rewrite < plus_n_O.rewrite < sym_times.reflexivity.
qed.
+lemma div_plus_times: \forall m,q,r:nat. r < m \to (q*m+r)/ m = q.
+intros.
+apply (div_mod_spec_to_eq (q*m+r) m ? ((q*m+r) \mod m) ? r)
+ [apply div_mod_spec_div_mod.
+ apply (le_to_lt_to_lt ? r)
+ [apply le_O_n|assumption]
+ |apply div_mod_spec_intro[assumption|reflexivity]
+ ]
+qed.
+
+lemma mod_plus_times: \forall m,q,r:nat. r < m \to (q*m+r) \mod m = r.
+intros.
+apply (div_mod_spec_to_eq2 (q*m+r) m ((q*m+r)/ m) ((q*m+r) \mod m) q r)
+ [apply div_mod_spec_div_mod.
+ apply (le_to_lt_to_lt ? r)
+ [apply le_O_n|assumption]
+ |apply div_mod_spec_intro[assumption|reflexivity]
+ ]
+qed.
(* some properties of div and mod *)
theorem div_times: \forall n,m:nat. ((S n)*m) / (S n) = m.
intros.
include "nat/map_iter_p.ma".
include "nat/totient.ma".
+lemma gcd_n_n: \forall n.gcd n n = n.
+intro.elim n
+ [reflexivity
+ |apply le_to_le_to_eq
+ [apply divides_to_le
+ [apply lt_O_S
+ |apply divides_gcd_n
+ ]
+ |apply divides_to_le
+ [apply lt_O_gcd.apply lt_O_S
+ |apply divides_d_gcd
+ [apply divides_n_n|apply divides_n_n]
+ ]
+ ]
+ ]
+qed.
+
+
+lemma count_card: \forall p.\forall n.
+p O = false \to count (S n) p = card n p.
+intros.elim n
+ [simplify.rewrite > H. reflexivity
+ |simplify.
+ rewrite < plus_n_O.
+ apply eq_f.assumption
+ ]
+qed.
+
+lemma count_card1: \forall p.\forall n.
+p O = false \to p n = false \to count n p = card n p.
+intros 3.apply (nat_case n)
+ [intro.simplify.rewrite > H. reflexivity
+ |intros.rewrite > (count_card ? ? H).
+ simplify.rewrite > H1.reflexivity
+ ]
+qed.
+
+
(* a reformulation of totient using card insted of count *)
+
lemma totient_card: \forall n.
totient n = card n (\lambda i.eqb (gcd i n) (S O)).
intro.apply (nat_case n)
set "baseuri" "cic:/matita/nat/exp".
include "nat/div_and_mod.ma".
+include "nat/lt_arith.ma".
let rec exp n m on m\def
match m with
variant inj_exp_r: \forall p:nat. (S O) < p \to \forall n,m:nat.
p \sup n = p \sup m \to n = m \def
injective_exp_r.
+
+theorem le_exp: \forall n,m,p:nat. O < p \to n \le m \to exp p n \le exp p m.
+apply nat_elim2
+ [intros.
+ apply lt_O_exp.assumption
+ |intros.
+ apply False_ind.
+ apply (le_to_not_lt ? ? ? H1).
+ apply le_O_n
+ |intros.
+ simplify.
+ apply le_times
+ [apply le_n
+ |apply H[assumption|apply le_S_S_to_le.assumption]
+ ]
+ ]
+qed.
+
+theorem lt_exp: \forall n,m,p:nat. S O < p \to n < m \to exp p n < exp p m.
+apply nat_elim2
+ [intros.
+ apply (lt_O_n_elim ? H1).intro.
+ simplify.unfold lt.
+ rewrite > times_n_SO.
+ apply le_times
+ [assumption
+ |apply lt_O_exp.
+ apply (trans_lt ? (S O))[apply le_n|assumption]
+ ]
+ |intros.
+ apply False_ind.
+ apply (le_to_not_lt ? ? ? H1).
+ apply le_O_n
+ |intros.simplify.
+ apply lt_times_r1
+ [apply (trans_lt ? (S O))[apply le_n|assumption]
+ |apply H
+ [apply H1
+ |apply le_S_S_to_le.assumption
+ ]
+ ]
+ ]
+qed.
+
+
+
+
+
\ No newline at end of file
theorem divides_max_prime_factor_n:
\forall n:nat. (S O) < n
\to nth_prime (max_prime_factor n) \divides n.
-intros; apply divides_b_true_to_divides;
-[ apply lt_O_nth_prime_n;
-| apply (f_max_true (\lambda p:nat.eqb (n \mod (nth_prime p)) O) n);
- cut (\exists i. nth_prime i = smallest_factor n);
+intros.
+apply divides_b_true_to_divides.
+apply (f_max_true (\lambda p:nat.eqb (n \mod (nth_prime p)) O) n);
+cut (\exists i. nth_prime i = smallest_factor n);
[ elim Hcut.
apply (ex_intro nat ? a);
split;
(*
apply prime_to_nth_prime;
apply prime_smallest_factor_n;
- assumption; *) ] ]
+ assumption; *) ]
qed.
theorem divides_to_max_prime_factor : \forall n,m. (S O) < n \to O < m \to n \divides m \to
*)
qed.
