda p.exp p (ord m p)).
+intro.
+elim n
+ [reflexivity
+ |apply (bool_elim ? (primeb n1));intro
+ [rewrite > true_to_pi_p_Sn in ⊢ (? ? ? %)
+ [apply (bool_elim ? (divides_b n1 m));intro
+ [rewrite > true_to_pi_p_Sn
+ [apply eq_f.
+ apply H
+ |apply true_to_true_to_andb_true;assumption
+ ]
+ |rewrite > false_to_pi_p_Sn
+ [rewrite > not_divides_to_ord_O
+ [simplify in ⊢ (? ? ? (? % ?)).
+ rewrite > sym_times.
+ rewrite < times_n_SO.
+ apply H
+ |apply primeb_true_to_prime.assumption
+ |apply divides_b_false_to_not_divides.
+ assumption
+ ]
+ |rewrite > H1.rewrite > H2.reflexivity
+ ]
+ ]
+ |assumption
+ ]
+ |rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_pi_p_Sn
+ [apply H
+ |assumption
+ ]
+ |rewrite > H1.reflexivity
+ ]
+ ]
+ ]
+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
+ [apply False_ind.
+ apply (not_le_Sn_O ? H1)
+ |apply (bool_elim ? (primeb n∧divides_b n m));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > (exp_ord n m) in ⊢ (? ? % ?)
+ [apply eq_f.
+ rewrite > (H (ord_rem m n))
+ [apply eq_pi_p1
+ [intros.
+ apply (bool_elim ? (primeb x));intro
+ [simplify.
+ apply (bool_elim ? (divides_b x (ord_rem m n)));intro
+ [apply sym_eq.
+ apply divides_to_divides_b_true
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |apply (trans_divides ? (ord_rem m n))
+ [apply divides_b_true_to_divides.
+ assumption
+ |apply divides_ord_rem
+ [apply (trans_lt ? x)
+ [apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+ |apply sym_eq.
+ apply not_divides_to_divides_b_false
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |apply ord_O_to_not_divides
+ [assumption
+ |apply primeb_true_to_prime.
+ assumption
+ |rewrite < (ord_ord_rem n)
+ [apply not_divides_to_ord_O
+ [apply primeb_true_to_prime.
+ assumption
+ |apply divides_b_false_to_not_divides.
+ assumption
+ ]
+ |assumption
+ |apply primeb_true_to_prime.
+ apply (andb_true_true ? ? H3)
+ |apply primeb_true_to_prime.
+ assumption
+ |assumption
+ ]
+ ]
+ ]
+ ]
+ |reflexivity
+ ]
+ |intros.
+ apply eq_f.
+ apply ord_ord_rem
+ [assumption
+ |apply primeb_true_to_prime.
+ apply (andb_true_true ? ? H3)
+ |apply primeb_true_to_prime.
+ apply (andb_true_true ? ? H5)
+ |assumption
+ ]
+ ]
+ |apply lt_O_ord_rem
+ [apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ apply (andb_true_true ? ? H3)
+ |assumption
+ ]
+ |apply not_eq_to_le_to_lt
+ [elim (exists_max_forall_false (λi:nat.primeb i∧divides_b i (ord_rem m n)) (ord_rem m n))
+ [elim H4.
+ intro.rewrite > H7 in H6.simplify in H6.
+ apply (not_divides_ord_rem m n)
+ [assumption
+ |apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ apply (andb_true_true ? ? H3)
+ |apply divides_b_true_to_divides.
+ apply (andb_true_true_r ? ? H6)
+ ]
+ |elim H4.rewrite > H6.
+ apply lt_to_not_eq.
+ apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ apply (andb_true_true ? ? H3)
+ ]
+ |apply (trans_le ? (max m (λi:nat.primeb i∧divides_b i (ord_rem m n))))
+ [apply le_to_le_max.
+ apply divides_to_le
+ [assumption
+ |apply divides_ord_rem
+ [apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ apply (andb_true_true ? ? H3)
+ |assumption
+ ]
+ ]
+ |apply (trans_le ? (max m (λi:nat.primeb i∧divides_b i m)))
+ [apply le_max_f_max_g.
+ intros.
+ apply (bool_elim ? (primeb i));intro
+ [simplify.
+ apply divides_to_divides_b_true
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |apply (trans_divides ? (ord_rem m n))
+ [apply divides_b_true_to_divides.
+ apply (andb_true_true_r ? ? H5)
+ |apply divides_ord_rem
+ [apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ apply (andb_true_true ? ? H3)
+ |assumption
+ ]
+ ]
+ ]
+ |rewrite > H6 in H5.
+ assumption
+ ]
+ |apply le_S_S_to_le.
+ assumption
+ ]
+ ]
+ ]
+ ]
+ |apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ apply (andb_true_true ? ? H3)
+ |assumption
+ ]
+ |assumption
+ ]
+ |elim (le_to_or_lt_eq ? ? H1)
+ [rewrite > false_to_pi_p_Sn
+ [apply H
+ [assumption
+ |apply false_to_lt_max
+ [apply (lt_to_le_to_lt ? (max m (λi:nat.primeb i∧divides_b i m)))
+ [apply lt_to_le.
+ apply lt_SO_max_prime.
+ assumption
+ |apply le_S_S_to_le.
+ assumption
+ ]
+ |assumption
+ |apply le_S_S_to_le.
+ assumption
+ ]
+ ]
+ |assumption
+ ]
+ |rewrite < H4.
+ rewrite < (pi_p_false (λp:nat.p\sup(ord (S O) p)) (S n) ) in ⊢ (? ? % ?).
+ apply eq_pi_p
+ [intros.
+ apply (bool_elim ? (primeb x));intro
+ [apply sym_eq.
+ change with (divides_b x (S O) =false).
+ apply not_divides_to_divides_b_false
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |intro.
+ apply (le_to_not_lt x (S O))
+ [apply divides_to_le
+ [apply lt_O_S.assumption
+ |assumption
+ ]
+ |elim (primeb_true_to_prime ? H6).
+ assumption
+ ]
+ ]
+ |reflexivity
+ ]
+ |intros.reflexivity
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem pi_p_primeb_divides_b: \forall n. O < n \to
+n = pi_p (S n) (\lambda i.primeb i \land divides_b i n) (\lambda p.exp p (ord n p)).
+intros.
+apply lt_max_to_pi_p_primeb
+ [assumption
+ |apply le_S_S.
+ apply le_max_n
+ ]
+qed.
+
+theorem pi_p_primeb: \forall n. O < n \to
+n = pi_p (S n) primeb (\lambda p.exp p (ord n p)).
+intros.
+rewrite < eq_pi_p_primeb_divides_b.
+apply pi_p_primeb_divides_b.
+assumption.
+qed.
+
+theorem le_ord_log: \forall n,p. O < n \to S O < p \to
+ord n p \le log p n.
+intros.
+rewrite > (exp_ord p) in ⊢ (? ? (? ? %))
+ [rewrite > log_exp
+ [apply le_plus_n_r
+ |assumption
+ |apply lt_O_ord_rem;assumption
+ ]
+ |assumption
+ |assumption
+ ]
+qed.
+
+theorem sigma_p_divides_b:
+\forall m,n,p. O < n \to prime p \to \lnot divides p n \to
+m = sigma_p m (λi:nat.divides_b (p\sup (S i)) ((exp p m)*n)) (λx:nat.S O).
+intro.elim m
+ [reflexivity
+ |simplify in ⊢ (? ? ? (? % ? ?)).
+ rewrite > true_to_sigma_p_Sn
+ [rewrite > sym_plus.rewrite < plus_n_SO.
+ apply eq_f.
+ rewrite > (H n1 p H1 H2 H3) in ⊢ (? ? % ?).
+ apply eq_sigma_p1
+ [intros.
+ apply (bool_elim ? (divides_b (p\sup(S x)) (p\sup n*n1)));intro
+ [apply sym_eq.
+ apply divides_to_divides_b_true
+ [apply lt_O_exp.
+ apply prime_to_lt_O.
+ assumption
+ |apply (witness ? ? ((exp p (n - x))*n1)).
+ rewrite < assoc_times.
+ rewrite < exp_plus_times.
+ apply eq_f2
+ [apply eq_f.simplify.
+ apply eq_f.
+ rewrite > sym_plus.
+ apply plus_minus_m_m.
+ apply lt_to_le.assumption
+ |reflexivity
+ ]
+ ]
+ |apply sym_eq.
+ apply False_ind.
+ apply (divides_b_false_to_not_divides ? ? H5).
+ apply (witness ? ? ((exp p (n - S x))*n1)).
+ rewrite < assoc_times.
+ rewrite < exp_plus_times.
+ apply eq_f2
+ [apply eq_f.
+ rewrite > sym_plus.
+ apply plus_minus_m_m.
+ assumption
+ |reflexivity
+ ]
+ ]
+ |intros.reflexivity
+ ]
+ |apply divides_to_divides_b_true
+ [apply lt_O_exp.
+ apply prime_to_lt_O.assumption
+ |apply (witness ? ? n1).reflexivity
+ ]
+ ]
+ ]
+qed.
+
+theorem sigma_p_divides_b1:
+\forall m,n,p,k. O < n \to prime p \to \lnot divides p n \to m \le k \to
+m = sigma_p k (λi:nat.divides_b (p\sup (S i)) ((exp p m)*n)) (λx:nat.S O).
+intros.
+lapply (prime_to_lt_SO ? H1) as H4.
+lapply (prime_to_lt_O ? H1) as H5.
+rewrite > (false_to_eq_sigma_p m k)
+ [apply sigma_p_divides_b;assumption
+ |assumption
+ |intros.
+ apply not_divides_to_divides_b_false
+ [apply lt_O_exp.assumption
+ |intro.apply (le_to_not_lt ? ? H6).
+ unfold lt.
+ rewrite < (ord_exp p)
+ [rewrite > (plus_n_O m).
+ rewrite < (not_divides_to_ord_O ? ? H1 H2).
+ rewrite < (ord_exp p ? H4) in ⊢ (? ? (? % ?)).
+ rewrite < ord_times
+ [apply divides_to_le_ord
+ [apply lt_O_exp.assumption
+ |rewrite > (times_n_O O).
+ apply lt_times
+ [apply lt_O_exp.assumption
+ |assumption
+ ]
+ |assumption
+ |assumption
+ ]
+ |apply lt_O_exp.assumption
+ |assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+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).
+intros.
+lapply (prime_to_lt_SO ? H1).
+rewrite > (exp_ord x n) in ⊢ (? ? ? (? ? (λi:?.? ? %) ?))
+ [apply sigma_p_divides_b1
+ [apply lt_O_ord_rem;assumption
+ |assumption
+ |apply not_divides_ord_rem;assumption
+ |apply lt_S_to_le.
+ apply (le_to_lt_to_lt ? (log x n))
+ [apply le_ord_log;assumption
+ |apply (lt_exp_to_lt x)
+ [assumption
+ |apply (le_to_lt_to_lt ? n ? ? H3).
+ apply le_exp_log.
+ assumption
+ ]
+ ]
+ ]
+ |assumption
+ |assumption
+ ]
+qed.
+
+theorem pi_p_primeb1: \forall n. O < n \to
+n = pi_p (S n) primeb
+ (\lambda p.(pi_p (log p n)
+ (\lambda i.divides_b (exp p (S i)) n) (\lambda i.p))).
+intros.
+rewrite > (pi_p_primeb n H) in ⊢ (? ? % ?).
+apply eq_pi_p1
+ [intros.reflexivity
+ |intros.
+ rewrite > exp_sigma_p.
+ apply eq_f.
+ apply eq_ord_sigma_p
+ [assumption
+ |apply primeb_true_to_prime.
+ assumption
+ |apply le_exp_log.assumption
+ |apply lt_exp_log.
+ apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ assumption
+ ]
+ ]
+qed.
+
+theorem eq_fact_pi_p:\forall n. fact n =
+pi_p (S n) (\lambda i.leb (S O) i) (\lambda i.i).
+intro.elim n
+ [reflexivity
+ |change with ((S n1)*n1! = pi_p (S (S n1)) (λi:nat.leb (S O) i) (λi:nat.i)).
+ rewrite > true_to_pi_p_Sn
+ [apply eq_f.assumption
+ |reflexivity
+ ]
+ ]
+qed.
+
+theorem eq_sigma_p_div: \forall n,q.O < q \to
+sigma_p (S n) (λm:nat.leb (S O) m∧divides_b q m) (λx:nat.S O)
+=n/q.
+intros.
