Library MathComp.prime
Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path fintype.
Require Import div bigop.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Fixpoint edivn2 q r := if r is r'.+2 then edivn2 q.+1 r' else (q, r).
Lemma edivn2P n : edivn_spec n 2 (edivn2 0 n).
Proof.
rewrite -[n]odd_double_half addnC -{1}[n./2]addn0 -{1}mul2n mulnC.
elim: n./2 {1 4}0 ⇒ [|r IHr] q; first by case (odd n) ⇒ /=.
rewrite addSnnS; exact: IHr.
Qed.
Fixpoint elogn2 e q r {struct q} :=
match q, r with
| 0, _ | _, 0 ⇒ (e, q)
| q'.+1, 1 ⇒ elogn2 e.+1 q' q'
| q'.+1, r'.+2 ⇒ elogn2 e q' r'
end.
CoInductive elogn2_spec n : nat × nat → Type :=
Elogn2Spec e m of n = 2 ^ e × m.*2.+1 : elogn2_spec n (e, m).
Lemma elogn2P n : elogn2_spec n.+1 (elogn2 0 n n).
Proof.
rewrite -{1}[n.+1]mul1n -[1]/(2 ^ 0) -{1}(addKn n n) addnn.
elim: n {1 4 6}n {2 3}0 (leqnn n) ⇒ [|q IHq] [|[|r]] e //=; last first.
by move/ltnW; exact: IHq.
clear 1; rewrite subn1 -[_.-1.+1]doubleS -mul2n mulnA -expnSr.
rewrite -{1}(addKn q q) addnn; exact: IHq.
Qed.
Definition ifnz T n (x y : T) := if n is 0 then y else x.
CoInductive ifnz_spec T n (x y : T) : T → Type :=
| IfnzPos of n > 0 : ifnz_spec n x y x
| IfnzZero of n = 0 : ifnz_spec n x y y.
Lemma ifnzP T n (x y : T) : ifnz_spec n x y (ifnz n x y).
Proof. by case: n ⇒ [|n]; [right | left]. Qed.
Definition NumFactor (f : nat × nat) := ([Num of f.1], f.2).
Definition pfactor p e := p ^ e.
Definition cons_pfactor (p e : nat) pd := ifnz e ((p, e) :: pd) pd.
Notation Local "p ^? e :: pd" := (cons_pfactor p e pd)
(at level 30, e at level 30, pd at level 60) : nat_scope.
Section prime_decomp.
Import NatTrec.
Fixpoint prime_decomp_rec m k a b c e :=
let p := k.*2.+1 in
if a is a'.+1 then
if b - (ifnz e 1 k - c) is b'.+1 then
[rec m, k, a', b', ifnz c c.-1 (ifnz e p.-2 1), e] else
if (b == 0) && (c == 0) then
let b' := k + a' in [rec b'.*2.+3, k, a', b', k.-1, e.+1] else
let bc' := ifnz e (ifnz b (k, 0) (edivn2 0 c)) (b, c) in
p ^? e :: ifnz a' [rec m, k.+1, a'.-1, bc'.1 + a', bc'.2, 0] [:: (m, 1)]
else if (b == 0) && (c == 0) then [:: (p, e.+2)] else p ^? e :: [:: (m, 1)]
where "[ 'rec' m , k , a , b , c , e ]" := (prime_decomp_rec m k a b c e).
Definition prime_decomp n :=
let: (e2, m2) := elogn2 0 n.-1 n.-1 in
if m2 < 2 then 2 ^? e2 :: 3 ^? m2 :: [::] else
let: (a, bc) := edivn m2.-2 3 in
let: (b, c) := edivn (2 - bc) 2 in
2 ^? e2 :: [rec m2.*2.+1, 1, a, b, c, 0].
Definition add_divisors f divs :=
let: (p, e) := f in
let add1 divs' := merge leq (map (NatTrec.mul p) divs') divs in
iter e add1 divs.
Definition add_totient_factor f m := let: (p, e) := f in p.-1 × p ^ e.-1 × m.
End prime_decomp.
Definition primes n := unzip1 (prime_decomp n).
Definition prime p := if prime_decomp p is [:: (_ , 1)] then true else false.
Definition nat_pred := simpl_pred nat.
Definition pi_unwrapped_arg := nat.
Definition pi_wrapped_arg := wrapped nat.
Coercion unwrap_pi_arg (wa : pi_wrapped_arg) : pi_unwrapped_arg := unwrap wa.
Coercion pi_arg_of_nat (n : nat) := Wrap n : pi_wrapped_arg.
Coercion pi_arg_of_fin_pred T pT (A : @fin_pred_sort T pT) : pi_wrapped_arg :=
Wrap #|A|.
Definition pi_of (n : pi_unwrapped_arg) : nat_pred := [pred p in primes n].
Notation "\pi ( n )" := (pi_of n)
(at level 2, format "\pi ( n )") : nat_scope.
Notation "\p 'i' ( A )" := \pi(#|A|)
(at level 2, format "\p 'i' ( A )") : nat_scope.
Definition pdiv n := head 1 (primes n).
Definition max_pdiv n := last 1 (primes n).
Definition divisors n := foldr add_divisors [:: 1] (prime_decomp n).
Definition totient n := foldr add_totient_factor (n > 0) (prime_decomp n).
Lemma prime_decomp_correct :
let pd_val pd := \prod_(f <- pd) pfactor f.1 f.2 in
let lb_dvd q m := ~~ has [pred d | d %| m] (index_iota 2 q) in
let pf_ok f := lb_dvd f.1 f.1 && (0 < f.2) in
let pd_ord q pd := path ltn q (unzip1 pd) in
let pd_ok q n pd := [/\ n = pd_val pd, all pf_ok pd & pd_ord q pd] in
∀ n, n > 0 → pd_ok 1 n (prime_decomp n).
Proof.
rewrite unlock ⇒ pd_val lb_dvd pf_ok pd_ord pd_ok.
have leq_pd_ok m p q pd: q ≤ p → pd_ok p m pd → pd_ok q m pd.
rewrite /pd_ok /pd_ord; case: pd ⇒ [|[r _] pd] //= leqp [<- ->].
by case/andP⇒ /(leq_trans _)->.
have apd_ok m e q p pd: lb_dvd p p || (e == 0) → q < p →
pd_ok p m pd → pd_ok q (p ^ e × m) (p ^? e :: pd).
- case: e ⇒ [|e]; rewrite orbC /= ⇒ pr_p ltqp.
rewrite mul1n; apply: leq_pd_ok; exact: ltnW.
by rewrite /pd_ok /pd_ord /pf_ok /= pr_p ltqp ⇒ [[<- → ->]].
case⇒ // n _; rewrite /prime_decomp.
case: elogn2P ⇒ e2 m2 → {n}; case: m2 ⇒ [|[|abc]]; try exact: apd_ok.
rewrite [_.-2]/= !ltnS ltn0 natTrecE; case: edivnP ⇒ a bc ->{abc}.
case: edivnP ⇒ b c def_bc /= ltc2 ltbc3; apply: (apd_ok) ⇒ //.
move def_m: _.*2.+1 ⇒ m; set k := {2}1; rewrite -[2]/k.*2; set e := 0.
pose p := k.*2.+1; rewrite -{1}[m]mul1n -[1]/(p ^ e)%N.
have{def_m bc def_bc ltc2 ltbc3}:
let kb := (ifnz e k 1).*2 in
[&& k > 0, p < m, lb_dvd p m, c < kb & lb_dvd p p || (e == 0)]
∧ m + (b × kb + c).*2 = p ^ 2 + (a × p).*2.