+theorem divides_to_max_prime_factor1 : \forall n,m. O < n \to O < m \to n \divides m \to
+max_prime_factor n \le max_prime_factor m.
+intros 3.
+elim (le_to_or_lt_eq ? ? H)
+ [apply divides_to_max_prime_factor
+ [assumption|assumption|assumption]
+ |rewrite < H1.
+ simplify.apply le_O_n.
+ ]
+qed.
+
theorem p_ord_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
p = max_prime_factor n \to
(pair nat nat q r) = p_ord n (nth_prime p) \to
--- /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/iteration2.ma".
+
+include "nat/primes.ma".
+include "nat/ord.ma".
+
+let rec sigma_p n p (g:nat \to nat) \def
+ match n with
+ [ O \Rightarrow O
+ | (S k) \Rightarrow
+ match p k with
+ [true \Rightarrow (g k)+(sigma_p k p g)
+ |false \Rightarrow sigma_p k p g]
+ ].
+
+theorem true_to_sigma_p_Sn:
+\forall n:nat. \forall p:nat \to bool. \forall g:nat \to nat.
+p n = true \to sigma_p (S n) p g =
+(g n)+(sigma_p n p g).
+intros.simplify.
+rewrite > H.reflexivity.
+qed.
+
+theorem false_to_sigma_p_Sn:
+\forall n:nat. \forall p:nat \to bool. \forall g:nat \to nat.
+p n = false \to sigma_p (S n) p g = sigma_p n p g.
+intros.simplify.
+rewrite > H.reflexivity.
+qed.
+
+theorem eq_sigma_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
+sigma_p n p1 g1 = sigma_p n p2 g2.
+intros 5.elim n
+ [reflexivity
+ |apply (bool_elim ? (p1 n1))
+ [intro.
+ rewrite > (true_to_sigma_p_Sn ? ? ? H3).
+ rewrite > true_to_sigma_p_Sn
+ [apply eq_f2
+ [apply H2.apply le_n.
+ |apply H
+ [intros.apply H1.apply le_S.assumption
+ |intros.apply H2.apply le_S.assumption
+ ]
+ ]
+ |rewrite < H3.apply sym_eq.apply H1.apply le_n
+ ]
+ |intro.
+ rewrite > (false_to_sigma_p_Sn ? ? ? H3).
+ rewrite > false_to_sigma_p_Sn
+ [apply H
+ [intros.apply H1.apply le_S.assumption
+ |intros.apply H2.apply le_S.assumption
+ ]
+ |rewrite < H3.apply sym_eq.apply H1.apply le_n
+ ]
+ ]
+ ]
+qed.
+
+theorem sigma_p_false:
+\forall g: nat \to nat.\forall n.sigma_p n (\lambda x.false) g = O.
+intros.
+elim n[reflexivity|simplify.assumption]
+qed.
+
+theorem sigma_p_plus: \forall n,k:nat.\forall p:nat \to bool.
+\forall g: nat \to nat.
+sigma_p (k+n) p g
+= sigma_p k (\lambda x.p (x+n)) (\lambda x.g (x+n)) + sigma_p n p g.
+intros.
+elim k
+ [reflexivity
+ |apply (bool_elim ? (p (n1+n)))
+ [intro.
+ simplify in \vdash (? ? (? % ? ?) ?).
+ rewrite > (true_to_sigma_p_Sn ? ? ? H1).
+ rewrite > (true_to_sigma_p_Sn n1 (\lambda x.p (x+n)) ? H1).
+ rewrite > assoc_plus.
+ rewrite < H.reflexivity
+ |intro.
+ simplify in \vdash (? ? (? % ? ?) ?).
+ rewrite > (false_to_sigma_p_Sn ? ? ? H1).
+ rewrite > (false_to_sigma_p_Sn n1 (\lambda x.p (x+n)) ? H1).
+ assumption.
+ ]
+ ]
+qed.
+
+theorem false_to_eq_sigma_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 sigma_p m p g = sigma_p n p g.
+intros 5.
+elim H
+ [reflexivity
+ |simplify.
+ rewrite > H3
+ [simplify.
+ apply H2.
+ intros.
+ apply H3[apply H4|apply le_S.assumption]
+ |assumption
+ |apply le_n
+ ]
+ ]
+qed.
+
+theorem sigma_p2 :
+\forall n,m:nat.
+\forall p1,p2:nat \to bool.
+\forall g: nat \to nat \to nat.
+sigma_p (n*m)
+ (\lambda x.andb (p1 (div x m)) (p2 (mod x m)))
+ (\lambda x.g (div x m) (mod x m)) =
+sigma_p n p1
+ (\lambda x.sigma_p m p2 (g x)).
+intros.
+elim n
+ [simplify.reflexivity
+ |apply (bool_elim ? (p1 n1))
+ [intro.
+ rewrite > (true_to_sigma_p_Sn ? ? ? H1).