+apply (div_mod_spec_to_eq n q ? (n \mod q) ? (n \mod q))
+ [apply div_mod_spec_intro
+ [apply lt_mod_m_m.assumption
+ |elim n
+ [simplify.elim q;reflexivity
+ |elim (or_div_mod1 n1 q)
+ [elim H2.clear H2.
+ rewrite > divides_to_mod_O
+ [rewrite < plus_n_O.
+ rewrite > true_to_sigma_p_Sn
+ [rewrite > H4 in ⊢ (? ? % ?).
+ apply eq_f2
+ [rewrite < sym_plus.
+ rewrite < plus_n_SO.
+ apply eq_f.
+ apply (div_mod_spec_to_eq n1 q (div n1 q) (mod n1 q) ? (mod n1 q))
+ [apply div_mod_spec_div_mod.
+ assumption
+ |apply div_mod_spec_intro
+ [apply lt_mod_m_m.assumption
+ |assumption
+ ]
+ ]
+ |reflexivity
+ ]
+ |apply true_to_true_to_andb_true
+ [reflexivity
+ |apply divides_to_divides_b_true;assumption
+ ]
+ ]
+ |assumption
+ |assumption
+ ]
+ |elim H2.clear H2.
+ rewrite > false_to_sigma_p_Sn
+ [rewrite > mod_S
+ [rewrite < plus_n_Sm.
+ apply eq_f.
+ assumption
+ |assumption
+ |elim (le_to_or_lt_eq (S (mod n1 q)) q)
+ [assumption
+ |apply False_ind.
+ apply H3.
+ apply (witness ? ? (S(div n1 q))).
+ rewrite < times_n_Sm.
+ rewrite < sym_plus.
+ rewrite < H2 in ⊢ (? ? ? (? ? %)).
+ rewrite > sym_times.
+ assumption
+ |apply lt_mod_m_m.
+ assumption
+ ]
+ ]
+ |rewrite > not_divides_to_divides_b_false
+ [rewrite < andb_sym in ⊢ (? ? % ?).reflexivity
+ |assumption
+ |assumption
+ ]
+ ]
+ |assumption
+ ]
+ ]
+ ]
+ |apply div_mod_spec_div_mod.
+ assumption
+ ]
+qed.
+
+theorem fact_pi_p: \forall n. fact n =
+pi_p (S n) primeb
+ (\lambda p.(pi_p (log p n)
+ (\lambda i.true) (\lambda i.(exp p (n /(exp p (S i))))))).
+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))))))
+ [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)))))
+ [apply eq_pi_p1
+ [intros.reflexivity
+ |intros.apply eq_pi_p1
+ [intros.elim (primeb x1)
+ [simplify.apply sym_eq.
+ apply le_to_leb_true.
+ apply le_S_S_to_le.
+ assumption
+ |reflexivity
+ ]
+ |intros.reflexivity
+ ]
+ ]
+ |apply (trans_eq ? ?
+ (pi_p (S n) (λi:nat.leb (S O) i)
+ (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)))))
+ [apply eq_pi_p1
+ [intros.reflexivity
+ |intros.
+ apply sym_eq.
+ apply false_to_eq_pi_p
+ [assumption
+ |intros.rewrite > lt_to_leb_false
+ [elim primeb;reflexivity|assumption]
+ ]
+ ]
+ |(* make a general theorem *)
+ apply (trans_eq ? ?
+ (pi_p (S n) primeb
+ (λp:nat
+ .pi_p (S n) (λm.leb p m)
+ (λm:nat.pi_p (log p m) (λi:nat.divides_b ((p)\sup(S i)) m) (λi:nat.p)))
+ ))
+ [apply pi_p_pi_p1.
+ intros.
+ apply (bool_elim ? (primeb y \land leb y x));intros
+ [rewrite > (le_to_leb_true (S O) x)
+ [reflexivity
+ |apply (trans_le ? y)
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ apply (andb_true_true ? ? H2)
+ |apply leb_true_to_le.
+ apply (andb_true_true_r ? ? H2)
+ ]
+ ]
+ |elim (leb (S O) x);reflexivity
+ ]
+ |apply eq_pi_p1
+ [intros.reflexivity
+ |intros.
+ apply (trans_eq ? ?
+ (pi_p (S n) (λm:nat.leb x m)
+ (λm:nat.pi_p (log x n) (λi:nat.divides_b (x\sup(S i)) m) (λi:nat.x))))
+ [apply eq_pi_p1
+ [intros.reflexivity
+ |intros.
+ apply sym_eq.
+ apply false_to_eq_pi_p
+ [apply le_log
+ [apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ assumption
+ |apply (lt_to_le_to_lt ? x)
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |apply leb_true_to_le.
+ assumption
+ ]
+ |apply le_S_S_to_le.
+ assumption
+ ]
+ |intros.
+ apply not_divides_to_divides_b_false
+ [apply lt_O_exp.
+ apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |intro.
+ apply (lt_to_not_le x1 (exp x (S i)))
+ [apply (lt_to_le_to_lt ? (exp x (S(log x x1))))
+ [apply lt_exp_log.
+ apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ assumption
+ |apply le_exp
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |apply le_S_S.
+ assumption
+ ]
+ ]
+ |apply divides_to_le
+ [apply (lt_to_le_to_lt ? x)
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |apply leb_true_to_le.
+ assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+ ]
+ ]
+ |apply
+ (trans_eq ? ?
+ (pi_p (log x n) (λi:nat.true)
+ (λi:nat.pi_p (S n) (λm:nat.leb x m \land divides_b (x\sup(S i)) m) (λm:nat.x))))
+ [apply (pi_p_pi_p1 (\lambda m,i.x)).
+ intros.
+ reflexivity
+ |apply eq_pi_p1
+ [intros.reflexivity
+ |intros.
+ rewrite > exp_sigma_p.
+ apply eq_f.
+ apply (trans_eq ? ?
+ (sigma_p (S n) (λm:nat.leb (S O) m∧divides_b (x\sup(S x1)) m) (λx:nat.S O)))
+ [apply eq_sigma_p1
+ [intros.
+ apply (bool_elim ? (divides_b (x\sup(S x1)) x2));intro
+ [apply (bool_elim ? (leb x x2));intro
+ [rewrite > le_to_leb_true
+ [reflexivity
+ |apply (trans_le ? x)
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |apply leb_true_to_le.
+ assumption
+ ]
+ ]
+ |rewrite > lt_to_leb_false
+ [reflexivity
+ |apply not_le_to_lt.intro.
+ apply (leb_false_to_not_le ? ? H6).
+ apply (trans_le ? (exp x (S x1)))
+ [rewrite > times_n_SO in ⊢ (? % ?).
+ change with (exp x (S O) \le exp x (S x1)).
+ apply le_exp
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |apply le_S_S.apply le_O_n.
+ ]
+ |apply divides_to_le
+ [assumption
+ |apply divides_b_true_to_divides.
+ assumption
+ ]
+ ]
+ ]
+ ]
+ |rewrite < andb_sym.
+ rewrite < andb_sym in ⊢ (? ? ? %).
+ reflexivity
+ ]
+ |intros.reflexivity
+ ]
+ |apply eq_sigma_p_div.
+ apply lt_O_exp.
+ apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
+
+(* odd n is just mod n (S(S O))
+let rec odd n \def
+ match n with
+ [ O \Rightarrow O
+ | S m \Rightarrow (S O) - odd m
+ ].
+
+theorem le_odd_SO: \forall n. odd n \le S O.
+intro.elim n
+ [apply le_O_n
+ |simplify.cases (odd n1)
+ [simplify.apply le_n.
+ |apply le_O_n
+ ]
+ ]
+qed.
+
+theorem SSO_odd: \forall n. n = (n/(S(S O)))*(S(S O)) + odd n.
+intro.elim n
+ [apply (lt_O_n_elim ? H).
+ intro.simplify.reflexivity
+ |
+*)
+
+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)))))))).
+intros.rewrite > fact_pi_p.
+apply eq_pi_p1
+ [intros.reflexivity
+ |intros.apply eq_pi_p
+ [intros.reflexivity
+ |intros.
+ rewrite < exp_plus_times.
+ apply eq_f.
+ rewrite > sym_times in ⊢ (? ? ? (? % ?)).
+ apply SSO_mod.
+ apply lt_O_exp.
+ apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ ]
+ ]
+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))))))).
+intros.
+rewrite < times_pi_p.
+rewrite > fact_pi_p2.
+apply eq_pi_p;intros
+ [reflexivity
+ |apply times_pi_p
+ ]
+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))))))))
+=
+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)))))))).
+intros.
+apply or_false_eq_SO_to_eq_pi_p
+ [apply le_S_S.
+ apply le_times_n.
+ apply lt_O_S
+ |intros.
+ right.
+ rewrite > log_i_SSOn
+ [change with (i\sup((S(S O))*(n/i\sup(S O)))*(S O) = S O).
+ rewrite < times_n_SO.
+ rewrite > eq_div_O
+ [reflexivity
+ |simplify.rewrite < times_n_SO.assumption
+ ]
+ |assumption
+ |assumption
+ |apply le_S_S_to_le.assumption
+ ]
+ ]
+qed.
+
+theorem pi_p_primeb5: \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))))))))
+=
+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)))))))).
+intros.
+rewrite > (pi_p_primeb4 ? H).
+apply eq_pi_p1;intros
+ [reflexivity
+ |apply or_false_eq_SO_to_eq_pi_p
+ [apply le_log
+ [apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ assumption
+ |apply lt_to_le.assumption
+ |apply le_times_n.
+ apply lt_O_S
+ ]
+ |intros.right.
+ rewrite > eq_div_O
+ [simplify.reflexivity
+ |apply (lt_to_le_to_lt ? (exp x (S(log x n))))
+ [apply lt_exp_log.
+ apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ assumption
+ |apply le_exp
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |apply le_S_S.
+ assumption
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem exp_fact_SSO: \forall n.
+exp (fact n) (S(S O))
+=
+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)))))))).
+intros.
+rewrite > fact_pi_p.
+rewrite < exp_pi_p.
+apply eq_pi_p;intros
+ [reflexivity
+ |apply sym_eq.
+ apply exp_times_pi_p
+ ]
+qed.
+
+definition B \def
+\lambda n.
+pi_p (S n) primeb
+ (\lambda p.(pi_p (log p n)
+ (\lambda i.true) (\lambda i.(exp p (mod (n /(exp p (S i))) (S(S O))))))).
+
+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).
+intros. unfold B.
+rewrite > fact_pi_p3.
+apply eq_f2
+ [apply sym_eq.
+ rewrite > pi_p_primeb5
+ [apply exp_fact_SSO
+ |assumption
+ ]
+ |reflexivity
+ ]
+qed.
+
+theorem le_B_A: \forall n. B n \le A n.
+intro.unfold B.
+rewrite > eq_A_A'.
+unfold A'.
+apply le_pi_p.intros.
+apply le_pi_p.intros.
+rewrite > exp_n_SO in ⊢ (? ? %).
+apply le_exp
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |apply le_S_S_to_le.
+ apply lt_mod_m_m.
+ apply lt_O_S
+ ]
+qed.
+
+theorem le_fact_A:\forall n.S O < n \to
+fact ((S(S O))*n) \le exp (fact n) (S(S O)) * A ((S(S O))*n).
+intros.
+rewrite > eq_fact_B
+ [apply le_times_r.
+ apply le_B_A
+ |assumption
+ ]
+qed.
+
+theorem le_B_exp: \forall n.S O < n \to
+B ((S(S O))*n) \le exp (S(S O)) ((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 < eq_fact_B
+ [rewrite < sym_times in ⊢ (? ? %).
+ rewrite > exp_SSO.
+ rewrite < assoc_times.
+ apply fact1
+ |assumption
+ ]
+ ]
+qed.
+
+theorem le_A_SSO_A: \forall n.
+A((S(S O))*n) \le
+ pi_p (S ((S(S O))*n)) primeb (λp:nat.p)*A n.
+unfold A.intros.
+cut (pi_p (S n) primeb (λp:nat.(exp p (log p n))) = pi_p (S ((S(S O))*n)) primeb (λp:nat.(p)\sup(log p n)))
+ [rewrite > Hcut.
+ rewrite < times_pi_p.
+ apply le_pi_p.intros.
+ lapply (prime_to_lt_SO ? (primeb_true_to_prime ? H1)) as H2.
+ change with (i\sup(log i ((S(S O))*n))≤i\sup(S(log i n))).
+ apply le_exp
+ [apply lt_to_le.assumption
+ |apply (trans_le ? (log i (i*n)))
+ [apply le_log
+ [assumption
+ |apply not_le_to_lt.intro.