- rewrite -{-2}def_m; split⇒ //=; last first.
by rewrite -def_bc addSn -doubleD 2!addSn -addnA subnKC // addnC.
rewrite ltc2 /lb_dvd /index_iota /= dvdn2 -def_m.
by rewrite [_.+2]lock /= odd_double.
move: {2}a.+1 (ltnSn a) ⇒ n; clearbody k e.
elim: n ⇒ // n IHn in a k p m b c e *; rewrite ltnS ⇒ le_a_n [].
set kb := _.*2; set d := _ + c ⇒ /and5P[lt0k ltpm leppm ltc pr_p def_m].
have def_k1: k.-1.+1 = k := ltn_predK lt0k.
have def_kb1: kb.-1.+1 = kb by rewrite /kb -def_k1; case e.
have eq_bc_0: (b == 0) && (c == 0) = (d == 0).
by rewrite addn_eq0 muln_eq0 orbC -def_kb1.
have lt1p: 1 < p by rewrite ltnS double_gt0.
have co_p_2: coprime p 2 by rewrite /coprime gcdnC gcdnE modn2 /= odd_double.
have if_d0: d = 0 → [/\ m = (p + a.*2) × p, lb_dvd p p & lb_dvd p (p + a.*2)].
move⇒ d0; have{d0 def_m} def_m: m = (p + a.*2) × p.
by rewrite d0 addn0 -mulnn -!mul2n mulnA -mulnDl in def_m ×.
split⇒ //; apply/hasPn⇒ r /(hasPn leppm); apply: contra ⇒ /= dv_r.
by rewrite def_m dvdn_mull.
by rewrite def_m dvdn_mulr.
case def_a: a ⇒ [|a'] /= in le_a_n *; rewrite !natTrecE -/p {}eq_bc_0.
case: d if_d0 def_m ⇒ [[//| def_m {pr_p}pr_p pr_m'] _ | d _ def_m] /=.
rewrite def_m def_a addn0 mulnA -2!expnSr.
by split; rewrite /pd_ord /pf_ok /= ?muln1 ?pr_p ?leqnn.
apply: apd_ok; rewrite // /pd_ok /= /pfactor expn1 muln1 /pd_ord /= ltpm.
rewrite /pf_ok !andbT /=; split⇒ //; apply: contra leppm.
case/hasP⇒ r /=; rewrite mem_index_iota ⇒ /andP[lt1r ltrm] dvrm; apply/hasP.
have [ltrp | lepr] := ltnP r p.
by ∃ r; rewrite // mem_index_iota lt1r.
case/dvdnP: dvrm ⇒ q def_q; ∃ q; last by rewrite def_q /= dvdn_mulr.
rewrite mem_index_iota -(ltn_pmul2r (ltnW lt1r)) -def_q mul1n ltrm.
move: def_m; rewrite def_a addn0 -(@ltn_pmul2r p) // mulnn ⇒ <-.
apply: (@leq_ltn_trans m); first by rewrite def_q leq_mul.
by rewrite -addn1 leq_add2l.
have def_k2: k.*2 = ifnz e 1 k × kb.
by rewrite /kb; case: (e) ⇒ [|e']; rewrite (mul1n, muln2).
case def_b': (b - _) ⇒ [|b']; last first.
have ->: ifnz e k.*2.-1 1 = kb.-1 by rewrite /kb; case e.
apply: IHn ⇒ {n le_a_n}//; rewrite -/p -/kb; split⇒ //.
rewrite lt0k ltpm leppm pr_p andbT /=.
by case: ifnzP; [move/ltn_predK->; exact: ltnW | rewrite def_kb1].
apply: (@addIn p.*2).
rewrite -2!addnA -!doubleD -addnA -mulSnr -def_a -def_m /d.
have ->: b × kb = b' × kb + (k.*2 - c × kb + kb).
rewrite addnCA addnC -mulSnr -def_b' def_k2 -mulnBl -mulnDl subnK //.
by rewrite ltnW // -subn_gt0 def_b'.
rewrite -addnA; congr (_ + (_ + _).*2).
case: (c) ltc; first by rewrite -addSnnS def_kb1 subn0 addn0 addnC.
rewrite /kb; case e ⇒ [[] // _ | e' c' _] /=; last first.
by rewrite subnDA subnn addnC addSnnS.
by rewrite mul1n -doubleB -doubleD subn1 !addn1 def_k1.
have ltdp: d < p.
move/eqP: def_b'; rewrite subn_eq0 -(@leq_pmul2r kb); last first.
by rewrite -def_kb1.
rewrite mulnBl -def_k2 ltnS -(leq_add2r c); move/leq_trans; apply.
have{ltc} ltc: c < k.*2.
by apply: (leq_trans ltc); rewrite leq_double /kb; case e.
rewrite -{2}(subnK (ltnW ltc)) leq_add2r leq_sub2l //.
by rewrite -def_kb1 mulnS leq_addr.
case def_d: d if_d0 ⇒ [|d'] ⇒ [[//|{def_m ltdp pr_p} def_m pr_p pr_m'] | _].
rewrite eqxx -doubleS -addnS -def_a doubleD -addSn -/p def_m.
rewrite mulnCA mulnC -expnSr.
apply: IHn ⇒ {n le_a_n}//; rewrite -/p -/kb; split.
rewrite lt0k -addn1 leq_add2l {1}def_a pr_m' pr_p /= def_k1 -addnn.
by rewrite leq_addr.
rewrite -addnA -doubleD addnCA def_a addSnnS def_k1 -(addnC k) -mulnSr.
rewrite -[_.*2.+1]/p mulnDl doubleD addnA -mul2n mulnA mul2n -mulSn.
by rewrite -/p mulnn.
have next_pm: lb_dvd p.+2 m.
rewrite /lb_dvd /index_iota 2!subSS subn0 -(subnK lt1p) iota_add.
rewrite has_cat; apply/norP; split⇒ //=; rewrite orbF subnKC // orbC.
apply/norP; split; apply/dvdnP⇒ [[q def_q]].
case/hasP: leppm; ∃ 2; first by rewrite /p -(subnKC lt0k).
by rewrite /= def_q dvdn_mull // dvdn2 /= odd_double.
move/(congr1 (dvdn p)): def_m; rewrite -mulnn -!mul2n mulnA -mulnDl.
rewrite dvdn_mull // dvdn_addr; last by rewrite def_q dvdn_mull.
case/dvdnP⇒ r; rewrite mul2n ⇒ def_r; move: ltdp (congr1 odd def_r).
rewrite odd_double -ltn_double {1}def_r -mul2n ltn_pmul2r //.
by case: r def_r ⇒ [|[|[]]] //; rewrite def_d // mul1n /= odd_double.
apply: apd_ok ⇒ //; case: a' def_a le_a_n ⇒ [|a'] def_a ⇒ [_ | lta] /=.
rewrite /pd_ok /= /pfactor expn1 muln1 /pd_ord /= ltpm /pf_ok !andbT /=.
split⇒ //; apply: contra next_pm.
case/hasP⇒ q; rewrite mem_index_iota ⇒ /andP[lt1q ltqm] dvqm; apply/hasP.
have [ltqp | lepq] := ltnP q p.+2.
by ∃ q; rewrite // mem_index_iota lt1q.
case/dvdnP: dvqm ⇒ r def_r; ∃ r; last by rewrite def_r /= dvdn_mulr.
rewrite mem_index_iota -(ltn_pmul2r (ltnW lt1q)) -def_r mul1n ltqm /=.
rewrite -(@ltn_pmul2l p.+2) //; apply: (@leq_ltn_trans m).
by rewrite def_r mulnC leq_mul.
rewrite -addn2 mulnn sqrnD mul2n muln2 -addnn addnCA -addnA addnCA addnA.
by rewrite def_a mul1n in def_m; rewrite -def_m addnS -addnA ltnS leq_addr.
set bc := ifnz _ _ _; apply: leq_pd_ok (leqnSn _) _.
rewrite -doubleS -{1}[m]mul1n -[1]/(k.+1.*2.+1 ^ 0)%N.
apply: IHn; first exact: ltnW.
rewrite doubleS -/p [ifnz 0 _ _]/=; do 2?split ⇒ //.
rewrite orbT next_pm /= -(leq_add2r d.*2) def_m 2!addSnnS -doubleS leq_add.
- move: ltc; rewrite /kb {}/bc andbT; case e ⇒ //= e' _; case: ifnzP ⇒ //.
by case: edivn2P.