+ simplify in \vdash (? ? (? % ? ?) ?);
+ rewrite > sigma_p_plus.
+ rewrite < H.
+ apply eq_f2
+ [apply eq_sigma_p
+ [intros.
+ rewrite > sym_plus.
+ rewrite > (div_plus_times ? ? ? H2).
+ rewrite > (mod_plus_times ? ? ? H2).
+ rewrite > H1.
+ simplify.reflexivity
+ |intros.
+ rewrite > sym_plus.
+ rewrite > (div_plus_times ? ? ? H2).
+ rewrite > (mod_plus_times ? ? ? H2).
+ rewrite > H1.
+ simplify.reflexivity.
+ ]
+ |reflexivity
+ ]
+ |intro.
+ rewrite > (false_to_sigma_p_Sn ? ? ? H1).
+ simplify in \vdash (? ? (? % ? ?) ?);
+ rewrite > sigma_p_plus.
+ rewrite > H.
+ apply (trans_eq ? ? (O+(sigma_p n1 p1 (\lambda x:nat.sigma_p m p2 (g x)))))
+ [apply eq_f2
+ [rewrite > (eq_sigma_p ? (\lambda x.false) ? (\lambda x:nat.g ((x+n1*m)/m) ((x+n1*m)\mod m)))
+ [apply sigma_p_false
+ |intros.
+ rewrite > sym_plus.
+ rewrite > (div_plus_times ? ? ? H2).
+ rewrite > (mod_plus_times ? ? ? H2).
+ rewrite > H1.
+ simplify.reflexivity
+ |intros.reflexivity.
+ ]
+ |reflexivity
+ ]
+ |reflexivity
+ ]
+ ]
+ ]
+qed.
+
+lemma sigma_p_gi: \forall g: nat \to nat.
+\forall n,i.\forall p:nat \to bool.i < n \to p i = true \to
+sigma_p n p g = g i + sigma_p n (\lambda x. andb (p x) (notb (eqb x i))) g.
+intros 2.
+elim n
+ [apply False_ind.
+ apply (not_le_Sn_O i).
+ assumption
+ |apply (bool_elim ? (p n1));intro
+ [elim (le_to_or_lt_eq i n1)
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite < assoc_plus.
+ rewrite < sym_plus in \vdash (? ? ? (? % ?)).
+ rewrite > assoc_plus.
+ apply eq_f2
+ [reflexivity
+ |apply H[assumption|assumption]
+ ]
+ |rewrite > H3.simplify.
+ change with (notb (eqb n1 i) = notb false).
+ apply eq_f.
+ apply not_eq_to_eqb_false.
+ unfold Not.intro.
+ apply (lt_to_not_eq ? ? H4).
+ apply sym_eq.assumption
+ ]
+ |assumption
+ ]
+ |rewrite > true_to_sigma_p_Sn
+ [rewrite > H4.
+ apply eq_f2
+ [reflexivity
+ |rewrite > false_to_sigma_p_Sn
+ [apply eq_sigma_p
+ [intros.
+ elim (p x)
+ [simplify.
+ change with (notb false = notb (eqb x n1)).
+ apply eq_f.
+ apply sym_eq.
+ apply not_eq_to_eqb_false.
+ apply (lt_to_not_eq ? ? H5)
+ |reflexivity
+ ]
+ |intros.reflexivity
+ ]
+ |rewrite > H3.
+ rewrite > (eq_to_eqb_true ? ? (refl_eq ? n1)).
+ reflexivity
+ ]
+ ]
+ |assumption
+ ]
+ |apply le_S_S_to_le.assumption
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [elim (le_to_or_lt_eq i n1)
+ [rewrite > false_to_sigma_p_Sn
+ [apply H[assumption|assumption]
+ |rewrite > H3.reflexivity
+ ]
+ |apply False_ind.
+ apply not_eq_true_false.
+ rewrite < H2.
+ rewrite > H4.
+ assumption
+ |apply le_S_S_to_le.assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+theorem eq_sigma_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
+sigma_p n p1 (\lambda x.g(h x)) = sigma_p n1 (\lambda x.p2 x) g.
+intros 4.
+elim n
+ [generalize in match H5.
+ elim n1
+ [reflexivity
+ |apply (bool_elim ? (p2 n2));intro
+ [apply False_ind.
+ apply (not_le_Sn_O (h1 n2)).
+ apply H7
+ [apply le_n|assumption]
+ |rewrite > false_to_sigma_p_Sn
+ [apply H6.
+ intros.
+ apply H7[apply le_S.apply H9|assumption]
+ |assumption
+ ]
+ ]
+ ]
+ |apply (bool_elim ? (p1 n1));intro
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > (sigma_p_gi g n2 (h n1))
+ [apply eq_f2
+ [reflexivity
+ |apply H
+ [intros.
+ rewrite > H1
+ [simplify.
+ change with ((\not eqb (h i) (h n1))= \not false).
+ apply eq_f.
+ apply not_eq_to_eqb_false.
+ unfold Not.intro.
+ apply (lt_to_not_eq ? ? H8).