+ apply (lt_to_not_le ? ? H).
+ apply (trans_le ? (S O))
+ [apply le_S_S.assumption
+ |apply lt_to_le.assumption
+ ]
+ |apply le_times_l.
+ assumption
+ ]
+ |rewrite > exp_n_SO in ⊢ (? (? ? (? % ?)) ?).
+ rewrite > log_exp
+ [apply le_n
+ |assumption
+ |apply not_le_to_lt.intro.
+ apply (lt_to_not_le ? ? H).
+ apply (le_n_O_elim ? H3).
+ apply lt_to_le.
+ assumption
+ ]
+ ]
+ ]
+ |apply sym_eq.
+ apply or_false_eq_SO_to_eq_pi_p
+ [apply le_S_S.
+ apply le_times_n.
+ apply lt_O_S
+ |intros.right.
+ change with (exp i (log i n) = (exp i O)).
+ apply eq_f.
+ apply antisymmetric_le
+ [cut (O < n)
+ [apply le_S_S_to_le.
+ apply (lt_exp_to_lt i)
+ [apply (le_to_lt_to_lt ? n);assumption
+ |apply (le_to_lt_to_lt ? n)
+ [apply le_exp_log.
+ assumption
+ |rewrite < exp_n_SO.
+ assumption
+ ]
+ ]
+ |apply not_le_to_lt.intro.
+ apply (lt_to_not_le ? ? H1).
+ generalize in match H.
+ apply (le_n_O_elim ? H2).
+ intro.assumption
+ ]
+ |apply le_O_n
+ ]
+ ]
+ ]
+qed.
+
+(* so far so good
+
+theorem le_A_BA: \forall n.
+A((S(S O))*n) \le B((S(S O))*n)*A n.
+(*
+ [simplify.reflexivity
+ |rewrite > times_SSO.
+ rewrite > times_SSO.
+ unfold A.
+apply (trans_le ? ((pi_p (S ((S(S O))*n)) primeb (λp:nat.p))*A n))
+ [apply le_A_SSO_A
+ |apply le_times_l.
+ unfold B.
+ apply le_pi_p.intros.
+*)
+intro.unfold A.unfold B.
+cut (pi_p (S n) primeb (λp:nat.(exp p (log p n))) = pi_p (S ((S(S O))*n)) primeb (λp:nat.(p)\sup(log p n)))
+ [rewrite > Hcut.
+ rewrite < times_pi_p.
+ apply le_pi_p.intros.
+ apply le_trans i.
+
+
+ change with (i\sup(log i ((S(S O))*n))≤i\sup(S(log i n))).
+ apply le_exp
+ [apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ |apply (trans_le ? (log i (i*n)))
+ [apply le_log
+ [apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ assumption
+ |apply not_le_to_lt.intro.
+ apply (lt_to_not_le ? ? H).
+ apply (trans_le ? (S O))
+ [apply le_S_S.assumption
+ |apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ ]
+ |apply le_times_l.
+ apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ assumption
+ ]
+ |rewrite > exp_n_SO in ⊢ (? (? ? (? % ?)) ?).
+ rewrite > log_exp
+ [apply le_n
+ |apply prime_to_lt_SO.
+ apply primeb_true_to_prime.
+ assumption
+ |apply not_le_to_lt.intro.
+ apply (lt_to_not_le ? ? H).
+ apply (le_n_O_elim ? H2).
+ apply prime_to_lt_O.
+ apply primeb_true_to_prime.
+ assumption
+ ]
+ ]
+ ]
+ |apply sym_eq.
+ apply or_false_eq_SO_to_eq_pi_p
+ [apply le_S_S.
+ apply le_times_n.
+ apply lt_O_S
+ |intros.right.
+ change with (exp i (log i n) = (exp i O)).
+ apply eq_f.
+ apply antisymmetric_le
+ [cut (O < n)
+ [apply le_S_S_to_le.
+ apply (lt_exp_to_lt i)
+ [apply (le_to_lt_to_lt ? n);assumption
+ |apply (le_to_lt_to_lt ? n)
+ [apply le_exp_log.
+ assumption
+ |rewrite < exp_n_SO.
+ assumption
+ ]
+ ]
+ |apply not_le_to_lt.intro.
+ apply (lt_to_not_le ? ? H1).
+ generalize in match H.
+ apply (le_n_O_elim ? H2).
+ intro.assumption
+ ]
+ |apply le_O_n
+ ]
+ ]
+ ]
+qed.
+
+
+unfold A.unfold B.
+rewrite > eq_A_A'.rewrite > eq_A_A'.
+unfold A'.unfold B.
+
+(* da spostare *)
+theorem times_exp: \forall n,m,p.exp n p * exp m p = exp (n*m) p.
+intros.elim p
+ [reflexivity
+ |simplify.autobatch
+ ]
+qed.
+
+theorem le_exp_log: \forall p,n. O < n \to
+exp p (
+log n) \le n.
+intros.
+apply leb_true_to_le.
+unfold log.
+apply (f_max_true (\lambda x.leb (exp p x) n)).
+apply (ex_intro ? ? O).
+split
+ [apply le_O_n
+ |apply le_to_leb_true.simplify.assumption
+ ]
+qed.
+
+theorem lt_log_n_n: \forall n. O < n \to log n < n.
+intros.
+cut (log n \le n)
+ [elim (le_to_or_lt_eq ? ? Hcut)
+ [assumption
+ |absurd (exp (S(S O)) n \le n)
+ [rewrite < H1 in ⊢ (? (? ? %) ?).
+ apply le_exp_log.
+ assumption
+ |apply lt_to_not_le.
+ apply lt_m_exp_nm.
+ apply le_n
+ ]
+ ]
+ |unfold log.apply le_max_n
+ ]
+qed.
+
+theorem le_log_n_n: \forall n. log n \le n.
+intro.
+cases n
+ [apply le_n
+ |apply lt_to_le.
+ apply lt_log_n_n.
+ apply lt_O_S
+ ]
+qed.
+
+theorem lt_exp_log: \forall n. n < exp (S(S O)) (S (log n)).
+intro.cases n
+ [simplify.apply le_S.apply le_n
+ |apply not_le_to_lt.
+ apply leb_false_to_not_le.
+ apply (lt_max_to_false ? (S n1) (S (log (S n1))))
+ [apply le_S_S.apply le_n
+ |apply lt_log_n_n.apply lt_O_S
+ ]
+ ]
+qed.
+
+theorem f_false_to_le_max: \forall f,n,p. (∃i:nat.i≤n∧f i=true) \to
+(\forall m. p < m \to f m = false)
+\to max n f \le p.
+intros.
+apply not_lt_to_le.intro.
+apply not_eq_true_false.
+rewrite < (H1 ? H2).
+apply sym_eq.
+apply f_max_true.
+assumption.
+qed.
+
+theorem log_times1: \forall n,m. O < n \to O < m \to
+log (n*m) \le S(log n+log m).
+intros.
+unfold in ⊢ (? (% ?) ?).
+apply f_false_to_le_max
+ [apply (ex_intro ? ? O).
+ split
+ [apply le_O_n
+ |apply le_to_leb_true.
+ simplify.
+ rewrite > times_n_SO.
+ apply le_times;assumption
+ ]
+ |intros.
+ apply lt_to_leb_false.
+ apply (lt_to_le_to_lt ? ((exp (S(S O)) (S(log n)))*(exp (S(S O)) (S(log m)))))
+ [apply lt_times;apply lt_exp_log
+ |rewrite < exp_plus_times.
+ apply le_exp
+ [apply lt_O_S
+ |simplify.
+ rewrite < plus_n_Sm.
+ assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem log_times: \forall n,m.log (n*m) \le S(log n+log m).
+intros.
+cases n
+ [apply le_O_n
+ |cases m
+ [rewrite < times_n_O.
+ apply le_O_n
+ |apply log_times1;apply lt_O_S
+ ]
+ ]
+qed.
+
+theorem log_exp: \forall n,m.O < m \to
+log ((exp (S(S O)) n)*m)=n+log m.
+intros.
+unfold log in ⊢ (? ? (% ?) ?).
+apply max_spec_to_max.
+unfold max_spec.
+split
+ [split
+ [elim n
+ [simplify.
+ rewrite < plus_n_O.
+ apply le_log_n_n
+ |simplify.
+ rewrite < plus_n_O.
+ rewrite > times_plus_l.
+ apply (trans_le ? (S((S(S O))\sup(n1)*m)))
+ [apply le_S_S.assumption
+ |apply (lt_O_n_elim ((S(S O))\sup(n1)*m))
+ [rewrite > (times_n_O O) in ⊢ (? % ?).
+ apply lt_times
+ [apply lt_O_exp.
+ apply lt_O_S
+ |assumption
+ ]
+ |intro.simplify.apply le_S_S.
+ apply le_plus_n
+ ]
+ ]
+ ]
+ |simplify.
+ apply le_to_leb_true.
+ rewrite > exp_plus_times.
+ apply le_times_r.
+ apply le_exp_log.
+ assumption
+ ]
+ |intros.
+ simplify.
+ apply lt_to_leb_false.
+ apply (lt_to_le_to_lt ? ((exp (S(S O)) n)*(exp (S(S O)) (S(log m)))))
+ [apply lt_times_r1
+ [apply lt_O_exp.apply lt_O_S
+ |apply lt_exp_log.
+ ]
+ |rewrite < exp_plus_times.
+ apply le_exp
+ [apply lt_O_S
+ |rewrite < plus_n_Sm.
+ assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem log_SSO: \forall n. O < n \to
+log ((S(S O))*n) = S (log n).
+intros.
+change with (log ((exp (S(S O)) (S O))*n)=S (log n)).
+rewrite > log_exp.reflexivity.assumption.
+qed.
+
+theorem or_eq_S: \forall n.\exists m.
+(n = (S(S O))*m) \lor (n = S((S(S O))*m)).
+intro.
+elim n
+ [apply (ex_intro ? ? O).left.apply times_n_O
+ |elim H.elim H1
+ [apply (ex_intro ? ? a).right.apply eq_f.assumption
+ |apply (ex_intro ? ? (S a)).left.
+ rewrite > H2.simplify.
+ rewrite < plus_n_O.
+ rewrite < plus_n_Sm.
+ reflexivity
+ ]
+ ]
+qed.
+
+theorem lt_O_to_or_eq_S: \forall n.O < n \to \exists m.m < n \land
+((n = (S(S O))*m) \lor (n = S((S(S O))*m))).
+intros.
+elim (or_eq_S n).
+elim H1
+ [apply (ex_intro ? ? a).split
+ [rewrite > H2.
+ rewrite > times_n_SO in ⊢ (? % ?).
+ rewrite > sym_times.
+ apply lt_times_l1
+ [apply (divides_to_lt_O a n ? ?)
+ [assumption
+ |apply (witness a n (S (S O))).
+ rewrite > sym_times.
+ assumption
+ ]
+ |apply le_n
+ ]
+ |left.assumption
+ ]
+ |apply (ex_intro ? ? a).split
+ [rewrite > H2.
+ apply (le_to_lt_to_lt ? ((S(S O))*a))
+ [rewrite > times_n_SO in ⊢ (? % ?).
+ rewrite > sym_times.
+ apply le_times_l.
+ apply le_n_Sn
+ |apply le_n
+ ]
+ |right.assumption
+ ]
+ ]
+qed.
+
+theorem factS: \forall n. fact (S n) = (S n)*(fact n).
+intro.simplify.reflexivity.
+qed.
+
+theorem exp_S: \forall n,m. exp m (S n) = m*exp m n.
+intros.reflexivity.
+qed.
+
+lemma times_SSO: \forall n.(S(S O))*(S n) = S(S((S(S O))*n)).
+intro.simplify.rewrite < plus_n_Sm.reflexivity.
+qed.
+
+theorem fact1: \forall n.
+fact ((S(S O))*n) \le (exp (S(S O)) ((S(S O))*n))*(fact n)*(fact n).
+intro.elim n
+ [rewrite < times_n_O.apply le_n
+ |rewrite > times_SSO.
+ rewrite > factS.
+ rewrite > factS.
+ rewrite < assoc_times.
+ rewrite > factS.
+ apply (trans_le ? (((S(S O))*(S n1))*((S(S O))*(S n1))*(fact (((S(S O))*n1)))))
+ [apply le_times_l.
+ rewrite > times_SSO.
+ apply le_times_r.
+ apply le_n_Sn
+ |rewrite > assoc_times.
+ rewrite > assoc_times.
+ rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > exp_S.
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite < assoc_times.