- by rewrite -{1}[p]muln1 -mulnn ltn_pmul2l.
by rewrite leq_double def_a mulSn (leq_trans ltdp) ?leq_addr.
rewrite mulnDl !muln2 -addnA addnCA doubleD addnCA.
rewrite (_ : _ + bc.2 = d); last first.
rewrite /d {}/bc /kb -muln2.
case: (e) (b) def_b' ⇒ //= _ []; first by case: edivn2P.
by case c; do 2?case; rewrite // mul1n /= muln2.
rewrite def_m 3!doubleS addnC -(addn2 p) sqrnD mul2n muln2 -3!addnA.
congr (_ + _); rewrite 4!addnS -!doubleD; congr _.*2.+2.+2.
by rewrite def_a -add2n mulnDl -addnA -muln2 -mulnDr mul2n.
Qed.
Lemma primePn n :
reflect (n < 2 ∨ exists2 d, 1 < d < n & d %| n) (~~ prime n).
Proof.
rewrite /prime; case: n ⇒ [|[|p2]]; try by do 2!left.
case: (@prime_decomp_correct p2.+2) ⇒ //; rewrite unlock.
case: prime_decomp ⇒ [|[q [|[|e]]] pd] //=; last first; last by rewrite andbF.
rewrite {1}/pfactor 2!expnS -!mulnA /=.
case: (_ ^ _ × _) ⇒ [|u → _ /andP[lt1q _]]; first by rewrite !muln0.
left; right; ∃ q; last by rewrite dvdn_mulr.
have lt0q := ltnW lt1q; rewrite lt1q -{1}[q]muln1 ltn_pmul2l //.
by rewrite -[2]muln1 leq_mul.
rewrite {1}/pfactor expn1; case: pd ⇒ [|[r e] pd] /=; last first.
case: e ⇒ [|e] /=; first by rewrite !andbF.
rewrite {1}/pfactor expnS -mulnA.
case: (_ ^ _ × _) ⇒ [|u → _ /and3P[lt1q ltqr _]]; first by rewrite !muln0.
left; right; ∃ q; last by rewrite dvdn_mulr.
by rewrite lt1q -{1}[q]mul1n ltn_mul // -[q.+1]muln1 leq_mul.
rewrite muln1 !andbT ⇒ def_q pr_q lt1q; right⇒ [[]] // [d].
by rewrite def_q -mem_index_iota ⇒ in_d_2q dv_d_q; case/hasP: pr_q; ∃ d.
Qed.
Lemma primeP p :
reflect (p > 1 ∧ ∀ d, d %| p → xpred2 1 p d) (prime p).
Proof.
rewrite -[prime p]negbK; have [npr_p | pr_p] := primePn p.
right⇒ [[lt1p pr_p]]; case: npr_p ⇒ [|[d n1pd]].
by rewrite ltnNge lt1p.
by move/pr_p⇒ /orP[] /eqP def_d; rewrite def_d ltnn ?andbF in n1pd.
have [lep1 | lt1p] := leqP; first by case: pr_p; left.
left; split⇒ // d dv_d_p; apply/norP⇒ [[nd1 ndp]]; case: pr_p; right.
∃ d; rewrite // andbC 2!ltn_neqAle ndp eq_sym nd1.
by have lt0p := ltnW lt1p; rewrite dvdn_leq // (dvdn_gt0 lt0p).
Qed.
Lemma prime_nt_dvdP d p : prime p → d != 1 → reflect (d = p) (d %| p).
Proof.
case/primeP⇒ _ min_p d_neq1; apply: (iffP idP) ⇒ [/min_p|-> //].
by rewrite (negPf d_neq1) /= ⇒ /eqP.
Qed.
Implicit Arguments primeP [p].
Implicit Arguments primePn [n].
Prenex Implicits primePn primeP.
Lemma prime_gt1 p : prime p → 1 < p.
Proof. by case/primeP. Qed.
Lemma prime_gt0 p : prime p → 0 < p.
Proof. by move/prime_gt1; exact: ltnW. Qed.
Hint Resolve prime_gt1 prime_gt0.
Lemma prod_prime_decomp n :
n > 0 → n = \prod_(f <- prime_decomp n) f.1 ^ f.2.
Proof. by case/prime_decomp_correct. Qed.
Lemma even_prime p : prime p → p = 2 ∨ odd p.
Proof.
move⇒ pr_p; case odd_p: (odd p); [by right | left].
have: 2 %| p by rewrite dvdn2 odd_p.
by case/primeP: pr_p ⇒ _ dv_p /dv_p/(2 =P p).
Qed.
Lemma prime_oddPn p : prime p → reflect (p = 2) (~~ odd p).
Proof.
by move⇒ p_pr; apply: (iffP idP) ⇒ [|-> //]; case/even_prime: p_pr ⇒ →.
Qed.
Lemma odd_prime_gt2 p : odd p → prime p → p > 2.
Proof. by move⇒ odd_p /prime_gt1; apply: odd_gt2. Qed.
Lemma mem_prime_decomp n p e :
(p, e) \in prime_decomp n → [/\ prime p, e > 0 & p ^ e %| n].
Proof.
case: (posnP n) ⇒ [-> //| /prime_decomp_correct[def_n mem_pd ord_pd pd_pe]].
have /andP[pr_p ->] := allP mem_pd _ pd_pe; split⇒ //; last first.
case/splitPr: pd_pe def_n ⇒ pd1 pd2 →.
by rewrite big_cat big_cons /= mulnCA dvdn_mulr.
have lt1p: 1 < p.
apply: (allP (order_path_min ltn_trans ord_pd)).
by apply/mapP; ∃ (p, e).
apply/primeP; split⇒ // d dv_d_p; apply/norP⇒ [[nd1 ndp]].
case/hasP: pr_p; ∃ d ⇒ //.
rewrite mem_index_iota andbC 2!ltn_neqAle ndp eq_sym nd1.
by have lt0p := ltnW lt1p; rewrite dvdn_leq // (dvdn_gt0 lt0p).
Qed.
Lemma prime_coprime p m : prime p → coprime p m = ~~ (p %| m).
Proof.
case/primeP⇒ p_gt1 p_pr; apply/eqP/negP⇒ [d1 | ndv_pm].
case/dvdnP⇒ k def_m; rewrite -(addn0 m) def_m gcdnMDl gcdn0 in d1.
by rewrite d1 in p_gt1.
by apply: gcdn_def ⇒ // d /p_pr /orP[] /eqP→.
Qed.
Lemma dvdn_prime2 p q : prime p → prime q → (p %| q) = (p == q).
Proof.
move⇒ pr_p pr_q; apply: negb_inj.
by rewrite eqn_dvd negb_and -!prime_coprime // coprime_sym orbb.
Qed.
Lemma Euclid_dvdM m n p : prime p → (p %| m × n) = (p %| m) || (p %| n).
Proof.
move⇒ pr_p; case dv_pm: (p %| m); first exact: dvdn_mulr.
by rewrite Gauss_dvdr // prime_coprime // dv_pm.
Qed.
Lemma Euclid_dvd1 p : prime p → (p %| 1) = false.
Proof. by rewrite dvdn1; case: eqP ⇒ // →. Qed.
Lemma Euclid_dvdX m n p : prime p → (p %| m ^ n) = (p %| m) && (n > 0).
Proof.
case: n ⇒ [|n] pr_p; first by rewrite andbF Euclid_dvd1.
by apply: (inv_inj negbK); rewrite !andbT -!prime_coprime // coprime_pexpr.
Qed.
Lemma mem_primes p n : (p \in primes n) = [&& prime p, n > 0 & p %| n].
Proof.
rewrite andbCA; case: posnP ⇒ [-> // | /= n_gt0].
apply/mapP/andP⇒ [[[q e]]|[pr_p]] /=.
case/mem_prime_decomp⇒ pr_q e_gt0; case/dvdnP⇒ u → → {p}.
by rewrite -(prednK e_gt0) expnS mulnCA dvdn_mulr.
rewrite {1}(prod_prime_decomp n_gt0) big_seq.
apply big_ind ⇒ [| u v IHu IHv | [q e] /= mem_qe dv_p_qe].