+ rewrite < H2
+ [rewrite < (H2 n1)
+ [apply eq_f.assumption|apply le_n|assumption]
+ |apply le_S.assumption
+ |assumption
+ ]
+ |apply le_S.assumption
+ |assumption
+ ]
+ |intros.
+ apply H2[apply le_S.assumption|assumption]
+ |intros.
+ apply H3[apply le_S.assumption|assumption]
+ |intros.
+ apply H4
+ [assumption
+ |generalize in match H9.
+ elim (p2 j)
+ [reflexivity|assumption]
+ ]
+ |intros.
+ apply H5
+ [assumption
+ |generalize in match H9.
+ elim (p2 j)
+ [reflexivity|assumption]
+ ]
+ |intros.
+ elim (le_to_or_lt_eq (h1 j) n1)
+ [assumption
+ |generalize in match H9.
+ elim (p2 j)
+ [simplify in H11.
+ absurd (j = (h n1))
+ [rewrite < H10.
+ rewrite > H5
+ [reflexivity|assumption|auto]
+ |apply eqb_false_to_not_eq.
+ generalize in match H11.
+ elim (eqb j (h n1))
+ [apply sym_eq.assumption|reflexivity]
+ ]
+ |simplify in H11.
+ apply False_ind.
+ apply not_eq_true_false.
+ apply sym_eq.assumption
+ ]
+ |apply le_S_S_to_le.
+ apply H6
+ [assumption
+ |generalize in match H9.
+ elim (p2 j)
+ [reflexivity|assumption]
+ ]
+ ]
+ ]
+ ]
+ |apply H3[apply le_n|assumption]
+ |apply H1[apply le_n|assumption]
+ ]
+ |assumption
+ ]
+ |rewrite > false_to_sigma_p_Sn
+ [apply H
+ [intros.apply H1[apply le_S.assumption|assumption]
+ |intros.apply H2[apply le_S.assumption|assumption]
+ |intros.apply H3[apply le_S.assumption|assumption]
+ |intros.apply H4[assumption|assumption]
+ |intros.apply H5[assumption|assumption]
+ |intros.
+ elim (le_to_or_lt_eq (h1 j) n1)
+ [assumption
+ |absurd (j = (h n1))
+ [rewrite < H10.
+ rewrite > H5
+ [reflexivity|assumption|assumption]
+ |unfold Not.intro.
+ apply not_eq_true_false.
+ rewrite < H7.
+ rewrite < H10.
+ rewrite > H4
+ [reflexivity|assumption|assumption]
+ ]
+ |apply le_S_S_to_le.
+ apply H6[assumption|assumption]
+ ]
+ ]
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+definition p_ord_times \def
+\lambda p,m,x.
+ match p_ord x p with
+ [pair q r \Rightarrow r*m+q].
+
+theorem eq_p_ord_times: \forall p,m,x.
+p_ord_times p m x = (ord_rem x p)*m+(ord x p).
+intros.unfold p_ord_times. unfold ord_rem.
+unfold ord.
+elim (p_ord x p).
+reflexivity.
+qed.
+
+theorem div_p_ord_times:
+\forall p,m,x. ord x p < m \to p_ord_times p m x / m = ord_rem x p.
+intros.rewrite > eq_p_ord_times.
+apply div_plus_times.
+assumption.
+qed.
+
+theorem mod_p_ord_times:
+\forall p,m,x. ord x p < m \to p_ord_times p m x \mod m = ord x p.
+intros.rewrite > eq_p_ord_times.
+apply mod_plus_times.
+assumption.
+qed.
+
+theorem sigma_p_divides:
+\forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to
+\forall g: nat \to nat.
+sigma_p (S (n*(exp p m))) (\lambda x.divides_b x (n*(exp p m))) g =
+sigma_p (S n) (\lambda x.divides_b x n)
+ (\lambda x.sigma_p (S m) (\lambda y.true) (\lambda y.g (x*(exp p y)))).
+intros.
+cut (O < p)
+ [rewrite < sigma_p2.
+ apply (trans_eq ? ?
+ (sigma_p (S n*S m) (\lambda x:nat.divides_b (x/S m) n)
+ (\lambda x:nat.g (x/S m*(p)\sup(x\mod S m)))))
+ [apply sym_eq.
+ apply (eq_sigma_p_gh g ? (p_ord_times p (S m)))
+ [intros.
+ lapply (divides_b_true_to_lt_O ? ? H H4).
+ apply divides_to_divides_b_true
+ [rewrite > (times_n_O O).
+ apply lt_times
+ [assumption
+ |apply lt_O_exp.assumption
+ ]
+ |apply divides_times
+ [apply divides_b_true_to_divides.assumption
+ |apply (witness ? ? (p \sup (m-i \mod (S m)))).
+ rewrite < exp_plus_times.
+ apply eq_f.
+ rewrite > sym_plus.
+ apply plus_minus_m_m.
+ auto
+ ]
+ ]
+ |intros.
+ lapply (divides_b_true_to_lt_O ? ? H H4).
+ unfold p_ord_times.