+ rewrite < assoc_times.
+ rewrite < sym_times in ⊢ (? (? (? % ?) ?) ?).
+ rewrite > assoc_times.
+ rewrite > assoc_times.
+ rewrite > exp_S.
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite > sym_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite < assoc_times in ⊢ (? ? %).
+ rewrite < assoc_times in ⊢ (? ? %).
+ rewrite < sym_times in ⊢ (? ? (? (? % ?) ?)).
+ rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite > sym_times in ⊢ (? ? (? ? %)).
+ rewrite > sym_times in ⊢ (? ? %).
+ assumption
+ ]
+ ]
+qed.
+
+theorem monotonic_log: monotonic nat le log.
+unfold monotonic.intros.
+elim (le_to_or_lt_eq ? ? H) 0
+ [cases x;intro
+ [apply le_O_n
+ |apply lt_S_to_le.
+ apply (lt_exp_to_lt (S(S O)))
+ [apply le_n
+ |apply (le_to_lt_to_lt ? (S n))
+ [apply le_exp_log.
+ apply lt_O_S
+ |apply (trans_lt ? y)
+ [assumption
+ |apply lt_exp_log
+ ]
+ ]
+ ]
+ ]
+ |intro.rewrite < H1.apply le_n
+ ]
+qed.
+
+theorem lt_O_fact: \forall n. O < fact n.
+intro.elim n
+ [simplify.apply lt_O_S
+ |rewrite > factS.
+ rewrite > (times_n_O O).
+ apply lt_times
+ [apply lt_O_S
+ |assumption
+ ]
+ ]
+qed.
+
+theorem fact2: \forall n.O < n \to
+(exp (S(S O)) ((S(S O))*n))*(fact n)*(fact n) < fact (S((S(S O))*n)).
+intros.elim H
+ [simplify.apply le_S.apply le_n
+ |rewrite > times_SSO.
+ rewrite > factS.
+ rewrite > factS.
+ rewrite < assoc_times.
+ rewrite > factS.
+ rewrite < times_SSO in ⊢ (? ? %).
+ apply (trans_lt ? (((S(S O))*S n1)*((S(S O))*S n1*(S ((S(S O))*n1))!)))
+ [rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > exp_S.
+ rewrite > assoc_times.
+ rewrite > assoc_times.
+ rewrite > assoc_times.
+ apply lt_times_r.
+ rewrite > exp_S.
+ rewrite > assoc_times.
+ rewrite > sym_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply lt_times_r.
+ rewrite > sym_times.
+ rewrite > assoc_times.
+ rewrite > assoc_times.
+ apply lt_times_r.
+ rewrite < assoc_times.
+ rewrite < assoc_times.
+ rewrite > sym_times in ⊢ (? (? (? % ?) ?) ?).
+ rewrite > assoc_times.
+ rewrite > assoc_times.
+ rewrite > sym_times in ⊢ (? ? %).
+ apply lt_times_r.
+ rewrite < assoc_times.
+ rewrite < sym_times.
+ rewrite < assoc_times.
+ assumption
+ |apply lt_times_l1
+ [rewrite > (times_n_O O) in ⊢ (? % ?).
+ apply lt_times
+ [rewrite > (times_n_O O) in ⊢ (? % ?).
+ apply lt_times
+ [apply lt_O_S
+ |apply lt_O_S
+ ]
+ |apply lt_O_fact
+ ]
+ |apply le_n
+ ]
+ ]
+ ]
+qed.
+
+theorem lt_O_log: \forall n. S O < n \to O < log n.
+intros.elim H
+ [simplify.apply lt_O_S
+ |apply (lt_to_le_to_lt ? (log n1))
+ [assumption
+ |apply monotonic_log.
+ apply le_n_Sn
+ ]
+ ]
+qed.
+
+theorem log_fact_SSSO: log (fact (S(S(S O)))) = S (S O).
+reflexivity.
+qed.
+
+lemma exp_SSO_SSO: exp (S(S O)) (S(S O)) = S(S(S(S O))).
+reflexivity.
+qed.
+(*
+theorem log_fact_SSSSO: log (fact (S(S(S(S O))))) = S(S(S(S O))).
+reflexivity.
+qed.
+*)
+theorem log_fact_SSSSSO: log (fact (S(S(S(S(S O)))))) = S(S(S(S O))).
+reflexivity.
+qed.
+
+theorem bof_bof: \forall n.(S(S(S(S O)))) < n \to
+n \le (S(S(S(S(S(S(S(S O)))))))) \to log (fact n) < n*log n - n.
+intros.
+elim H
+ [rewrite > factS in ⊢ (? (? %) ?).
+ rewrite > factS in ⊢ (? (? (? ? %)) ?).
+ rewrite < assoc_times in ⊢ (? (? %) ?).
+ rewrite < sym_times in ⊢ (? (? (? % ?)) ?).
+ rewrite > assoc_times in ⊢ (? (? %) ?).
+ change with (exp (S(S O)) (S(S O))) in ⊢ (? (? (? % ?)) ?).
+
+theorem bof: \forall n.(S(S(S(S O))))) < n \to log (fact n) < n*log n - n.
+intro.apply (nat_elim1 n).
+intros.
+elim (lt_O_to_or_eq_S m)
+ [elim H2.clear H2.
+ elim H4.clear H4.
+ rewrite > H2.
+ apply (le_to_lt_to_lt ? (log ((exp (S(S O)) ((S(S O))*a))*(fact a)*(fact a))))
+ [apply monotonic_log.
+ apply fact1
+ |rewrite > assoc_times in ⊢ (? (? %) ?).
+ rewrite > log_exp.
+ apply (le_to_lt_to_lt ? ((S(S O))*a+S(log a!+log a!)))
+ [apply le_plus_r.
+ apply log_times
+ |rewrite > plus_n_Sm.
+ rewrite > log_SSO.
+ rewrite < times_n_Sm.
+ apply (le_to_lt_to_lt ? ((S(S O))*a+(log a!+(a*log a-a))))
+ [apply le_plus_r.
+ apply le_plus_r.
+ apply H.assumption
+ |apply (lt_to_le_to_lt ? ((S(S O))*a+((a*log a-a)+(a*log a-a))))
+ [apply lt_plus_r.
+ apply lt_plus_l.
+ apply H.
+ assumption.
+ |rewrite > times_SSO_n.
+ rewrite > distr_times_minus.
+ rewrite > sym_plus.
+ rewrite > plus_minus
+ [rewrite > sym_plus.
+ rewrite < assoc_times.
+ apply le_n
+ |rewrite < assoc_times.
+ rewrite > times_n_SO in ⊢ (? % ?).
+ apply le_times
+ [apply le_n
+ |apply lt_O_log.
+ apply (lt_times_n_to_lt_r (S(S O)))
+ [apply lt_O_S
+ |rewrite < times_n_SO.
+ rewrite < H2.
+ assumption
+ ]
+ ]
+ ]
+ ]
+
+ .
+ ]
+ |
+ rewrite < eq_plus_minus_minus_plus.
+
+ rewrite > assoc_plus.
+ apply lt_plus_r.
+ rewrite > plus_n_O in ⊢ (? (? (? ? (? ? %))) ?).
+ change with
+ (S((S(S O))*a+((S(S O))*log a!)) < (S(S O))*a*log ((S(S O))*a)-(S(S O))*a+k*log ((S(S O))*a)).
+ apply (trans_lt ? (S ((S(S O))*a+(S(S O))*(a*log a-a+k*log a))))
+ [apply le_S_S.
+ apply lt_plus_r.
+ apply lt_times_r.
+ apply H.
+ assumption
+ |
+
+theorem stirling: \forall n,k.k \le n \to
+log (fact n) < n*log n - n + k*log n.
+intro.
+apply (nat_elim1 n).
+intros.
+elim (lt_O_to_or_eq_S m)
+ [elim H2.clear H2.
+ elim H4.clear H4.
+ rewrite > H2.
+ apply (le_to_lt_to_lt ? (log ((exp (S(S O)) ((S(S O))*a))*(fact a)*(fact a))))
+ [apply monotonic_log.
+ apply fact1
+ |rewrite > assoc_times in ⊢ (? (? %) ?).
+ rewrite > log_exp.
+ apply (le_to_lt_to_lt ? ((S(S O))*a+S(log a!+log a!)))
+ [apply le_plus_r.
+ apply log_times
+ |rewrite < plus_n_Sm.
+ rewrite > plus_n_O in ⊢ (? (? (? ? (? ? %))) ?).
+ change with
+ (S((S(S O))*a+((S(S O))*log a!)) < (S(S O))*a*log ((S(S O))*a)-(S(S O))*a+k*log ((S(S O))*a)).
+ apply (trans_lt ? (S ((S(S O))*a+(S(S O))*(a*log a-a+k*log a))))
+ [apply le_S_S.
+ apply lt_plus_r.
+ apply lt_times_r.
+ apply H.
+ assumption
+ |
+
+ [
+
+ a*log a-a+k*log a
+
]
qed.
+theorem leb_true_to_le:\forall n,m.
+leb n m = true \to (n \le m).
+intros 2.
+apply leb_elim
+ [intros.assumption
+ |intros.destruct H1.
+ ]
+qed.
+
+theorem leb_false_to_not_le:\forall n,m.
+leb n m = false \to \lnot (n \le m).
+intros 2.
+apply leb_elim
+ [intros.destruct H1
+ |intros.assumption
+ ]
+qed.
(*
theorem decidable_le: \forall n,m. n \leq m \lor n \nleq m.
intros.
]
qed.
+theorem or_div_mod: \forall n,q. O < q \to
+((S (n \mod q)=q) \land S n = (S (div n q)) * q \lor
+((S (n \mod q)<q) \land S n= (div n q) * q + S (n\mod q))).
+intros.
+elim (le_to_or_lt_eq ? ? (lt_mod_m_m n q H))
+ [right.split
+ [assumption
+ |rewrite < plus_n_Sm.
+ apply eq_f.
+ apply div_mod.
+ assumption
+ ]
+ |left.split
+ [assumption
+ |simplify.
+ rewrite > sym_plus.
+ rewrite < H1 in ⊢ (? ? ? (? ? %)).
+ rewrite < plus_n_Sm.
+ apply eq_f.
+ apply div_mod.
+ assumption
+ ]
+ ]
+qed.
+
(* injectivity *)
theorem injective_times_r: \forall n:nat.injective nat nat (\lambda m:nat.(S n)*m).
change with (\forall n,p,q:nat.(S n)*p = (S n)*q \to p=q).
qed.
*)
-
(*this obvious property is useful because simplify, sometimes,
"simplifies too much", and doesn't allow to obtain this simple result.
*)
]
qed.
-
lemma totient_card: \forall n.
totient n = card n (\lambda i.eqb (gcd i n) (S O)).
intros.
intro.simplify.rewrite < times_n_SO.reflexivity.
qed.
+theorem exp_SSO: \forall n. exp n (S(S O)) = n*n.
+intro.simplify.
+rewrite < times_n_SO.
+reflexivity.
+qed.
+
theorem exp_exp_times : \forall n,p,q:nat.
(n \sup p) \sup q = n \sup (p * q).
intros.
]
qed.
-
+theorem lt_exp_to_lt:
+\forall a,n,m. S O < a \to exp a n < exp a m \to n < m.
+intros.
+elim (le_to_or_lt_eq n m)
+ [assumption
+ |apply False_ind.
+ apply (lt_to_not_eq ? ? H1).
+ rewrite < H2.
+ reflexivity
+ |apply (le_exp_to_le a)
+ [assumption
+ |apply lt_to_le.
+ assumption
+ ]
+ ]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / Matita is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/nat/factorial2".
+
+include "nat/exp.ma".
+include "nat/factorial.ma".
+
+theorem factS: \forall n. fact (S n) = (S n)*(fact n).
+intro.simplify.reflexivity.
+qed.
+
+theorem exp_S: \forall n,m. exp m (S n) = m*exp m n.
+intros.reflexivity.
+qed.
+
+lemma times_SSO: \forall n.(S(S O))*(S n) = S(S((S(S O))*n)).
+intro.simplify.rewrite < plus_n_Sm.reflexivity.
+qed.
+
+theorem fact1: \forall n.
+fact ((S(S O))*n) \le (exp (S(S O)) ((S(S O))*n))*(fact n)*(fact n).
+intro.elim n
+ [rewrite < times_n_O.apply le_n
+ |rewrite > times_SSO.
+ rewrite > factS.
+ rewrite > factS.
+ rewrite < assoc_times.
+ rewrite > factS.