- by rewrite Euclid_dvd1.
- by rewrite Euclid_dvdM // ⇒ /orP[].
∃ (q, e) ⇒ //=; case/mem_prime_decomp: mem_qe ⇒ pr_q _ _.
by rewrite Euclid_dvdX // dvdn_prime2 // in dv_p_qe; case: eqP dv_p_qe.
Qed.
Lemma sorted_primes n : sorted ltn (primes n).
Proof.
by case: (posnP n) ⇒ [-> // | /prime_decomp_correct[_ _]]; exact: path_sorted.
Qed.
Lemma eq_primes m n : (primes m =i primes n) ↔ (primes m = primes n).
Proof.
split⇒ [eqpr| → //].
by apply: (eq_sorted_irr ltn_trans ltnn); rewrite ?sorted_primes.
Qed.
Lemma primes_uniq n : uniq (primes n).
Proof. exact: (sorted_uniq ltn_trans ltnn (sorted_primes n)). Qed.
Lemma pi_pdiv n : (pdiv n \in \pi(n)) = (n > 1).
Proof.
case: n ⇒ [|[|n]] //; rewrite /pdiv !inE /primes.
have:= prod_prime_decomp (ltn0Sn n.+1); rewrite unlock.
by case: prime_decomp ⇒ //= pf pd _; rewrite mem_head.
Qed.
Lemma pdiv_prime n : 1 < n → prime (pdiv n).
Proof. by rewrite -pi_pdiv mem_primes; case/and3P. Qed.
Lemma pdiv_dvd n : pdiv n %| n.
Proof.
by case: n (pi_pdiv n) ⇒ [|[|n]] //; rewrite mem_primes⇒ /and3P[].
Qed.
Lemma pi_max_pdiv n : (max_pdiv n \in \pi(n)) = (n > 1).
Proof.
rewrite !inE -pi_pdiv /max_pdiv /pdiv !inE.
by case: (primes n) ⇒ //= p ps; rewrite mem_head mem_last.
Qed.
Lemma max_pdiv_prime n : n > 1 → prime (max_pdiv n).
Proof. by rewrite -pi_max_pdiv mem_primes ⇒ /andP[]. Qed.
Lemma max_pdiv_dvd n : max_pdiv n %| n.
Proof.
by case: n (pi_max_pdiv n) ⇒ [|[|n]] //; rewrite mem_primes ⇒ /andP[].
Qed.
Lemma pdiv_leq n : 0 < n → pdiv n ≤ n.
Proof. by move⇒ n_gt0; rewrite dvdn_leq // pdiv_dvd. Qed.
Lemma max_pdiv_leq n : 0 < n → max_pdiv n ≤ n.
Proof. by move⇒ n_gt0; rewrite dvdn_leq // max_pdiv_dvd. Qed.
Lemma pdiv_gt0 n : 0 < pdiv n.
Proof. by case: n ⇒ [|[|n]] //; rewrite prime_gt0 ?pdiv_prime. Qed.
Lemma max_pdiv_gt0 n : 0 < max_pdiv n.
Proof. by case: n ⇒ [|[|n]] //; rewrite prime_gt0 ?max_pdiv_prime. Qed.
Hint Resolve pdiv_gt0 max_pdiv_gt0.
Lemma pdiv_min_dvd m d : 1 < d → d %| m → pdiv m ≤ d.
Proof.
move⇒ lt1d dv_d_m; case: (posnP m) ⇒ [->|mpos]; first exact: ltnW.
rewrite /pdiv; apply: leq_trans (pdiv_leq (ltnW lt1d)).
have: pdiv d \in primes m.
by rewrite mem_primes mpos pdiv_prime // (dvdn_trans (pdiv_dvd d)).
case: (primes m) (sorted_primes m) ⇒ //= p pm ord_pm.
rewrite inE ⇒ /predU1P[-> //|].
move/(allP (order_path_min ltn_trans ord_pm)); exact: ltnW.
Qed.
Lemma max_pdiv_max n p : p \in \pi(n) → p ≤ max_pdiv n.
Proof.
rewrite /max_pdiv !inE ⇒ n_p.
case/splitPr: n_p (sorted_primes n) ⇒ p1 p2; rewrite last_cat -cat_rcons /=.
rewrite headI /= cat_path -(last_cons 0) -headI last_rcons; case/andP⇒ _.
move/(order_path_min ltn_trans); case/lastP: p2 ⇒ //= p2 q.
by rewrite all_rcons last_rcons ltn_neqAle -andbA ⇒ /and3P[].
Qed.
Lemma ltn_pdiv2_prime n : 0 < n → n < pdiv n ^ 2 → prime n.
Proof.
case def_n: n ⇒ [|[|n']] // _; rewrite -def_n ⇒ lt_n_p2.
suffices ->: n = pdiv n by rewrite pdiv_prime ?def_n.
apply/eqP; rewrite eqn_leq leqNgt andbC pdiv_leq; last by rewrite def_n.
move: lt_n_p2; rewrite ltnNge; apply: contra ⇒ lt_pm_m.
case/dvdnP: (pdiv_dvd n) ⇒ q def_q.
rewrite {2}def_q -mulnn leq_pmul2r // pdiv_min_dvd //.
by rewrite -[pdiv n]mul1n {2}def_q ltn_pmul2r in lt_pm_m.
by rewrite def_q dvdn_mulr.
Qed.
Lemma primePns n :
reflect (n < 2 ∨ ∃ p, [/\ prime p, p ^ 2 ≤ n & p %| n]) (~~ prime n).
Proof.
apply: (iffP idP) ⇒ [npr_p|]; last first.
case⇒ [|[p [pr_p le_p2_n dv_p_n]]]; first by case: n ⇒ [|[]].
apply/negP⇒ pr_n; move: dv_p_n le_p2_n; rewrite dvdn_prime2 //; move/eqP→.
by rewrite leqNgt -{1}[n]muln1 -mulnn ltn_pmul2l ?prime_gt1 ?prime_gt0.
case: leqP ⇒ [lt1p|]; [right | by left].
∃ (pdiv n); rewrite pdiv_dvd pdiv_prime //; split⇒ //.
by case: leqP npr_p ⇒ //; move/ltn_pdiv2_prime->; auto.
Qed.
Implicit Arguments primePns [n].
Prenex Implicits primePns.
Lemma pdivP n : n > 1 → {p | prime p & p %| n}.
Proof. by move⇒ lt1n; ∃ (pdiv n); rewrite ?pdiv_dvd ?pdiv_prime. Qed.
Lemma primes_mul m n p : m > 0 → n > 0 →
(p \in primes (m × n)) = (p \in primes m) || (p \in primes n).
Proof.
move⇒ m_gt0 n_gt0; rewrite !mem_primes muln_gt0 m_gt0 n_gt0.
by case pr_p: (prime p); rewrite // Euclid_dvdM.
Qed.
Lemma primes_exp m n : n > 0 → primes (m ^ n) = primes m.
Proof.
case: n ⇒ // n _; rewrite expnS; case: (posnP m) ⇒ [-> //| m_gt0].
apply/eq_primes ⇒ /= p; elim: n ⇒ [|n IHn]; first by rewrite muln1.
by rewrite primes_mul ?(expn_gt0, expnS, IHn, orbb, m_gt0).
Qed.
Lemma primes_prime p : prime p → primes p = [::p].
Proof.
move⇒ pr_p; apply: (eq_sorted_irr ltn_trans ltnn) ⇒ // [|q].
exact: sorted_primes.
rewrite mem_seq1 mem_primes prime_gt0 //=.
by apply/andP/idP⇒ [[pr_q q_p] | /eqP→ //]; rewrite -dvdn_prime2.
Qed.
Lemma coprime_has_primes m n : m > 0 → n > 0 →
coprime m n = ~~ has (mem (primes m)) (primes n).