+ rewrite > (p_ord_exp1 p ? (i \mod (S m)) (i/S m))
+ [change with ((i/S m)*S m+i \mod S m=i).
+ apply sym_eq.
+ apply div_mod.
+ apply lt_O_S
+ |assumption
+ |unfold Not.intro.
+ apply H2.
+ apply (trans_divides ? (i/ S m))
+ [assumption|
+ apply divides_b_true_to_divides;assumption]
+ |apply sym_times.
+ ]
+ |intros.
+ apply le_S_S.
+ apply le_times
+ [apply le_S_S_to_le.
+ change with ((i/S m) < S n).
+ apply (lt_times_to_lt_l m).
+ apply (le_to_lt_to_lt ? i)
+ [auto|assumption]
+ |apply le_exp
+ [assumption
+ |apply le_S_S_to_le.
+ apply lt_mod_m_m.
+ apply lt_O_S
+ ]
+ ]
+ |intros.
+ cut (ord j p < S m)
+ [rewrite > div_p_ord_times
+ [apply divides_to_divides_b_true
+ [apply lt_O_ord_rem
+ [elim H1.assumption
+ |apply (divides_b_true_to_lt_O ? ? ? H4).
+ rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ ]
+ |cut (n = ord_rem (n*(exp p m)) p)
+ [rewrite > Hcut2.
+ apply divides_to_divides_ord_rem
+ [apply (divides_b_true_to_lt_O ? ? ? H4).
+ rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |assumption
+ |apply divides_b_true_to_divides.
+ assumption
+ ]
+ |unfold ord_rem.
+ rewrite > (p_ord_exp1 p ? m n)
+ [reflexivity
+ |assumption
+ |assumption
+ |apply sym_times
+ ]
+ ]
+ ]
+ |assumption
+ ]
+ |cut (m = ord (n*(exp p m)) p)
+ [apply le_S_S.
+ rewrite > Hcut1.
+ apply divides_to_le_ord
+ [apply (divides_b_true_to_lt_O ? ? ? H4).
+ rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |assumption
+ |apply divides_b_true_to_divides.
+ assumption
+ ]
+ |unfold ord.
+ rewrite > (p_ord_exp1 p ? m n)
+ [reflexivity
+ |assumption
+ |assumption
+ |apply sym_times
+ ]
+ ]
+ ]
+ |intros.
+ cut (ord j p < S m)
+ [rewrite > div_p_ord_times
+ [rewrite > mod_p_ord_times
+ [rewrite > sym_times.
+ apply sym_eq.
+ apply exp_ord
+ [elim H1.assumption
+ |apply (divides_b_true_to_lt_O ? ? ? H4).
+ rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ ]
+ |cut (m = ord (n*(exp p m)) p)
+ [apply le_S_S.
+ rewrite > Hcut2.
+ apply divides_to_le_ord
+ [apply (divides_b_true_to_lt_O ? ? ? H4).
+ rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |assumption
+ |apply divides_b_true_to_divides.
+ assumption
+ ]
+ |unfold ord.
+ rewrite > (p_ord_exp1 p ? m n)
+ [reflexivity
+ |assumption
+ |assumption
+ |apply sym_times
+ ]
+ ]
+ ]
+ |assumption
+ ]
+ |cut (m = ord (n*(exp p m)) p)
+ [apply le_S_S.
+ rewrite > Hcut1.
+ apply divides_to_le_ord
+ [apply (divides_b_true_to_lt_O ? ? ? H4).
+ rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |assumption
+ |apply divides_b_true_to_divides.
+ assumption
+ ]
+ |unfold ord.
+ rewrite > (p_ord_exp1 p ? m n)
+ [reflexivity
+ |assumption
+ |assumption
+ |apply sym_times
+ ]
+ ]
+ ]
+ |intros.
+ rewrite > eq_p_ord_times.
+ rewrite > sym_plus.
+ apply (lt_to_le_to_lt ? (S m +ord_rem j p*S m))
+ [apply lt_plus_l.
+ apply le_S_S.
+ cut (m = ord (n*(p \sup m)) p)
+ [rewrite > Hcut1.
+ apply divides_to_le_ord
+ [apply (divides_b_true_to_lt_O ? ? ? H4).
+ rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |assumption
+ |apply divides_b_true_to_divides.
+ assumption
+ ]
+ |unfold ord.
+ rewrite > sym_times.
+ rewrite > (p_ord_exp1 p ? m n)
+ [reflexivity
+ |assumption
+ |assumption
+ |reflexivity
+ ]
+ ]
+ |change with (S (ord_rem j p)*S m \le S n*S m).
+ apply le_times_l.
+ apply le_S_S.
+ cut (n = ord_rem (n*(p \sup m)) p)
+ [rewrite > Hcut1.
+ apply divides_to_le
+ [apply lt_O_ord_rem
+ [elim H1.assumption
+ |rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ ]
+ |apply divides_to_divides_ord_rem
+ [apply (divides_b_true_to_lt_O ? ? ? H4).
+ rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |rewrite > (times_n_O O).
+ apply lt_times
+ [assumption|apply lt_O_exp.assumption]
+ |assumption
+ |apply divides_b_true_to_divides.