+ apply (trans_le ? (((S(S O))*(S n1))*((S(S O))*(S n1))*(fact (((S(S O))*n1)))))
+ [apply le_times_l.
+ rewrite > times_SSO.
+ apply le_times_r.
+ apply le_n_Sn
+ |rewrite > assoc_times.
+ rewrite > assoc_times.
+ rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > exp_S.
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite < assoc_times.
+ rewrite < assoc_times.
+ rewrite < sym_times in ⊢ (? (? (? % ?) ?) ?).
+ rewrite > assoc_times.
+ rewrite > assoc_times.
+ rewrite > exp_S.
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite > sym_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite < assoc_times in ⊢ (? ? %).
+ rewrite < assoc_times in ⊢ (? ? %).
+ rewrite < sym_times in ⊢ (? ? (? (? % ?) ?)).
+ rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply le_times_r.
+ rewrite > sym_times in ⊢ (? ? (? ? %)).
+ rewrite > sym_times in ⊢ (? ? %).
+ assumption
+ ]
+ ]
+qed.
+
+theorem lt_O_fact: \forall n. O < fact n.
+intro.elim n
+ [simplify.apply lt_O_S
+ |rewrite > factS.
+ rewrite > (times_n_O O).
+ apply lt_times
+ [apply lt_O_S
+ |assumption
+ ]
+ ]
+qed.
+
+theorem fact2: \forall n.O < n \to
+(exp (S(S O)) ((S(S O))*n))*(fact n)*(fact n) < fact (S((S(S O))*n)).
+intros.elim H
+ [simplify.apply le_S.apply le_n
+ |rewrite > times_SSO.
+ rewrite > factS.
+ rewrite > factS.
+ rewrite < assoc_times.
+ rewrite > factS.
+ rewrite < times_SSO in ⊢ (? ? %).
+ apply (trans_lt ? (((S(S O))*S n1)*((S(S O))*S n1*(S ((S(S O))*n1))!)))
+ [rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > exp_S.
+ rewrite > assoc_times.
+ rewrite > assoc_times.
+ rewrite > assoc_times.
+ apply lt_times_r.
+ rewrite > exp_S.
+ rewrite > assoc_times.
+ rewrite > sym_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ rewrite > assoc_times in ⊢ (? ? %).
+ apply lt_times_r.
+ rewrite > sym_times.
+ rewrite > assoc_times.
+ rewrite > assoc_times.
+ apply lt_times_r.
+ rewrite < assoc_times.
+ rewrite < assoc_times.
+ rewrite > sym_times in ⊢ (? (? (? % ?) ?) ?).
+ rewrite > assoc_times.
+ rewrite > assoc_times.
+ rewrite > sym_times in ⊢ (? ? %).
+ apply lt_times_r.
+ rewrite < assoc_times.
+ rewrite < sym_times.
+ rewrite < assoc_times.
+ assumption
+ |apply lt_times_l1
+ [rewrite > (times_n_O O) in ⊢ (? % ?).
+ apply lt_times
+ [rewrite > (times_n_O O) in ⊢ (? % ?).
+ apply lt_times
+ [apply lt_O_S
+ |apply lt_O_S
+ ]
+ |apply lt_O_fact
+ ]
+ |apply le_n
+ ]
+ ]
+ ]
+qed.
+
+(*
+theorem stirling: \forall n,k.k \le n \to
+log (fact n) < n*log n - n + k*log n.
+intro.
+apply (nat_elim1 n).
+intros.
+elim (lt_O_to_or_eq_S m)
+ [elim H2.clear H2.
+ elim H4.clear H4.
+ rewrite > H2.
+ apply (le_to_lt_to_lt ? (log ((exp (S(S O)) ((S(S O))*a))*(fact a)*(fact a))))
+ [apply monotonic_log.
+ apply fact1
+ |rewrite > assoc_times in ⊢ (? (? %) ?).
+ rewrite > log_exp.
+ apply (le_to_lt_to_lt ? ((S(S O))*a+S(log a!+log a!)))
+ [apply le_plus_r.
+ apply log_times
+ |rewrite < plus_n_Sm.
+ rewrite > plus_n_O in ⊢ (? (? (? ? (? ? %))) ?).
+ change with
+ (S((S(S O))*a+((S(S O))*log a!)) < (S(S O))*a*log ((S(S O))*a)-(S(S O))*a+k*log ((S(S O))*a)).
+ apply (trans_lt ? (S ((S(S O))*a+(S(S O))*(a*log a-a+k*log a))))
+ [apply le_S_S.
+ apply lt_plus_r.
+ apply lt_times_r.
+ apply H.
+ assumption
+ |
+
+ [
+
+ a*log a-a+k*log a
+
+*)
\ No newline at end of file
set "baseuri" "cic:/matita/nat/factorization".
include "nat/ord.ma".
-include "nat/gcd.ma".
-include "nat/nth_prime.ma".
(* the following factorization algorithm looks for the largest prime
factor. *)
definition max_prime_factor \def \lambda n:nat.
(max n (\lambda p:nat.eqb (n \mod (nth_prime p)) O)).
+theorem lt_SO_max_prime: \forall m. S O < m \to
+S O < max m (λi:nat.primeb i∧divides_b i m).
+intros.
+apply (lt_to_le_to_lt ? (smallest_factor m))
+ [apply lt_SO_smallest_factor.assumption
+ |apply f_m_to_le_max
+ [apply le_smallest_factor_n
+ |apply true_to_true_to_andb_true
+ [apply prime_to_primeb_true.
+ apply prime_smallest_factor_n.
+ assumption
+ |apply divides_to_divides_b_true
+ [apply lt_O_smallest_factor.apply lt_to_le.assumption
+ |apply divides_smallest_factor_n.
+ apply lt_to_le.assumption
+ ]
+ ]
+ ]
+ ]
+qed.
(* max_prime_factor is indeed a factor *)
theorem divides_max_prime_factor_n:
\forall n:nat. (S O) < n
| rewrite > H1;
apply le_smallest_factor_n; ]
| rewrite > H1;
- (*CSC: simplify here does something nasty! *)
change with (divides_b (smallest_factor n) n = true);
apply divides_to_divides_b_true;
[ apply (trans_lt ? (S O));
intro.
apply (nat_case n).reflexivity.
intro.apply (nat_case m).reflexivity.
-intro.(*CSC: simplify here does something really nasty *)
+intro.
change with
(let p \def (max (S(S m1)) (\lambda p:nat.eqb ((S(S m1)) \mod (nth_prime p)) O)) in
defactorize (match p_ord (S(S m1)) (nth_prime p) with
cut ((S(S m1)) = (nth_prime p) \sup q *r).
cut (O<r).
rewrite > defactorize_aux_factorize_aux.
-(*CSC: simplify here does something really nasty *)
change with (r*(nth_prime p) \sup (S (pred q)) = (S(S m1))).
cut ((S (pred q)) = q).
rewrite > Hcut2.
unfold lt.apply le_S_S.apply le_S_S. apply le_O_n.
cut ((S(S m1)) = r).
rewrite > Hcut3 in \vdash (? (? ? %)).
-(*CSC: simplify here does something really nasty *)
change with (nth_prime p \divides r \to False).
intro.
apply (p_ord_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r).
reflexivity.
qed.
-theorem divides_exp_to_divides:
-\forall p,n,m:nat. prime p \to
-p \divides n \sup m \to p \divides n.
-intros 3.elim m.simplify in H1.
-apply (transitive_divides p (S O)).assumption.
-apply divides_SO_n.
-cut (p \divides n \lor p \divides n \sup n1).
-elim Hcut.assumption.
-apply H.assumption.assumption.
-apply divides_times_to_divides.assumption.
-exact H2.
-qed.
-
-theorem divides_exp_to_eq:
-\forall p,q,m:nat. prime p \to prime q \to
-p \divides q \sup m \to p = q.
-intros.
-unfold prime in H1.
-elim H1.apply H4.
-apply (divides_exp_to_divides p q m).
-assumption.assumption.
-unfold prime in H.elim H.assumption.
-qed.
-
lemma eq_p_max: \forall n,p,r:nat. O < n \to
O < r \to
r = (S O) \lor (max r (\lambda p:nat.eqb (r \mod (nth_prime p)) O)) < p \to
qed.
theorem factorize_defactorize:
-\forall f,g: nat_fact_all. factorize (defactorize f) = f.
+\forall f: nat_fact_all. factorize (defactorize f) = f.
intros.
apply injective_defactorize.
apply defactorize_factorize.
apply le_to_or_lt_eq.apply le_O_n.
qed.
+(* primes and divides *)
theorem divides_times_to_divides: \forall n,p,q:nat.prime n \to n \divides p*q \to
n \divides p \lor n \divides q.
intros.
]
qed.
+theorem divides_exp_to_divides:
+\forall p,n,m:nat. prime p \to
+p \divides n \sup m \to p \divides n.
+intros 3.elim m.simplify in H1.
+apply (transitive_divides p (S O)).assumption.
+apply divides_SO_n.
+cut (p \divides n \lor p \divides n \sup n1).
+elim Hcut.assumption.
+apply H.assumption.assumption.
+apply divides_times_to_divides.assumption.
+exact H2.
+qed.
+
+theorem divides_exp_to_eq:
+\forall p,q,m:nat. prime p \to prime q \to
+p \divides q \sup m \to p = q.
+intros.
+unfold prime in H1.
+elim H1.apply H4.
+apply (divides_exp_to_divides p q m).
+assumption.assumption.
+unfold prime in H.elim H.assumption.
+qed.
+
theorem eq_gcd_times_SO: \forall m,n,p:nat. O < n \to O < p \to
gcd m n = (S O) \to gcd m p = (S O) \to gcd m (n*p) = (S O).
intros.
|apply (decidable_divides n p).
assumption.
]
-qed.
\ No newline at end of file
+qed.
+
]
qed.
+(* a therem slightly more general than the previous one *)
+theorem or_false_eq_baseA_to_eq_iter_p_gen: \forall A:Type. \forall n,m:nat.\forall p:nat \to bool.
+\forall g: nat \to A. \forall baseA:A. \forall plusA: A \to A \to A.
+(\forall a. plusA baseA a = a) \to
+n \le m \to (\forall i:nat. n \le i \to i < m \to p i = false \lor g i = baseA)
+\to iter_p_gen m p A g baseA plusA = iter_p_gen n p A g baseA plusA.
+intros 9.
+elim H1
+[reflexivity
+|apply (bool_elim ? (p n1));intro
+ [elim (H4 n1)
+ [apply False_ind.
+ apply not_eq_true_false.
+ rewrite < H5.
+ rewrite < H6.
+ reflexivity
+ |rewrite > true_to_iter_p_gen_Sn
+ [rewrite > H6.
+ rewrite > H.
+ apply H3.intros.
+ apply H4
+ [assumption
+ |apply le_S.assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ |apply le_n
+ ]
+ |rewrite > false_to_iter_p_gen_Sn
+ [apply H3.intros.
+ apply H4
+ [assumption
+ |apply le_S.assumption
+ ]
+ |assumption
+ ]
+ ]
+]
+qed.
+
theorem iter_p_gen2 :
\forall n,m:nat.
\forall p1,p2:nat \to bool.
]
qed.
-
+theorem iter_p_gen_iter_p_gen:
+\forall A:Type.
+\forall baseA: A.
+\forall plusA: A \to A \to A.
+(symmetric A plusA) \to
+(associative A plusA) \to
+(\forall a:A.(plusA a baseA) = a)\to
+\forall g: nat \to nat \to A.
+\forall n,m.
+\forall p11,p21:nat \to bool.
+\forall p12,p22:nat \to nat \to bool.
+(\forall x,y. x < n \to y < m \to
+ (p11 x \land p12 x y) = (p21 y \land p22 y x)) \to
+iter_p_gen n p11 A
+ (\lambda x:nat.iter_p_gen m (p12 x) A (\lambda y. g x y) baseA plusA)
+ baseA plusA =
+iter_p_gen m p21 A
+ (\lambda y:nat.iter_p_gen n (p22 y) A (\lambda x. g x y) baseA plusA )
+ baseA plusA.
+intros.
+apply (iter_p_gen_2_eq A baseA plusA H H1 H2 (\lambda x,y. g x y) (\lambda x,y.y) (\lambda x,y.x) (\lambda x,y.y) (\lambda x,y.x)
+ n m m n p11 p21 p12 p22)
+ [intros.split
+ [split
+ [split
+ [split
+ [split
+ [apply (andb_true_true ? (p12 j i)).
+ rewrite > H3
+ [rewrite > H6.rewrite > H7.reflexivity
+ |assumption
+ |assumption
+ ]
+ |apply (andb_true_true_r (p11 j)).