Proof.
move⇒ m_gt0 n_gt0; apply/eqnP/hasPn⇒ [mn1 p | no_p_mn].
rewrite /= !mem_primes m_gt0 n_gt0 /= ⇒ /andP[pr_p p_n].
have:= prime_gt1 pr_p; rewrite pr_p ltnNge -mn1 /=; apply: contra ⇒ p_m.
by rewrite dvdn_leq ?gcdn_gt0 ?m_gt0 // dvdn_gcd ?p_m.
case: (ltngtP (gcdn m n) 1) ⇒ //; first by rewrite ltnNge gcdn_gt0 ?m_gt0.
move/pdiv_prime; set p := pdiv _ ⇒ pr_p.
move/implyP: (no_p_mn p); rewrite /= !mem_primes m_gt0 n_gt0 pr_p /=.
by rewrite !(dvdn_trans (pdiv_dvd _)) // (dvdn_gcdl, dvdn_gcdr).
Qed.
Lemma pdiv_id p : prime p → pdiv p = p.
Proof. by move⇒ p_pr; rewrite /pdiv primes_prime. Qed.
Lemma pdiv_pfactor p k : prime p → pdiv (p ^ k.+1) = p.
Proof. by move⇒ p_pr; rewrite /pdiv primes_exp ?primes_prime. Qed.
Fixpoint logn_rec d m r :=
match r, edivn m d with
| r'.+1, (_.+1 as m', 0) ⇒ (logn_rec d m' r').+1
| _, _ ⇒ 0
end.
Definition logn p m := if prime p then logn_rec p m m else 0.
Lemma lognE p m :
logn p m = if [&& prime p, 0 < m & p %| m] then (logn p (m %/ p)).+1 else 0.
Proof.
rewrite /logn /dvdn; case p_pr: (prime p) ⇒ //.
rewrite /divn modn_def; case def_m: {2 3}m ⇒ [|m'] //=.
case: edivnP def_m ⇒ [[|q] [|r] → _] // def_m; congr _.+1; rewrite [_.1]/=.
have{m def_m}: q < m'.
by rewrite -ltnS -def_m addn0 mulnC -{1}[q.+1]mul1n ltn_pmul2r // prime_gt1.
elim: {m' q}_.+1 {-2}m' q.+1 (ltnSn m') (ltn0Sn q) ⇒ // s IHs.
case⇒ [[]|r] //= m; rewrite ltnS ⇒ lt_rs m_gt0 le_mr.
rewrite -{3}[m]prednK //=; case: edivnP ⇒ [[|q] [|_] def_q _] //.
have{def_q} lt_qm': q < m.-1.
by rewrite -[q.+1]muln1 -ltnS prednK // def_q addn0 ltn_pmul2l // prime_gt1.
have{le_mr} le_m'r: m.-1 ≤ r by rewrite -ltnS prednK.
by rewrite (IHs r) ?(IHs m.-1) // ?(leq_trans lt_qm', leq_trans _ lt_rs).
Qed.
Lemma logn_gt0 p n : (0 < logn p n) = (p \in primes n).
Proof. by rewrite lognE -mem_primes; case: {+}(p \in _). Qed.
Lemma ltn_log0 p n : n < p → logn p n = 0.
Proof. by case: n ⇒ [|n] ltnp; rewrite lognE ?andbF // gtnNdvd ?andbF. Qed.
Lemma logn0 p : logn p 0 = 0.
Proof. by rewrite /logn if_same. Qed.
Lemma logn1 p : logn p 1 = 0.
Proof. by rewrite lognE dvdn1 /= andbC; case: eqP ⇒ // →. Qed.
Lemma pfactor_gt0 p n : 0 < p ^ logn p n.
Proof. by rewrite expn_gt0 lognE; case: (posnP p) ⇒ // →. Qed.
Hint Resolve pfactor_gt0.
Lemma pfactor_dvdn p n m : prime p → m > 0 → (p ^ n %| m) = (n ≤ logn p m).
Proof.
move⇒ p_pr; elim: n m ⇒ [|n IHn] m m_gt0; first exact: dvd1n.
rewrite lognE p_pr m_gt0 /=; case dv_pm: (p %| m); last first.
apply/dvdnP⇒ [] [/= q def_m].
by rewrite def_m expnS mulnCA dvdn_mulr in dv_pm.
case/dvdnP: dv_pm m_gt0 ⇒ q ->{m}; rewrite muln_gt0 ⇒ /andP[p_gt0 q_gt0].
by rewrite expnSr dvdn_pmul2r // mulnK // IHn.
Qed.
Lemma pfactor_dvdnn p n : p ^ logn p n %| n.
Proof.
case: n ⇒ // n; case pr_p: (prime p); first by rewrite pfactor_dvdn.
by rewrite lognE pr_p dvd1n.
Qed.
Lemma logn_prime p q : prime q → logn p q = (p == q).
Proof.
move⇒ pr_q; have q_gt0 := prime_gt0 pr_q; rewrite lognE q_gt0 /=.
case pr_p: (prime p); last by case: eqP pr_p pr_q ⇒ // → →.
by rewrite dvdn_prime2 //; case: eqP ⇒ // ->; rewrite divnn q_gt0 logn1.
Qed.
Lemma pfactor_coprime p n :
prime p → n > 0 → {m | coprime p m & n = m × p ^ logn p n}.
Proof.
move⇒ p_pr n_gt0; set k := logn p n.
have dv_pk_n: p ^ k %| n by rewrite pfactor_dvdn.
∃ (n %/ p ^ k); last by rewrite divnK.
rewrite prime_coprime // -(@dvdn_pmul2r (p ^ k)) ?expn_gt0 ?prime_gt0 //.
by rewrite -expnS divnK // pfactor_dvdn // ltnn.
Qed.
Lemma pfactorK p n : prime p → logn p (p ^ n) = n.
Proof.
move⇒ p_pr; have pn_gt0: p ^ n > 0 by rewrite expn_gt0 prime_gt0.
apply/eqP; rewrite eqn_leq -pfactor_dvdn // dvdnn andbT.
by rewrite -(leq_exp2l _ _ (prime_gt1 p_pr)) dvdn_leq // pfactor_dvdn.
Qed.
Lemma pfactorKpdiv p n : prime p → logn (pdiv (p ^ n)) (p ^ n) = n.
Proof. by case: n ⇒ // n p_pr; rewrite pdiv_pfactor ?pfactorK. Qed.
Lemma dvdn_leq_log p m n : 0 < n → m %| n → logn p m ≤ logn p n.
Proof.
move⇒ n_gt0 dv_m_n; have m_gt0 := dvdn_gt0 n_gt0 dv_m_n.
case p_pr: (prime p); last by do 2!rewrite lognE p_pr /=.
by rewrite -pfactor_dvdn //; apply: dvdn_trans dv_m_n; rewrite pfactor_dvdn.
Qed.
Lemma ltn_logl p n : 0 < n → logn p n < n.
Proof.
move⇒ n_gt0; have [p_gt1 | p_le1] := boolP (1 < p).
by rewrite (leq_trans (ltn_expl _ p_gt1)) // dvdn_leq ?pfactor_dvdnn.
by rewrite lognE (contraNF (@prime_gt1 _)).
Qed.
Lemma logn_Gauss p m n : coprime p m → logn p (m × n) = logn p n.
Proof.
move⇒ co_pm; case p_pr: (prime p); last by rewrite /logn p_pr.
have [-> | n_gt0] := posnP n; first by rewrite muln0.
have [m0 | m_gt0] := posnP m; first by rewrite m0 prime_coprime ?dvdn0 in co_pm.
have mn_gt0: m × n > 0 by rewrite muln_gt0 m_gt0.
apply/eqP; rewrite eqn_leq andbC dvdn_leq_log ?dvdn_mull //.
set k := logn p _; have: p ^ k %| m × n by rewrite pfactor_dvdn.
by rewrite Gauss_dvdr ?coprime_expl // -pfactor_dvdn.
Qed.
Lemma lognM p m n : 0 < m → 0 < n → logn p (m × n) = logn p m + logn p n.