+ assumption
+ ]
+ ]
+ |unfold ord_rem.
+ rewrite > sym_times.
+ rewrite > (p_ord_exp1 p ? m n)
+ [reflexivity
+ |assumption
+ |assumption
+ |reflexivity
+ ]
+ ]
+ ]
+ ]
+ |apply eq_sigma_p
+ [intros.
+ elim (divides_b (x/S m) n);reflexivity
+ |intros.reflexivity
+ ]
+ ]
+|elim H1.apply lt_to_le.assumption
+]
+qed.
+
+
apply (ltn_to_ltO p q H2).
qed.
+theorem lt_times_r1:
+\forall n,m,p. O < n \to m < p \to n*m < n*p.
+intros.
+elim H;apply lt_times_r;assumption.
+qed.
+
+theorem lt_times_l1:
+\forall n,m,p. O < n \to m < p \to m*n < p*n.
+intros.
+elim H;apply lt_times_l;assumption.
+qed.
+
+theorem lt_to_le_to_lt_times :
+\forall n,n1,m,m1. n < n1 \to m \le m1 \to O < m1 \to n*m < n1*m1.
+intros.
+apply (le_to_lt_to_lt ? (n*m1))
+ [apply le_times_r.assumption
+ |apply lt_times_l1
+ [assumption|assumption]
+ ]
+qed.
+
theorem lt_times_to_lt_l:
\forall n,p,q:nat. p*(S n) < q*(S n) \to p < q.
intros.
rewrite > H2.simplify.apply le_n.
qed.
-
-definition max_spec \def \lambda f:nat \to bool.\lambda n: nat.
-\exists i. (le i n) \land (f i = true) \to
-(f n) = true \land (\forall i. i < n \to (f i = false)).
-
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.
intro.rewrite > H7.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)).
+
+theorem max_spec_to_max: \forall f:nat \to bool.\forall n,m:nat.
+max_spec f n m \to max n f = m.
+intros 2.
+elim n
+ [elim H.elim H1.apply (le_n_O_elim ? H3).
+ apply max_O_f
+ |elim H1.
+ elim (max_S_max f n1)
+ [elim H4.
+ rewrite > H6.
+ apply le_to_le_to_eq
+ [apply not_lt_to_le.
+ unfold Not.intro.
+ apply not_eq_true_false.
+ rewrite < H5.
+ apply H3
+ [assumption|apply le_n]
+ |elim H2.assumption
+ ]
+ |elim H4.
+ rewrite > H6.
+ apply H.
+ elim H2.
+ split
+ [split
+ [elim (le_to_or_lt_eq ? ? H7)
+ [apply le_S_S_to_le.assumption
+ |apply False_ind.
+ apply not_eq_true_false.
+ rewrite < H8.
+ rewrite > H9.
+ assumption
+ ]
+ |assumption
+ ]
+ |intros.
+ apply H3
+ [assumption|apply le_S.assumption]
+ ]
+ ]
+ ]
+qed.
+
let rec min_aux off n f \def
match f (n-off) with
[ true \Rightarrow (n-off)
unfold lt. apply le_n.apply lt_SO_nth_prime_n.
qed.
+theorem lt_n_nth_prime_n : \forall n:nat. n \lt nth_prime n.
+intro.
+elim n
+ [apply lt_O_nth_prime_n
+ |apply (lt_to_le_to_lt ? (S (nth_prime n1)))
+ [unfold.apply le_S_S.assumption
+ |apply lt_nth_prime_n_nth_prime_Sn
+ ]
+ ]
+qed.
+
theorem ex_m_le_n_nth_prime_m:
\forall n: nat. nth_prime O \le n \to
\exists m. nth_prime m \le n \land n < nth_prime (S m).
include "nat/gcd.ma".
include "nat/relevant_equations.ma". (* required by auto paramod *)
-(* this definition of log is based on pairs, with a remainder *)
-
let rec p_ord_aux p n m \def
match n \mod m with
[ O \Rightarrow
apply divides_to_le.unfold.apply le_n.assumption.
rewrite < times_n_SO.
apply exp_n_SO.
-qed.
\ No newline at end of file
+qed.
+
+(* spostare *)
+theorem le_plus_to_le:
+\forall a,n,m. a + n \le a + m \to n \le m.
+intro.
+elim a
+ [assumption
+ |apply H.
+ apply le_S_S_to_le.assumption
+ ]
+qed.
+
+theorem le_times_to_le:
+\forall a,n,m. O < a \to a * n \le a * m \to n \le m.
+intro.
+apply nat_elim2;intros
+ [apply le_O_n
+ |apply False_ind.
+ rewrite < times_n_O in H1.
+ generalize in match H1.
+ apply (lt_O_n_elim ? H).
+ intros.
+ simplify in H2.
+ apply (le_to_not_lt ? ? H2).
+ apply lt_O_S
+ |apply le_S_S.
+ apply H
+ [assumption
+ |rewrite < times_n_Sm in H2.
+ rewrite < times_n_Sm in H2.
+ apply (le_plus_to_le a).