+ rewrite > H3
+ [rewrite > H6.rewrite > H7.reflexivity
+ |assumption
+ |assumption
+ ]
+ ]
+ |reflexivity
+ ]
+ |reflexivity
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |intros.split
+ [split
+ [split
+ [split
+ [split
+ [apply (andb_true_true ? (p22 j i)).
+ rewrite < H3
+ [rewrite > H6.rewrite > H7.reflexivity
+ |assumption
+ |assumption
+ ]
+ |apply (andb_true_true_r (p21 j)).
+ rewrite < H3
+ [rewrite > H6.rewrite > H7.reflexivity
+ |assumption
+ |assumption
+ ]
+ ]
+ |reflexivity
+ ]
+ |reflexivity
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+qed.
]
qed.
+theorem le_S_times_SSO: \forall n,m.O < m \to
+n \le m \to S n \le (S(S O))*m.
+intros.
+simplify.
+rewrite > plus_n_O.
+simplify.rewrite > plus_n_Sm.
+apply le_plus
+ [assumption
+ |rewrite < plus_n_O.
+ assumption
+ ]
+qed.
(*0 and times *)
theorem O_lt_const_to_le_times_const: \forall a,c:nat.
O \lt c \to a \le a*c.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / Matita is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/nat/log".
+
+include "datatypes/constructors.ma".
+include "nat/minimization.ma".
+include "nat/relevant_equations.ma".
+include "nat/primes.ma".
+
+definition log \def \lambda p,n.
+max n (\lambda x.leb (exp p x) n).
+
+theorem le_exp_log: \forall p,n. O < n \to
+exp p (log p n) \le n.
+intros.
+apply leb_true_to_le.
+unfold log.
+apply (f_max_true (\lambda x.leb (exp p x) n)).
+apply (ex_intro ? ? O).
+split
+ [apply le_O_n
+ |apply le_to_leb_true.simplify.assumption
+ ]
+qed.
+
+theorem log_SO: \forall n. S O < n \to log n (S O) = O.
+intros.
+apply sym_eq.apply le_n_O_to_eq.
+apply (le_exp_to_le n)
+ [assumption
+ |simplify in ⊢ (? ? %).
+ apply le_exp_log.
+ apply le_n
+ ]
+qed.
+
+theorem lt_log_n_n: \forall p, n. S O < p \to O < n \to log p n < n.
+intros.
+cut (log p n \le n)
+ [elim (le_to_or_lt_eq ? ? Hcut)
+ [assumption
+ |absurd (exp p n \le n)
+ [rewrite < H2 in ⊢ (? (? ? %) ?).
+ apply le_exp_log.
+ assumption
+ |apply lt_to_not_le.
+ apply lt_m_exp_nm.
+ assumption
+ ]
+ ]
+ |unfold log.apply le_max_n
+ ]
+qed.
+
+theorem lt_O_log: \forall p,n. O < n \to p \le n \to O < log p n.
+intros.
+unfold log.
+apply not_lt_to_le.
+intro.
+apply (leb_false_to_not_le ? ? ? H1).
+rewrite > (exp_n_SO p).
+apply (lt_max_to_false ? ? ? H2).
+assumption.
+qed.
+
+theorem le_log_n_n: \forall p,n. S O < p \to log p n \le n.
+intros.
+cases n
+ [apply le_n
+ |apply lt_to_le.
+ apply lt_log_n_n
+ [assumption|apply lt_O_S]
+ ]
+qed.
+
+theorem lt_exp_log: \forall p,n. S O < p \to n < exp p (S (log p n)).
+intros.cases n
+ [simplify.rewrite < times_n_SO.apply lt_to_le.assumption
+ |apply not_le_to_lt.
+ apply leb_false_to_not_le.
+ apply (lt_max_to_false ? (S n1) (S (log p (S n1))))
+ [apply le_S_S.apply le_n
+ |apply lt_log_n_n
+ [assumption|apply lt_O_S]
+ ]
+ ]
+qed.
+
+theorem log_times1: \forall p,n,m. S O < p \to O < n \to O < m \to
+log p (n*m) \le S(log p n+log p m).
+intros.
+unfold in ⊢ (? (% ? ?) ?).
+apply f_false_to_le_max
+ [apply (ex_intro ? ? O).
+ split
+ [apply le_O_n
+ |apply le_to_leb_true.
+ simplify.
+ rewrite > times_n_SO.
+ apply le_times;assumption
+ ]
+ |intros.
+ apply lt_to_leb_false.
+ apply (lt_to_le_to_lt ? ((exp p (S(log p n)))*(exp p (S(log p m)))))
+ [apply lt_times;apply lt_exp_log;assumption
+ |rewrite < exp_plus_times.
+ apply le_exp
+ [apply lt_to_le.assumption
+ |simplify.
+ rewrite < plus_n_Sm.
+ assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem log_times: \forall p,n,m.S O < p \to log p (n*m) \le S(log p n+log p m).
+intros.
+cases n
+ [apply le_O_n
+ |cases m
+ [rewrite < times_n_O.
+ apply le_O_n
+ |apply log_times1
+ [assumption
+ |apply lt_O_S
+ |apply lt_O_S
+ ]
+ ]
+ ]
+qed.
+
+theorem log_exp: \forall p,n,m.S O < p \to O < m \to
+log p ((exp p n)*m)=n+log p m.
+intros.
+unfold log in ⊢ (? ? (% ? ?) ?).
+apply max_spec_to_max.
+unfold max_spec.
+split
+ [split
+ [elim n
+ [simplify.
+ rewrite < plus_n_O.
+ apply le_log_n_n.
+ assumption
+ |simplify.
+ rewrite > assoc_times.
+ apply (trans_le ? ((S(S O))*(p\sup n1*m)))
+ [apply le_S_times_SSO
+ [rewrite > (times_n_O O) in ⊢ (? % ?).
+ apply lt_times
+ [apply lt_O_exp.
+ apply lt_to_le.
+ assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ |apply le_times
+ [assumption
+ |apply le_n
+ ]
+ ]
+ ]
+ |simplify.
+ apply le_to_leb_true.
+ rewrite > exp_plus_times.
+ apply le_times_r.
+ apply le_exp_log.
+ assumption
+ ]
+ |intros.
+ simplify.
+ apply lt_to_leb_false.
+ apply (lt_to_le_to_lt ? ((exp p n)*(exp p (S(log p m)))))
+ [apply lt_times_r1
+ [apply lt_O_exp.apply lt_to_le.assumption
+ |apply lt_exp_log.assumption
+ ]
+ |rewrite < exp_plus_times.
+ apply le_exp
+ [apply lt_to_le.assumption
+ |rewrite < plus_n_Sm.
+ assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem le_log: \forall p,n,m. S O < p \to O < n \to n \le m \to
+log p n \le log p m.
+intros.
+apply le_S_S_to_le.
+apply (lt_exp_to_lt p)
+ [assumption
+ |apply (le_to_lt_to_lt ? n)
+ [apply le_exp_log.
+ assumption
+ |apply (le_to_lt_to_lt ? m)
+ [assumption
+ |apply lt_exp_log.
+ assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem log_n_n: \forall n. S O < n \to log n n = S O.
+intros.
+rewrite > exp_n_SO in ⊢ (? ? (? ? %) ?).
+rewrite > times_n_SO in ⊢ (? ? (? ? %) ?).
+rewrite > log_exp
+ [rewrite > log_SO
+ [reflexivity
+ |assumption
+ ]
+ |assumption
+ |apply le_n
+ ]
+qed.
+
+theorem log_i_SSOn: \forall n,i. S O < n \to n < i \to i \le ((S(S O))*n) \to
+log i ((S(S O))*n) = S O.
+intros.
+apply antisymmetric_le
+ [apply not_lt_to_le.intro.
+ apply (lt_to_not_le ((S(S O)) * n) (exp i (S(S O))))
+ [rewrite > exp_SSO.
+ apply lt_times
+ [apply (le_to_lt_to_lt ? n);assumption
+ |assumption
+ ]
+ |apply (trans_le ? (exp i (log i ((S(S O))*n))))
+ [apply le_exp
+ [apply (ltn_to_ltO ? ? H1)
+ |assumption
+ ]
+ |apply le_exp_log.
+ rewrite > (times_n_O O) in ⊢ (? % ?).
+ apply lt_times
+ [apply lt_O_S
+ |apply lt_to_le.assumption
+ ]
+ ]
+ ]
+ |apply (trans_le ? (log i i))
+ [rewrite < (log_n_n i) in ⊢ (? % ?)
+ [apply le_log
+ [apply (trans_lt ? n);assumption
+ |apply (ltn_to_ltO ? ? H1)
+ |apply le_n
+ ]
+ |apply (trans_lt ? n);assumption
+ ]
+ |apply le_log
+ [apply (trans_lt ? n);assumption
+ |apply (ltn_to_ltO ? ? H1)
+ |assumption
+ ]
+ ]
+ ]
+qed.
]
qed.
+theorem le_times_to_le_div: \forall a,b,c:nat.
+O \lt b \to (b*c) \le a \to c \le (a /b).
+intros.
+apply lt_S_to_le.
+apply (lt_times_n_to_lt b)
+ [assumption
+ |rewrite > sym_times.
+ apply (le_to_lt_to_lt ? a)
+ [assumption
+ |simplify.
+ rewrite > sym_plus.
+ rewrite > (div_mod a b) in ⊢ (? % ?)
+ [apply lt_plus_r.
+ apply lt_mod_m_m.
+ assumption
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
theorem nat_compare_times_l : \forall n,p,q:nat.
nat_compare p q = nat_compare ((S n) * p) ((S n) * q).
intros.apply nat_compare_elim.intro.
unfold lt. apply le_n.assumption.
qed.
+theorem eq_div_div_div_times: \forall n,m,q. O < n \to O < m \to
+q/n/m = q/(n*m).
+intros.
+apply (div_mod_spec_to_eq q (n*m) ? (q\mod n+n*(q/n\mod m)) ? (mod q (n*m)))
+ [apply div_mod_spec_intro
+ [apply (lt_to_le_to_lt ? (n*(S (q/n\mod m))))
+ [rewrite < times_n_Sm.
+ apply lt_plus_l.
+ apply lt_mod_m_m.
+ assumption
+ |apply le_times_r.
+ apply lt_mod_m_m.
+ assumption
+ ]
+ |rewrite > sym_times in ⊢ (? ? ? (? (? ? %) ?)).
+ rewrite < assoc_times.
+ rewrite > (eq_times_div_minus_mod ? ? H1).
+ rewrite > sym_times.
+ rewrite > distributive_times_minus.
+ rewrite > sym_times.
+ rewrite > (eq_times_div_minus_mod ? ? H).
+ rewrite < sym_plus in ⊢ (? ? ? (? ? %)).
+ rewrite < assoc_plus.
+ rewrite < plus_minus_m_m
+ [rewrite < plus_minus_m_m
+ [reflexivity
+ |apply (eq_plus_to_le ? ? ((q/n)*n)).
+ rewrite < sym_plus.
+ apply div_mod.
+ assumption
+ ]
+ |apply (trans_le ? (n*(q/n)))
+ [apply le_times_r.
+ apply (eq_plus_to_le ? ? ((q/n)/m*m)).
+ rewrite < sym_plus.
+ apply div_mod.
+ assumption
+ |rewrite > sym_times.
+ rewrite > eq_times_div_minus_mod
+ [apply le_n
+ |assumption
+ ]
+ ]
+ ]
+ ]
+ |apply div_mod_spec_div_mod.
+ rewrite > (times_n_O O).
+ apply lt_times;assumption
+ ]
+qed.
+
+theorem eq_div_div_div_div: \forall n,m,q. O < n \to O < m \to
+q/n/m = q/m/n.
+intros.
+apply (trans_eq ? ? (q/(n*m)))
+ [apply eq_div_div_div_times;assumption
+ |rewrite > sym_times.
+ apply sym_eq.
+ apply eq_div_div_div_times;assumption
+ ]
+qed.
+theorem SSO_mod: \forall n,m. O < m \to (S(S O))*n/m = (n/m)*(S(S O)) + mod ((S(S O))*n/m) (S(S O)).
+intros.
+rewrite < (lt_O_to_div_times n (S(S O))) in ⊢ (? ? ? (? (? (? % ?) ?) ?))
+ [rewrite > eq_div_div_div_div
+ [rewrite > sym_times in ⊢ (? ? ? (? (? (? (? % ?) ?) ?) ?)).
+ apply div_mod.