Proof.
case p_pr: (prime p); last by rewrite /logn p_pr.
have xlp := pfactor_coprime p_pr.
case/xlp⇒ m' co_m' def_m /xlp[n' co_n' def_n] {xlp}.
by rewrite {1}def_m {1}def_n mulnCA -mulnA -expnD !logn_Gauss // pfactorK.
Qed.
Lemma lognX p m n : logn p (m ^ n) = n × logn p m.
Proof.
case p_pr: (prime p); last by rewrite /logn p_pr muln0.
elim: n ⇒ [|n IHn]; first by rewrite logn1.
have [->|m_gt0] := posnP m; first by rewrite exp0n // lognE andbF muln0.
by rewrite expnS lognM ?IHn // expn_gt0 m_gt0.
Qed.
Lemma logn_div p m n : m %| n → logn p (n %/ m) = logn p n - logn p m.
Proof.
rewrite dvdn_eq ⇒ /eqP def_n.
case: (posnP n) ⇒ [-> |]; first by rewrite div0n logn0.
by rewrite -{1 3}def_n muln_gt0 ⇒ /andP[q_gt0 m_gt0]; rewrite lognM ?addnK.
Qed.
Lemma dvdn_pfactor p d n : prime p →
reflect (exists2 m, m ≤ n & d = p ^ m) (d %| p ^ n).
Proof.
move⇒ p_pr; have pn_gt0: p ^ n > 0 by rewrite expn_gt0 prime_gt0.
apply: (iffP idP) ⇒ [dv_d_pn|[m le_m_n ->]]; last first.
by rewrite -(subnK le_m_n) expnD dvdn_mull.
∃ (logn p d); first by rewrite -(pfactorK n p_pr) dvdn_leq_log.
have d_gt0: d > 0 by exact: dvdn_gt0 dv_d_pn.
case: (pfactor_coprime p_pr d_gt0) ⇒ q co_p_q def_d.
rewrite {1}def_d ((q =P 1) _) ?mul1n // -dvdn1.
suff: q %| p ^ n × 1 by rewrite Gauss_dvdr // coprime_sym coprime_expl.
by rewrite muln1 (dvdn_trans _ dv_d_pn) // def_d dvdn_mulr.
Qed.
Lemma prime_decompE n : prime_decomp n = [seq (p, logn p n) | p <- primes n].
Proof.
case: n ⇒ // n; pose f0 := (0, 0); rewrite -map_comp.
apply: (@eq_from_nth _ f0) ⇒ [|i lt_i_n]; first by rewrite size_map.
rewrite (nth_map f0) //; case def_f: (nth _ _ i) ⇒ [p e] /=.
congr (_, _); rewrite [n.+1]prod_prime_decomp //.
have: (p, e) \in prime_decomp n.+1 by rewrite -def_f mem_nth.
case/mem_prime_decomp⇒ pr_p _ _.
rewrite (big_nth f0) big_mkord (bigD1 (Ordinal lt_i_n)) //=.
rewrite def_f mulnC logn_Gauss ?pfactorK //.
apply big_ind ⇒ [|m1 m2 com1 com2| [j ltj] /=]; first exact: coprimen1.
by rewrite coprime_mulr com1.
rewrite -val_eqE /= ⇒ nji; case def_j: (nth _ _ j) ⇒ [q e1] /=.
have: (q, e1) \in prime_decomp n.+1 by rewrite -def_j mem_nth.
case/mem_prime_decomp⇒ pr_q e1_gt0 _; rewrite coprime_pexpr //.
rewrite prime_coprime // dvdn_prime2 //; apply: contra nji ⇒ eq_pq.
rewrite -(nth_uniq 0 _ _ (primes_uniq n.+1)) ?size_map //=.
by rewrite !(nth_map f0) // def_f def_j /= eq_sym.
Qed.
Lemma divn_count_dvd d n : n %/ d = \sum_(1 ≤ i < n.+1) (d %| i).
Proof.
have [-> | d_gt0] := posnP d; first by rewrite big_add1 divn0 big1.
apply: (@addnI (d %| 0)); rewrite -(@big_ltn _ 0 _ 0 _ (dvdn d)) // big_mkord.
rewrite (partition_big (fun i : 'I_n.+1 ⇒ inord (i %/ d)) 'I_(n %/ d).+1) //=.
rewrite dvdn0 add1n -{1}[_.+1]card_ord -sum1_card; apply: eq_bigr ⇒ [[q ?] _].
rewrite (bigD1 (inord (q × d))) /eq_op /= !inordK ?ltnS -?leq_divRL ?mulnK //.
rewrite dvdn_mull ?big1 // ⇒ [[i /= ?] /andP[/eqP <- /negPf]].
by rewrite eq_sym dvdn_eq inordK ?ltnS ?leq_div2r // ⇒ →.
Qed.
Lemma logn_count_dvd p n : prime p → logn p n = \sum_(1 ≤ k < n) (p ^ k %| n).
Proof.
rewrite big_add1 ⇒ p_prime; case: n ⇒ [|n]; first by rewrite logn0 big_geq.
rewrite big_mkord -big_mkcond (eq_bigl _ _ (fun _ ⇒ pfactor_dvdn _ _ _)) //=.
by rewrite big_ord_narrow ?sum1_card ?card_ord // -ltnS ltn_logl.
Qed.
Definition trunc_log p n :=
let fix loop n k :=
if k is k'.+1 then if p ≤ n then (loop (n %/ p) k').+1 else 0 else 0
in loop n n.
Lemma trunc_log_bounds p n :
1 < p → 0 < n → let k := trunc_log p n in p ^ k ≤ n < p ^ k.+1.
Proof.
rewrite {+}/trunc_log ⇒ p_gt1; have p_gt0 := ltnW p_gt1.
elim: n {-2 5}n (leqnn n) ⇒ [|m IHm] [|n] //=; rewrite ltnS ⇒ le_n_m _.
have [le_p_n | // ] := leqP p _; rewrite 2!expnSr -leq_divRL -?ltn_divLR //.
by apply: IHm; rewrite ?divn_gt0 // -ltnS (leq_trans (ltn_Pdiv _ _)).
Qed.
Lemma trunc_log_ltn p n : 1 < p → n < p ^ (trunc_log p n).+1.
Proof.
have [-> | n_gt0] := posnP n; first by move⇒ /ltnW; rewrite expn_gt0.
by case/trunc_log_bounds/(_ n_gt0)/andP.
Qed.
Lemma trunc_logP p n : 1 < p → 0 < n → p ^ trunc_log p n ≤ n.
Proof. by move⇒ p_gt1 /(trunc_log_bounds p_gt1)/andP[]. Qed.
Lemma trunc_log_max p k j : 1 < p → p ^ j ≤ k → j ≤ trunc_log p k.
Proof.
move⇒ p_gt1 le_pj_k; rewrite -ltnS -(@ltn_exp2l p) //.
exact: leq_ltn_trans (trunc_log_ltn _ _).
Qed.
Canonical nat_pred_pred := Eval hnf in [predType of nat_pred].
Coercion nat_pred_of_nat (p : nat) : nat_pred := pred1 p.
Section NatPreds.
Variables (n : nat) (pi : nat_pred).
Definition negn : nat_pred := [predC pi].
Definition pnat : pred nat := fun m ⇒ (m > 0) && all (mem pi) (primes m).
Definition partn := \prod_(0 ≤ p < n.+1 | p \in pi) p ^ logn p n.
End NatPreds.
Notation "pi ^'" := (negn pi) (at level 2, format "pi ^'") : nat_scope.
Notation "pi .-nat" := (pnat pi) (at level 2, format "pi .-nat") : nat_scope.
Notation "n `_ pi" := (partn n pi) : nat_scope.
Section PnatTheory.
Implicit Types (n p : nat) (pi rho : nat_pred).
Lemma negnK pi : pi^'^' =i pi.
Proof. move⇒ p; exact: negbK. Qed.
Lemma eq_negn pi1 pi2 : pi1 =i pi2 → pi1^' =i pi2^'.
Proof. by move⇒ eq_pi n; rewrite 3!inE /= eq_pi. Qed.
Lemma eq_piP m n : \pi(m) =i \pi(n) ↔ \pi(m) = \pi(n).