+ assumption
+ ]
+ ]
+qed.
+
+theorem le_exp_to_le:
+\forall a,n,m. S O < a \to exp a n \le exp a m \to n \le m.
+intro.
+apply nat_elim2;intros
+ [apply le_O_n
+ |apply False_ind.
+ apply (le_to_not_lt ? ? H1).
+ simplify.
+ rewrite > times_n_SO.
+ apply lt_to_le_to_lt_times
+ [assumption
+ |apply lt_O_exp.apply lt_to_le.assumption
+ |apply lt_O_exp.apply lt_to_le.assumption
+ ]
+ |simplify in H2.
+ apply le_S_S.
+ apply H
+ [assumption
+ |apply (le_times_to_le a)
+ [apply lt_to_le.assumption|assumption]
+ ]
+ ]
+qed.
+
+theorem divides_to_p_ord: \forall p,a,b,c,d,n,m:nat.
+O < n \to O < m \to prime p
+\to divides n m \to p_ord n p = pair ? ? a b \to
+p_ord m p = pair ? ? c d \to divides b d \land a \le c.
+intros.
+cut (S O < p)
+ [lapply (p_ord_to_exp1 ? ? ? ? Hcut H H4).
+ lapply (p_ord_to_exp1 ? ? ? ? Hcut H1 H5).
+ elim Hletin. clear Hletin.
+ elim Hletin1. clear Hletin1.
+ rewrite > H9 in H3.
+ split
+ [apply (gcd_SO_to_divides_times_to_divides (exp p c))
+ [elim (le_to_or_lt_eq ? ? (le_O_n b))
+ [assumption
+ |apply False_ind.
+ apply (lt_to_not_eq O ? H).
+ rewrite > H7.
+ rewrite < H10.
+ auto
+ ]
+ |elim c
+ [rewrite > sym_gcd.
+ apply gcd_SO_n
+ |simplify.
+ apply eq_gcd_times_SO
+ [apply lt_to_le.assumption
+ |apply lt_O_exp.apply lt_to_le.assumption
+ |rewrite > sym_gcd.
+ (* hint non trova prime_to_gcd_SO e
+ auto non chiude il goal *)
+ apply prime_to_gcd_SO
+ [assumption|assumption]
+ |assumption
+ ]
+ ]
+ |apply (trans_divides ? n)
+ [apply (witness ? ? (exp p a)).
+ rewrite > sym_times.
+ assumption
+ |assumption
+ ]
+ ]
+ |apply (le_exp_to_le p)
+ [assumption
+ |apply divides_to_le
+ [apply lt_O_exp.apply lt_to_le.assumption
+ |apply (gcd_SO_to_divides_times_to_divides d)
+ [apply lt_O_exp.apply lt_to_le.assumption
+ |elim a
+ [apply gcd_SO_n
+ |simplify.rewrite < sym_gcd.
+ apply eq_gcd_times_SO
+ [apply lt_to_le.assumption
+ |apply lt_O_exp.apply lt_to_le.assumption
+ |rewrite > sym_gcd.
+ (* hint non trova prime_to_gcd_SO e
+ auto non chiude il goal *)
+ apply prime_to_gcd_SO
+ [assumption|assumption]
+ |rewrite > sym_gcd. assumption
+ ]
+ ]
+ |apply (trans_divides ? n)
+ [apply (witness ? ? b).assumption
+ |rewrite > sym_times.assumption
+ ]
+ ]
+ ]
+ ]
+ ]
+ |elim H2.assumption
+ ]
+qed.
+
+definition ord :nat \to nat \to nat \def
+\lambda n,p. fst ? ? (p_ord n p).
+
+definition ord_rem :nat \to nat \to nat \def
+\lambda n,p. snd ? ? (p_ord n p).
+
+theorem divides_to_ord: \forall p,n,m:nat.
+O < n \to O < m \to prime p
+\to divides n m
+\to divides (ord_rem n p) (ord_rem m p) \land (ord n p) \le (ord m p).
+intros.
+apply (divides_to_p_ord p ? ? ? ? n m H H1 H2 H3)
+ [unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+ |unfold ord.unfold ord_rem.apply eq_pair_fst_snd
+ ]
+qed.
+
+theorem divides_to_divides_ord_rem: \forall p,n,m:nat.
+O < n \to O < m \to prime p \to divides n m \to
+divides (ord_rem n p) (ord_rem m p).
+intros.
+elim (divides_to_ord p n m H H1 H2 H3).assumption.
+qed.
+
+theorem divides_to_le_ord: \forall p,n,m:nat.
+O < n \to O < m \to prime p \to divides n m \to
+(ord n p) \le (ord m p).
+intros.
+elim (divides_to_ord p n m H H1 H2 H3).assumption.
+qed.
+
+theorem exp_ord: \forall p,n. (S O) \lt p
+\to O \lt n \to n = p \sup (ord n p) * (ord_rem n p).
+intros.
+elim (p_ord_to_exp1 p n (ord n p) (ord_rem n p))
+ [assumption
+ |assumption
+ |assumption
+ |unfold ord.unfold ord_rem.