+ apply lt_O_S
+ |apply lt_O_S
+ |assumption
+ ]
+ |apply lt_O_S
+ ]
+qed.
(* Forall a,b : N. 0 < b \to b * (a/b) <= a < b * (a/b +1) *)
(* The theorem is shown in two different parts: *)
rewrite > H2.simplify.apply le_n.
qed.
+theorem max_f_g: \forall f,g,n. (\forall i. i \le n \to f i = g i) \to
+max n f = max n g.
+intros 3.
+elim n
+ [simplify.
+ rewrite > (H O)
+ [reflexivity
+ |apply le_n
+ ]
+ |simplify.
+ rewrite > H
+ [rewrite > H1
+ [reflexivity
+ |apply le_n
+ ]
+ |intros.
+ apply H1.
+ apply le_S.
+ assumption
+ ]
+ ]
+qed.
+
+theorem le_max_f_max_g: \forall f,g,n. (\forall i. i \le n \to f i = true \to g i =true) \to
+max n f \le max n g.
+intros 3.
+elim n
+ [simplify.
+ elim (f O);apply le_O_n
+ |simplify.
+ apply (bool_elim ? (f (S n1)));intro
+ [rewrite > (H1 (S n1) ? H2)
+ [apply le_n
+ |apply le_n
+ ]
+ |cases (g(S n1))
+ [simplify.
+ apply le_S.
+ apply le_max_n
+ |simplify.
+ apply H.
+ intros.
+ apply H1
+ [apply le_S.assumption
+ |assumption
+ ]
+ ]
+ ]
+ ]
+qed.
+
+
+theorem max_O : \forall f:nat \to bool. \forall n:nat.
+(\forall i:nat. le i n \to f i = false) \to max n f = O.
+intros 2.elim n
+ [simplify.rewrite > H
+ [reflexivity
+ |apply le_O_n
+ ]
+ |simplify.rewrite > H1
+ [simplify.apply H.
+ intros.
+ apply H1.
+ apply le_S.
+ assumption
+ |apply le_n
+ ]
+ ]
+qed.
+
theorem f_max_true : \forall f:nat \to bool. \forall n:nat.
(\exists i:nat. le i n \land f i = true) \to f (max n f) = true.
intros 2.
reflexivity.
qed.
+theorem exists_forall_le:\forall f,n.
+(\exists i. i \le n \land f i = true) \lor
+(\forall i. i \le n \to f i = false).
+intros.
+elim n
+ [apply (bool_elim ? (f O));intro
+ [left.apply (ex_intro ? ? O).
+ split[apply le_n|assumption]
+ |right.intros.
+ apply (le_n_O_elim ? H1).
+ assumption
+ ]
+ |elim H
+ [elim H1.elim H2.
+ left.apply (ex_intro ? ? a).
+ split[apply le_S.assumption|assumption]
+ |apply (bool_elim ? (f (S n1)));intro
+ [left.apply (ex_intro ? ? (S n1)).
+ split[apply le_n|assumption]
+ |right.intros.
+ elim (le_to_or_lt_eq ? ? H3)
+ [apply H1.
+ apply le_S_S_to_le.
+ apply H4
+ |rewrite > H4.
+ assumption
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem exists_max_forall_false:\forall f,n.
+((\exists i. i \le n \land f i = true) \land (f (max n f) = true))\lor
+((\forall i. i \le n \to f i = false) \land (max n f) = O).
+intros.
+elim (exists_forall_le f n)
+ [left.split
+ [assumption
+ |apply f_max_true.assumption
+ ]
+ |right.split
+ [assumption
+ |apply max_O.assumption
+ ]
+ ]
+qed.
+
+theorem false_to_lt_max: \forall f,n,m.O < n \to
+f n = false \to max m f \le n \to max m f < n.
+intros.
+elim (le_to_or_lt_eq ? ? H2)
+ [assumption
+ |elim (exists_max_forall_false f m)
+ [elim H4.
+ apply False_ind.
+ apply not_eq_true_false.
+ rewrite < H6.
+ rewrite > H3.
+ assumption
+ |elim H4.
+ rewrite > H6.
+ assumption
+ ]
+ ]
+qed.
+
theorem lt_max_to_false : \forall f:nat \to bool.
\forall n,m:nat. (max n f) < m \to m \leq n \to f m = false.
intros 2.
intro.rewrite > H7.assumption.
qed.
+theorem f_false_to_le_max: \forall f,n,p. (∃i:nat.i≤n∧f i=true) \to
+(\forall m. p < m \to f m = false)
+\to max n f \le p.
+intros.
+apply not_lt_to_le.intro.
+apply not_eq_true_false.
+rewrite < (H1 ? H2).
+apply sym_eq.
+apply f_max_true.
+assumption.
+qed.
+
definition max_spec \def \lambda f:nat \to bool.\lambda n,m: nat.
m \le n \land (f m)=true \land (\forall i. m < i \to i \le n \to (f i = false)).
include "datatypes/constructors.ma".
include "nat/exp.ma".
include "nat/nth_prime.ma".
+include "nat/gcd.ma". (* for some properties of divides *)
include "nat/relevant_equations.ma". (* required by autobatch paramod *)
let rec p_ord_aux p n m \def
]
qed.
-(* p_ord_inv is the inverse of ord *)
+(* properties of ord e ord_rem *)
+
+theorem ord_times: \forall p,m,n.O<m \to O<n \to prime p \to
+ord (m*n) p = ord m p+ord n p.
+intros.unfold ord.
+rewrite > (p_ord_times ? ? ? (ord m p) (ord_rem m p) (ord n p) (ord_rem n p))
+ [reflexivity
+ |assumption
+ |assumption
+ |assumption
+ |unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+ |unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+ ]
+qed.
+
+theorem ord_exp: \forall p,m.S O < p \to
+ord (exp p m) p = m.
+intros.
+unfold ord.
+rewrite > (p_ord_exp1 p (exp p m) m (S O))
+ [reflexivity
+ |apply lt_to_le.assumption
+ |intro.apply (lt_to_not_le ? ? H).
+ apply divides_to_le
+ [apply lt_O_S
+ |assumption
+ ]
+ |apply times_n_SO
+ ]
+qed.
+
+theorem not_divides_to_ord_O:
+\forall p,m. prime p \to \lnot (divides p m) \to
+ord m p = O.
+intros.unfold ord.
+rewrite > (p_ord_exp1 p m O m)
+ [reflexivity
+ |apply prime_to_lt_O.assumption
+ |assumption
+ |simplify.apply plus_n_O
+ ]
+qed.
+
+theorem ord_O_to_not_divides:
+\forall p,m. O < m \to prime p \to ord m p = O \to
+\lnot (divides p m).
+intros.
+lapply (p_ord_to_exp1 p m (ord m p) (ord_rem m p))
+ [elim Hletin.
+ rewrite > H4.
+ rewrite > H2.
+ rewrite > sym_times.
+ rewrite < times_n_SO.
+ assumption
+ |apply prime_to_lt_SO.assumption
+ |assumption
+ |unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+ ]
+qed.
+
+theorem divides_to_not_ord_O:
+\forall p,m. O < m \to prime p \to divides p m \to
+\lnot(ord m p = O).
+intros.intro.
+apply (ord_O_to_not_divides ? ? H H1 H3).
+assumption.
+qed.
+
+theorem not_ord_O_to_divides:
+\forall p,m. O < m \to prime p \to \lnot (ord m p = O) \to
+divides p m.
+intros.
+rewrite > (exp_ord p) in ⊢ (? ? %)
+ [apply (trans_divides ? (exp p (ord m p)))
+ [generalize in match H2.
+ cases (ord m p);intro
+ [apply False_ind.apply H3.reflexivity
+ |simplify.autobatch
+ ]
+ |autobatch
+ ]
+ |apply prime_to_lt_SO.
+ assumption
+ |assumption
+ ]
+qed.
+
+theorem not_divides_ord_rem: \forall m,p.O< m \to S O < p \to
+\lnot (divides p (ord_rem m p)).
+intros.
+elim (p_ord_to_exp1 p m (ord m p) (ord_rem m p))
+ [assumption
+ |assumption
+ |assumption
+ |unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+ ]
+qed.
+
+theorem ord_ord_rem: \forall p,q,m. O < m \to
+prime p \to prime q \to
+q < p \to ord (ord_rem m p) q = ord m q.
+intros.
+rewrite > (exp_ord p) in ⊢ (? ? ? (? % ?))
+ [rewrite > ord_times
+ [rewrite > not_divides_to_ord_O in ⊢ (? ? ? (? % ?))
+ [reflexivity
+ |assumption
+ |intro.
+ apply (lt_to_not_eq ? ? H3).
+ apply (divides_exp_to_eq ? ? ? H2 H1 H4)
+ ]
+ |apply lt_O_exp.
+ apply (ltn_to_ltO ? ? H3)
+ |apply lt_O_ord_rem
+ [elim H1.assumption
+ |assumption
+
+ ]
+ |assumption
+ ]
+ |elim H1.assumption
+ |assumption
+ ]
+qed.
+
+theorem lt_ord_rem: \forall n,m. prime n \to O < m \to
+divides n m \to ord_rem m n < m.
+intros.
+elim (le_to_or_lt_eq (ord_rem m n) m)
+ [assumption
+ |apply False_ind.
+ apply (ord_O_to_not_divides ? ? H1 H ? H2).
+ apply (inj_exp_r n)
+ [apply prime_to_lt_SO.assumption
+ |apply (inj_times_l1 m)
+ [assumption
+ |rewrite > sym_times in ⊢ (? ? ? %).
+ rewrite < times_n_SO.
+ rewrite < H3 in ⊢ (? ? (? ? %) ?).
+ apply sym_eq.
+ apply exp_ord
+ [apply prime_to_lt_SO.assumption
+ |assumption
+ ]
+ ]
+ ]
+ |apply divides_to_le
+ [assumption
+ |apply divides_ord_rem
+ [apply prime_to_lt_SO.assumption
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+(* p_ord_inv is used to encode the pair ord and rem into
+ a single natural number. *)
+
definition p_ord_inv \def
\lambda p,m,x.
match p_ord x p with
apply le_O_n.assumption.
qed.
+theorem lt_SO_n_to_lt_O_pred_n: \forall n:nat.
+(S O) \lt n \to O \lt (pred n).
+intros.
+apply (ltn_to_ltO (pred (S O)) (pred n) ?).
+ apply (lt_pred (S O) n);
+ [ apply (lt_O_S O)
+ | assumption
+ ]
+qed.
+
theorem lt_O_n_elim: \forall n:nat. lt O n \to
\forall P:nat\to Prop. (\forall m:nat.P (S m)) \to P n.
intro.elim n.apply False_ind.exact (not_le_Sn_O O H).
--- /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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/nat/pi_p".
+
+include "nat/primes.ma".
+(* include "nat/ord.ma". *)
+include "nat/generic_iter_p.ma".
+(* include "nat/count.ma". necessary just to use bool_to_nat and bool_to_nat_andb*)
+include "nat/iteration2.ma".
+
+(* pi_p on nautral numbers is a specialization of iter_p_gen *)
+definition pi_p: nat \to (nat \to bool) \to (nat \to nat) \to nat \def
+\lambda n, p, g. (iter_p_gen n p nat g (S O) times).
+
+theorem true_to_pi_p_Sn:
+\forall n:nat. \forall p:nat \to bool. \forall g:nat \to nat.
+p n = true \to pi_p (S n) p g = (g n)*(pi_p n p g).
+intros.
+unfold pi_p.
+apply true_to_iter_p_gen_Sn.
+assumption.
+qed.
+
+theorem false_to_pi_p_Sn:
+\forall n:nat. \forall p:nat \to bool. \forall g:nat \to nat.
+p n = false \to pi_p (S n) p g = pi_p n p g.
+intros.
+unfold pi_p.
+apply false_to_iter_p_gen_Sn.
+assumption.
+qed.
+
+theorem eq_pi_p: \forall p1,p2:nat \to bool.
+\forall g1,g2: nat \to nat.\forall n.
+(\forall x. x < n \to p1 x = p2 x) \to
+(\forall x. x < n \to g1 x = g2 x) \to
+pi_p n p1 g1 = pi_p n p2 g2.
+intros.
+unfold pi_p.
+apply eq_iter_p_gen;
+assumption.
+qed.
+
+theorem eq_pi_p1: \forall p1,p2:nat \to bool.
+\forall g1,g2: nat \to nat.\forall n.
+(\forall x. x < n \to p1 x = p2 x) \to
+(\forall x. x < n \to p1 x = true \to g1 x = g2 x) \to
+pi_p n p1 g1 = pi_p n p2 g2.