Proof.
rewrite /pi_of; have eqs := eq_sorted_irr ltn_trans ltnn.
by split⇒ [|-> //]; move/(eqs _ _ (sorted_primes m) (sorted_primes n)) →.
Qed.
Lemma part_gt0 pi n : 0 < n`_pi.
Proof. exact: prodn_gt0. Qed.
Hint Resolve part_gt0.
Lemma sub_in_partn pi1 pi2 n :
{in \pi(n), {subset pi1 ≤ pi2}} → n`_pi1 %| n`_pi2.
Proof.
move⇒ pi12; rewrite ![n`__]big_mkcond /=.
apply (big_ind2 (fun m1 m2 ⇒ m1 %| m2)) ⇒ // [*|p _]; first exact: dvdn_mul.
rewrite lognE -mem_primes; case: ifP ⇒ pi1p; last exact: dvd1n.
by case: ifP ⇒ pr_p; [rewrite pi12 | rewrite if_same].
Qed.
Lemma eq_in_partn pi1 pi2 n : {in \pi(n), pi1 =i pi2} → n`_pi1 = n`_pi2.
Proof.
by move⇒ pi12; apply/eqP; rewrite eqn_dvd ?sub_in_partn // ⇒ p /pi12→.
Qed.
Lemma eq_partn pi1 pi2 n : pi1 =i pi2 → n`_pi1 = n`_pi2.
Proof. by move⇒ pi12; apply: eq_in_partn ⇒ p _. Qed.
Lemma partnNK pi n : n`_pi^'^' = n`_pi.
Proof. by apply: eq_partn; exact: negnK. Qed.
Lemma widen_partn m pi n :
n ≤ m → n`_pi = \prod_(0 ≤ p < m.+1 | p \in pi) p ^ logn p n.
Proof.
move⇒ le_n_m; rewrite big_mkcond /=.
rewrite [n`_pi](big_nat_widen _ _ m.+1) // big_mkcond /=.
apply: eq_bigr ⇒ p _; rewrite ltnS lognE.
by case: and3P ⇒ [[_ n_gt0 p_dv_n]|]; rewrite ?if_same // andbC dvdn_leq.
Qed.
Lemma partn0 pi : 0`_pi = 1.
Proof. by apply: big1_seq ⇒ [] [|n]; rewrite andbC. Qed.
Lemma partn1 pi : 1`_pi = 1.
Proof. by apply: big1_seq ⇒ [] [|[|n]]; rewrite andbC. Qed.
Lemma partnM pi m n : m > 0 → n > 0 → (m × n)`_pi = m`_pi × n`_pi.
Proof.
have le_pmul m' n': m' > 0 → n' ≤ m' × n' by move/prednK <-; exact: leq_addr.
move⇒ mpos npos; rewrite !(@widen_partn (n × m)) 3?(le_pmul, mulnC) //.
rewrite !big_mkord -big_split; apply: eq_bigr ⇒ p _ /=.
by rewrite lognM // expnD.
Qed.
Lemma partnX pi m n : (m ^ n)`_pi = m`_pi ^ n.
Proof.
elim: n ⇒ [|n IHn]; first exact: partn1.
rewrite expnS; case: (posnP m) ⇒ [->|m_gt0]; first by rewrite partn0 exp1n.
by rewrite expnS partnM ?IHn // expn_gt0 m_gt0.
Qed.
Lemma partn_dvd pi m n : n > 0 → m %| n → m`_pi %| n`_pi.
Proof.
move⇒ n_gt0 dvmn; case/dvdnP: dvmn n_gt0 ⇒ q ->{n}.
by rewrite muln_gt0 ⇒ /andP[q_gt0 m_gt0]; rewrite partnM ?dvdn_mull.
Qed.
Lemma p_part p n : n`_p = p ^ logn p n.
Proof.
case (posnP (logn p n)) ⇒ [log0 |].
by rewrite log0 [n`_p]big1_seq // ⇒ q; case/andP; move/eqnP->; rewrite log0.
rewrite logn_gt0 mem_primes; case/and3P⇒ _ n_gt0 dv_p_n.
have le_p_n: p < n.+1 by rewrite ltnS dvdn_leq.
by rewrite [n`_p]big_mkord (big_pred1 (Ordinal le_p_n)).
Qed.
Lemma p_part_eq1 p n : (n`_p == 1) = (p \notin \pi(n)).
Proof.
rewrite mem_primes p_part lognE; case: and3P ⇒ // [[p_pr _ _]].
by rewrite -dvdn1 pfactor_dvdn // logn1.
Qed.
Lemma p_part_gt1 p n : (n`_p > 1) = (p \in \pi(n)).
Proof. by rewrite ltn_neqAle part_gt0 andbT eq_sym p_part_eq1 negbK. Qed.
Lemma primes_part pi n : primes n`_pi = filter (mem pi) (primes n).
Proof.
have ltnT := ltn_trans.
case: (posnP n) ⇒ [-> | n_gt0]; first by rewrite partn0.
apply: (eq_sorted_irr ltnT ltnn); rewrite ?(sorted_primes, sorted_filter) //.
move⇒ p; rewrite mem_filter /= !mem_primes n_gt0 part_gt0 /=.
apply/andP/and3P⇒ [[p_pr] | [pi_p p_pr dv_p_n]].
rewrite /partn; apply big_ind ⇒ [|n1 n2 IHn1 IHn2|q pi_q].
- by rewrite dvdn1; case: eqP p_pr ⇒ // →.
- by rewrite Euclid_dvdM //; case/orP.
rewrite -{1}(expn1 p) pfactor_dvdn // lognX muln_gt0.
rewrite logn_gt0 mem_primes n_gt0 - andbA /=; case/and3P⇒ pr_q dv_q_n.
by rewrite logn_prime //; case: eqP ⇒ // →.
have le_p_n: p < n.+1 by rewrite ltnS dvdn_leq.
rewrite [n`_pi]big_mkord (bigD1 (Ordinal le_p_n)) //= dvdn_mulr //.
by rewrite lognE p_pr n_gt0 dv_p_n expnS dvdn_mulr.
Qed.
Lemma filter_pi_of n m : n < m → filter \pi(n) (index_iota 0 m) = primes n.
Proof.
move⇒ lt_n_m; have ltnT := ltn_trans; apply: (eq_sorted_irr ltnT ltnn).
- by rewrite sorted_filter // iota_ltn_sorted.
- exact: sorted_primes.
move⇒ p; rewrite mem_filter mem_index_iota /= mem_primes; case: and3P ⇒ //.
case⇒ _ n_gt0 dv_p_n; apply: leq_ltn_trans lt_n_m; exact: dvdn_leq.
Qed.
Lemma partn_pi n : n > 0 → n`_\pi(n) = n.
Proof.
move⇒ n_gt0; rewrite {3}(prod_prime_decomp n_gt0) prime_decompE big_map.
by rewrite -[n`__]big_filter filter_pi_of.
Qed.
Lemma partnT n : n > 0 → n`_predT = n.
Proof.
move⇒ n_gt0; rewrite -{2}(partn_pi n_gt0) {2}/partn big_mkcond /=.
by apply: eq_bigr ⇒ p _; rewrite -logn_gt0; case: (logn p _).
Qed.
Lemma partnC pi n : n > 0 → n`_pi × n`_pi^' = n.
Proof.
move⇒ n_gt0; rewrite -{3}(partnT n_gt0) /partn.
do 2!rewrite mulnC big_mkcond /=; rewrite -big_split; apply: eq_bigr ⇒ p _ /=.
by rewrite mulnC inE /=; case: (p \in pi); rewrite /= (muln1, mul1n).
Qed.
Lemma dvdn_part pi n : n`_pi %| n.
Proof. by case: n ⇒ // n; rewrite -{2}[n.+1](@partnC pi) // dvdn_mulr. Qed.
Lemma logn_part p m : logn p m`_p = logn p m.
Proof.
case p_pr: (prime p); first by rewrite p_part pfactorK.
by rewrite lognE (lognE p m) p_pr.