+ apply eq_pair_fst_snd
+ ]
+qed.
+
+theorem divides_ord_rem: \forall p,n. (S O) < p \to O < n
+\to divides (ord_rem n p) n.
+intros.
+apply (witness ? ? (p \sup (ord n p))).
+rewrite > sym_times.
+apply exp_ord[assumption|assumption]
+qed.
+
+theorem lt_O_ord_rem: \forall p,n. (S O) < p \to O < n \to O < ord_rem n p.
+intros.
+elim (le_to_or_lt_eq O (ord_rem n p))
+ [assumption
+ |apply False_ind.
+ apply (lt_to_not_eq ? ? H1).
+ lapply (divides_ord_rem ? ? H H1).
+ rewrite < H2 in Hletin.
+ elim Hletin.
+ rewrite > H3.
+ reflexivity
+ |apply le_O_n
+ ]
+qed.
apply H3. apply le_S_S. assumption.
qed.
-(* le to eq *)
+(* le and eq *)
lemma le_to_le_to_eq: \forall n,m. n \le m \to m \le n \to n = m.
apply nat_elim2
[intros.apply le_n_O_to_eq.assumption
(* boolean divides *)
definition divides_b : nat \to nat \to bool \def
\lambda n,m :nat. (eqb (m \mod n) O).
-
+
theorem divides_b_to_Prop :
\forall n,m:nat. O < n \to
match divides_b n m with
intro.simplify.unfold Not.intro.apply H1.apply divides_to_mod_O.assumption.assumption.
qed.
-theorem divides_b_true_to_divides :
+theorem divides_b_true_to_divides1:
\forall n,m:nat. O < n \to
(divides_b n m = true ) \to n \divides m.
intros.
assumption.
qed.
-theorem divides_b_false_to_not_divides :
+theorem divides_b_true_to_divides:
+\forall n,m:nat. divides_b n m = true \to n \divides m.
+intros 2.apply (nat_case n)
+ [apply (nat_case m)
+ [intro.apply divides_n_n
+ |simplify.intros.apply False_ind.
+ apply not_eq_true_false.apply sym_eq.assumption
+ ]
+ |intros.
+ apply divides_b_true_to_divides1
+ [apply lt_O_S|assumption]
+ ]
+qed.
+
+theorem divides_b_false_to_not_divides1:
\forall n,m:nat. O < n \to
(divides_b n m = false ) \to n \ndivides m.
intros.
assumption.
qed.
+theorem divides_b_false_to_not_divides:
+\forall n,m:nat. divides_b n m = false \to n \ndivides m.
+intros 2.apply (nat_case n)
+ [apply (nat_case m)
+ [simplify.unfold Not.intros.
+ apply not_eq_true_false.assumption
+ |unfold Not.intros.elim H1.
+ apply (not_eq_O_S m1).apply sym_eq.
+ assumption
+ ]
+ |intros.
+ apply divides_b_false_to_not_divides1
+ [apply lt_O_S|assumption]
+ ]
+qed.
+
theorem decidable_divides: \forall n,m:nat.O < n \to
decidable (n \divides m).
intros.unfold decidable.
reflexivity.
qed.
+theorem divides_b_true_to_lt_O: \forall n,m. O < n \to divides_b m n = true \to O < m.
+intros.
+elim (le_to_or_lt_eq ? ? (le_O_n m))
+ [assumption
+ |apply False_ind.
+ elim H1.
+ rewrite < H2 in H1.
+ simplify in H1.
+ apply (lt_to_not_eq O n H).
+ apply sym_eq.
+ apply eqb_true_to_eq.
+ assumption
+ ]
+qed.
+
(* divides and pi *)
theorem divides_f_pi_f : \forall f:nat \to nat.\forall n,m,i:nat.
m \le i \to i \le n+m \to f i \divides pi n f m.
(S O) < i \to i \le n \to i \ndivides S n!.
intros.
apply divides_b_false_to_not_divides.
-apply (trans_lt O (S O)).apply (le_n (S O)).assumption.
unfold divides_b.
-rewrite > mod_S_fact.simplify.reflexivity.
-assumption.assumption.
+rewrite > mod_S_fact[simplify.reflexivity|assumption|assumption].
qed.
(* prime *)
apply (witness ? ? (S O)). simplify.reflexivity.
intros.
apply divides_b_true_to_divides.
-apply (lt_O_smallest_factor ? H).
change with
(eqb ((S(S m1)) \mod (min_aux m1 (S(S m1))
(\lambda m.(eqb ((S(S m1)) \mod m) O)))) O = true).
intro.apply (nat_case m).intro. apply False_ind.apply (not_le_Sn_n (S O) H).
intros.
apply divides_b_false_to_not_divides.
-apply (trans_lt O (S O)).apply (le_n (S O)).assumption.unfold divides_b.
apply (lt_min_aux_to_false
(\lambda i:nat.eqb ((S(S m1)) \mod i) O) (S(S m1)) m1 i).
cut ((S(S O)) = (S(S m1)-m1)).
projects / helm.git / commitdiff