+intros.
+unfold pi_p.
+apply eq_iter_p_gen1;
+assumption.
+qed.
+
+theorem pi_p_false:
+\forall g: nat \to nat.\forall n.pi_p n (\lambda x.false) g = S O.
+intros.
+unfold pi_p.
+apply iter_p_gen_false.
+qed.
+
+theorem pi_p_times: \forall n,k:nat.\forall p:nat \to bool.
+\forall g: nat \to nat.
+pi_p (k+n) p g
+= pi_p k (\lambda x.p (x+n)) (\lambda x.g (x+n)) * pi_p n p g.
+intros.
+unfold pi_p.
+apply (iter_p_gen_plusA nat n k p g (S O) times)
+[ apply sym_times.
+| intros.
+ apply sym_eq.
+ apply times_n_SO
+| apply associative_times
+]
+qed.
+
+theorem false_to_eq_pi_p: \forall n,m:nat.n \le m \to
+\forall p:nat \to bool.
+\forall g: nat \to nat. (\forall i:nat. n \le i \to i < m \to
+p i = false) \to pi_p m p g = pi_p n p g.
+intros.
+unfold pi_p.
+apply (false_to_eq_iter_p_gen);
+assumption.
+qed.
+
+theorem or_false_eq_SO_to_eq_pi_p:
+\forall n,m:nat.\forall p:nat \to bool.
+\forall g: nat \to nat.
+n \le m \to (\forall i:nat. n \le i \to i < m \to p i = false \lor g i = S O)
+\to pi_p m p g = pi_p n p g.
+intros.
+unfold pi_p.
+apply or_false_eq_baseA_to_eq_iter_p_gen
+ [intros.simplify.rewrite < plus_n_O.reflexivity
+ |assumption
+ |assumption
+ ]
+qed.
+
+theorem pi_p2 :
+\forall n,m:nat.
+\forall p1,p2:nat \to bool.
+\forall g: nat \to nat \to nat.
+pi_p (n*m)
+ (\lambda x.andb (p1 (div x m)) (p2 (mod x m)))
+ (\lambda x.g (div x m) (mod x m)) =
+pi_p n p1
+ (\lambda x.pi_p m p2 (g x)).
+intros.
+unfold pi_p.
+apply (iter_p_gen2 n m p1 p2 nat g (S O) times)
+[ apply sym_times
+| apply associative_times
+| intros.
+ apply sym_eq.
+ apply times_n_SO
+]
+qed.
+
+theorem pi_p2' :
+\forall n,m:nat.
+\forall p1:nat \to bool.
+\forall p2:nat \to nat \to bool.
+\forall g: nat \to nat \to nat.
+pi_p (n*m)
+ (\lambda x.andb (p1 (div x m)) (p2 (div x m) (mod x m)))
+ (\lambda x.g (div x m) (mod x m)) =
+pi_p n p1
+ (\lambda x.pi_p m (p2 x) (g x)).
+intros.
+unfold pi_p.
+apply (iter_p_gen2' n m p1 p2 nat g (S O) times)
+[ apply sym_times
+| apply associative_times
+| intros.
+ apply sym_eq.
+ apply times_n_SO
+]
+qed.
+
+lemma pi_p_gi: \forall g: nat \to nat.
+\forall n,i.\forall p:nat \to bool.i < n \to p i = true \to
+pi_p n p g = g i * pi_p n (\lambda x. andb (p x) (notb (eqb x i))) g.
+intros.
+unfold pi_p.
+apply (iter_p_gen_gi)
+[ apply sym_times
+| apply associative_times
+| intros.
+ apply sym_eq.
+ apply times_n_SO
+| assumption
+| assumption
+]
+qed.
+
+theorem eq_pi_p_gh:
+\forall g,h,h1: nat \to nat.\forall n,n1.
+\forall p1,p2:nat \to bool.
+(\forall i. i < n \to p1 i = true \to p2 (h i) = true) \to
+(\forall i. i < n \to p1 i = true \to h1 (h i) = i) \to
+(\forall i. i < n \to p1 i = true \to h i < n1) \to
+(\forall j. j < n1 \to p2 j = true \to p1 (h1 j) = true) \to
+(\forall j. j < n1 \to p2 j = true \to h (h1 j) = j) \to
+(\forall j. j < n1 \to p2 j = true \to h1 j < n) \to
+pi_p n p1 (\lambda x.g(h x)) = pi_p n1 p2 g.
+intros.
+unfold pi_p.
+apply (eq_iter_p_gen_gh nat (S O) times ? ? ? g h h1 n n1 p1 p2)
+[ apply sym_times
+| apply associative_times
+| intros.
+ apply sym_eq.
+ apply times_n_SO
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+| assumption
+]
+qed.
+
+(* monotonicity *)
+theorem le_pi_p:
+\forall n:nat. \forall p:nat \to bool. \forall g1,g2:nat \to nat.
+(\forall i. i < n \to p i = true \to g1 i \le g2 i ) \to
+pi_p n p g1 \le pi_p n p g2.
+intros.
+generalize in match H.
+elim n
+ [apply le_n.
+ |apply (bool_elim ? (p n1));intros
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_pi_p_Sn in ⊢ (? ? %)
+ [apply le_times
+ [apply H2[apply le_n|assumption]
+ |apply H1.
+ intros.
+ apply H2[apply le_S.assumption|assumption]
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_pi_p_Sn in ⊢ (? ? %)
+ [apply H1.
+ intros.
+ apply H2[apply le_S.assumption|assumption]
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem exp_sigma_p: \forall n,a,p.
+pi_p n p (\lambda x.a) = (exp a (sigma_p n p (\lambda x.S O))).
+intros.
+elim n
+ [reflexivity
+ |apply (bool_elim ? (p n1))
+ [intro.
+ rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_sigma_p_Sn
+ [simplify.
+ rewrite > H.
+ reflexivity.
+ |assumption
+ ]
+ |assumption
+ ]
+ |intro.
+ rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_sigma_p_Sn
+ [simplify.assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem times_pi_p: \forall n,p,f,g.
+pi_p n p (\lambda x.f x*g x) = pi_p n p f * pi_p n p g.
+intros.
+elim n
+ [simplify.reflexivity
+ |apply (bool_elim ? (p n1))
+ [intro.
+ rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > H.autobatch
+ |assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+ |intro.
+ rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_pi_p_Sn;assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem pi_p_SO: \forall n,p.
+pi_p n p (\lambda i.S O) = S O.
+intros.elim n
+ [reflexivity
+ |simplify.elim (p n1)
+ [simplify.rewrite < plus_n_O.assumption
+ |simplify.assumption
+ ]
+ ]
+qed.
+
+theorem exp_pi_p: \forall n,m,p,f.
+pi_p n p (\lambda x.exp (f x) m) = exp (pi_p n p f) m.
+intros.
+elim m
+ [simplify.apply pi_p_SO
+ |simplify.
+ rewrite > times_pi_p.
+ rewrite < H.
+ reflexivity
+ ]
+qed.
+
+theorem exp_times_pi_p: \forall n,m,k,p,f.
+pi_p n p (\lambda x.exp k (m*(f x))) =
+exp (pi_p n p (\lambda x.exp k (f x))) m.
+intros.
+apply (trans_eq ? ? (pi_p n p (\lambda x.(exp (exp k (f x)) m))))
+ [apply eq_pi_p;intros
+ [reflexivity
+ |apply sym_eq.rewrite > sym_times.
+ apply exp_exp_times
+ ]
+ |apply exp_pi_p
+ ]
+qed.
+
+
+theorem pi_p_knm:
+\forall g: nat \to nat.
+\forall h2:nat \to nat \to nat.
+\forall h11,h12:nat \to nat.
+\forall k,n,m.
+\forall p1,p21:nat \to bool.
+\forall p22:nat \to nat \to bool.
+(\forall x. x < k \to p1 x = true \to
+p21 (h11 x) = true \land p22 (h11 x) (h12 x) = true
+\land h2 (h11 x) (h12 x) = x
+\land (h11 x) < n \land (h12 x) < m) \to
+(\forall i,j. i < n \to j < m \to p21 i = true \to p22 i j = true \to
+p1 (h2 i j) = true \land
+h11 (h2 i j) = i \land h12 (h2 i j) = j
+\land h2 i j < k) \to
+pi_p k p1 g =
+pi_p n p21 (\lambda x:nat.pi_p m (p22 x) (\lambda y. g (h2 x y))).
+intros.
+unfold pi_p.unfold pi_p.
+apply (iter_p_gen_knm nat (S O) times sym_times assoc_times ? ? ? h11 h12)
+ [intros.apply sym_eq.apply times_n_SO.
+ |assumption
+ |assumption
+ ]
+qed.
+
+theorem pi_p_pi_p:
+\forall g: nat \to nat \to nat.
+\forall h11,h12,h21,h22: nat \to nat \to nat.
+\forall n1,m1,n2,m2.
+\forall p11,p21:nat \to bool.
+\forall p12,p22:nat \to nat \to bool.
+(\forall i,j. i < n2 \to j < m2 \to p21 i = true \to p22 i j = true \to
+p11 (h11 i j) = true \land p12 (h11 i j) (h12 i j) = true
+\land h21 (h11 i j) (h12 i j) = i \land h22 (h11 i j) (h12 i j) = j
+\land h11 i j < n1 \land h12 i j < m1) \to
+(\forall i,j. i < n1 \to j < m1 \to p11 i = true \to p12 i j = true \to
+p21 (h21 i j) = true \land p22 (h21 i j) (h22 i j) = true
+\land h11 (h21 i j) (h22 i j) = i \land h12 (h21 i j) (h22 i j) = j
+\land (h21 i j) < n2 \land (h22 i j) < m2) \to
+pi_p n1 p11
+ (\lambda x:nat .pi_p m1 (p12 x) (\lambda y. g x y)) =
+pi_p n2 p21
+ (\lambda x:nat .pi_p m2 (p22 x) (\lambda y. g (h11 x y) (h12 x y))).
+intros.
+unfold pi_p.unfold pi_p.
+apply (iter_p_gen_2_eq ? ? ? sym_times assoc_times ? ? ? ? h21 h22)
+ [intros.apply sym_eq.apply times_n_SO.
+ |assumption
+ |assumption
+ ]
+qed.
+
+theorem pi_p_pi_p1:
+\forall g: nat \to nat \to nat.
+\forall n,m.
+\forall p11,p21:nat \to bool.
+\forall p12,p22:nat \to nat \to bool.
+(\forall x,y. x < n \to y < m \to
+ (p11 x \land p12 x y) = (p21 y \land p22 y x)) \to
+pi_p n p11 (\lambda x:nat.pi_p m (p12 x) (\lambda y. g x y)) =
+pi_p m p21 (\lambda y:nat.pi_p n (p22 y) (\lambda x. g x y)).
+intros.
+unfold pi_p.unfold pi_p.
+apply (iter_p_gen_iter_p_gen ? ? ? sym_times assoc_times)
+ [intros.apply sym_eq.apply times_n_SO.
+ |assumption
+ ]
+qed.
\ No newline at end of file
simplify.exact (not_le_Sn_n O).
qed.
-
-(*divides and div*)
+(*a variant of or_div_mod *)
+theorem or_div_mod1: \forall n,q. O < q \to
+(divides q (S n)) \land S n = (S (div n q)) * q \lor
+(\lnot (divides q (S n)) \land S n= (div n q) * q + S (n\mod q)).
+intros.elim (or_div_mod n q H);elim H1
+ [left.split
+ [apply (witness ? ? (S (n/q))).
+ rewrite > sym_times.assumption
+ |assumption
+ ]
+ |right.split
+ [intro.
+ apply (not_eq_O_S (n \mod q)).
+ (* come faccio a fare unfold nelleipotesi ? *)
+ cut ((S n) \mod q = O)
+ [rewrite < Hcut.
+ apply (div_mod_spec_to_eq2 (S n) q (div (S n) q) (mod (S n) q) (div n q) (S (mod n q)))
+ [apply div_mod_spec_div_mod.
+ assumption
+ |apply div_mod_spec_intro;assumption
+ ]
+ |apply divides_to_mod_O;assumption
+ ]
+ |assumption
+ ]
+ ]
+qed.
theorem divides_to_div: \forall n,m.divides n m \to m/n*n = m.
intro.
intros.elim H.apply lt_to_le.assumption.
qed.
+theorem prime_to_lt_SO: \forall p. prime p \to S O < p.
+intros.elim H.
+assumption.
+qed.
+
(* smallest factor *)
definition smallest_factor : nat \to nat \def
\lambda n:nat.
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