Qed.
Lemma partn_lcm pi m n : m > 0 → n > 0 → (lcmn m n)`_pi = lcmn m`_pi n`_pi.
Proof.
move⇒ m_gt0 n_gt0; have p_gt0: lcmn m n > 0 by rewrite lcmn_gt0 m_gt0.
apply/eqP; rewrite eqn_dvd dvdn_lcm !partn_dvd ?dvdn_lcml ?dvdn_lcmr //.
rewrite -(dvdn_pmul2r (part_gt0 pi^' (lcmn m n))) partnC // dvdn_lcm !andbT.
rewrite -{1}(partnC pi m_gt0) andbC -{1}(partnC pi n_gt0).
by rewrite !dvdn_mul ?partn_dvd ?dvdn_lcml ?dvdn_lcmr.
Qed.
Lemma partn_gcd pi m n : m > 0 → n > 0 → (gcdn m n)`_pi = gcdn m`_pi n`_pi.
Proof.
move⇒ m_gt0 n_gt0; have p_gt0: gcdn m n > 0 by rewrite gcdn_gt0 m_gt0.
apply/eqP; rewrite eqn_dvd dvdn_gcd !partn_dvd ?dvdn_gcdl ?dvdn_gcdr //=.
rewrite -(dvdn_pmul2r (part_gt0 pi^' (gcdn m n))) partnC // dvdn_gcd.
rewrite -{3}(partnC pi m_gt0) andbC -{3}(partnC pi n_gt0).
by rewrite !dvdn_mul ?partn_dvd ?dvdn_gcdl ?dvdn_gcdr.
Qed.
Lemma partn_biglcm (I : finType) (P : pred I) F pi :
(∀ i, P i → F i > 0) →
(\big[lcmn/1%N]_(i | P i) F i)`_pi = \big[lcmn/1%N]_(i | P i) (F i)`_pi.
Proof.
move⇒ F_gt0; set m := \big[lcmn/1%N]_(i | P i) F i.
have m_gt0: 0 < m by apply big_ind ⇒ // p q p_gt0; rewrite lcmn_gt0 p_gt0.
apply/eqP; rewrite eqn_dvd andbC; apply/andP; split.
by apply/dvdn_biglcmP⇒ i Pi; rewrite partn_dvd // (@biglcmn_sup _ i).
rewrite -(dvdn_pmul2r (part_gt0 pi^' m)) partnC //.
apply/dvdn_biglcmP⇒ i Pi; rewrite -(partnC pi (F_gt0 i Pi)) dvdn_mul //.
by rewrite (@biglcmn_sup _ i).
by rewrite partn_dvd // (@biglcmn_sup _ i).
Qed.
Lemma partn_biggcd (I : finType) (P : pred I) F pi :
#|SimplPred P| > 0 → (∀ i, P i → F i > 0) →
(\big[gcdn/0]_(i | P i) F i)`_pi = \big[gcdn/0]_(i | P i) (F i)`_pi.
Proof.
move⇒ ntP F_gt0; set d := \big[gcdn/0]_(i | P i) F i.
have d_gt0: 0 < d.
case/card_gt0P: ntP ⇒ i /= Pi; have:= F_gt0 i Pi.
rewrite !lt0n -!dvd0n; apply: contra ⇒ dv0d.
by rewrite (dvdn_trans dv0d) // (@biggcdn_inf _ i).
apply/eqP; rewrite eqn_dvd; apply/andP; split.
by apply/dvdn_biggcdP⇒ i Pi; rewrite partn_dvd ?F_gt0 // (@biggcdn_inf _ i).
rewrite -(dvdn_pmul2r (part_gt0 pi^' d)) partnC //.
apply/dvdn_biggcdP⇒ i Pi; rewrite -(partnC pi (F_gt0 i Pi)) dvdn_mul //.
by rewrite (@biggcdn_inf _ i).
by rewrite partn_dvd ?F_gt0 // (@biggcdn_inf _ i).
Qed.
Lemma sub_in_pnat pi rho n :
{in \pi(n), {subset pi ≤ rho}} → pi.-nat n → rho.-nat n.
Proof.
rewrite /pnat ⇒ subpi /andP[-> pi_n].
apply/allP⇒ p pr_p; apply: subpi ⇒ //; exact: (allP pi_n).
Qed.
Lemma eq_in_pnat pi rho n : {in \pi(n), pi =i rho} → pi.-nat n = rho.-nat n.
Proof. by move⇒ eqpi; apply/idP/idP; apply: sub_in_pnat ⇒ p /eqpi→. Qed.
Lemma eq_pnat pi rho n : pi =i rho → pi.-nat n = rho.-nat n.
Proof. by move⇒ eqpi; apply: eq_in_pnat ⇒ p _. Qed.
Lemma pnatNK pi n : pi^'^'.-nat n = pi.-nat n.
Proof. exact: eq_pnat (negnK pi). Qed.
Lemma pnatI pi rho n : [predI pi & rho].-nat n = pi.-nat n && rho.-nat n.
Proof. by rewrite /pnat andbCA all_predI !andbA andbb. Qed.
Lemma pnat_mul pi m n : pi.-nat (m × n) = pi.-nat m && pi.-nat n.
Proof.
rewrite /pnat muln_gt0 andbCA -andbA andbCA.
case: posnP ⇒ // n_gt0; case: posnP ⇒ //= m_gt0.
apply/allP/andP⇒ [pi_mn | [pi_m pi_n] p].
by split; apply/allP⇒ p m_p; apply: pi_mn; rewrite primes_mul // m_p ?orbT.
rewrite primes_mul // ⇒ /orP[]; [exact: (allP pi_m) | exact: (allP pi_n)].
Qed.
Lemma pnat_exp pi m n : pi.-nat (m ^ n) = pi.-nat m || (n == 0).
Proof. by case: n ⇒ [|n]; rewrite orbC // /pnat expn_gt0 orbC primes_exp. Qed.
Lemma part_pnat pi n : pi.-nat n`_pi.
Proof.
rewrite /pnat primes_part part_gt0.
by apply/allP⇒ p; rewrite mem_filter ⇒ /andP[].
Qed.
Lemma pnatE pi p : prime p → pi.-nat p = (p \in pi).
Proof. by move⇒ pr_p; rewrite /pnat prime_gt0 ?primes_prime //= andbT. Qed.
Lemma pnat_id p : prime p → p.-nat p.
Proof. by move⇒ pr_p; rewrite pnatE ?inE /=. Qed.
Lemma coprime_pi' m n : m > 0 → n > 0 → coprime m n = \pi(m)^'.-nat n.
Proof.
by move⇒ m_gt0 n_gt0; rewrite /pnat n_gt0 all_predC coprime_has_primes.
Qed.
Lemma pnat_pi n : n > 0 → \pi(n).-nat n.
Proof. rewrite /pnat ⇒ ->; exact/allP. Qed.
Lemma pi_of_dvd m n : m %| n → n > 0 → {subset \pi(m) ≤ \pi(n)}.
Proof.
move⇒ m_dv_n n_gt0 p; rewrite !mem_primes n_gt0 ⇒ /and3P[-> _ p_dv_m].
exact: dvdn_trans p_dv_m m_dv_n.
Qed.
Lemma pi_ofM m n : m > 0 → n > 0 → \pi(m × n) =i [predU \pi(m) & \pi(n)].
Proof. move⇒ m_gt0 n_gt0 p; exact: primes_mul. Qed.
Lemma pi_of_part pi n : n > 0 → \pi(n`_pi) =i [predI \pi(n) & pi].
Proof. by move⇒ n_gt0 p; rewrite /pi_of primes_part mem_filter andbC. Qed.
Lemma pi_of_exp p n : n > 0 → \pi(p ^ n) = \pi(p).
Proof. by move⇒ n_gt0; rewrite /pi_of primes_exp. Qed.
Lemma pi_of_prime p : prime p → \pi(p) =i (p : nat_pred).
Proof. by move⇒ pr_p q; rewrite /pi_of primes_prime // mem_seq1. Qed.
Lemma p'natEpi p n : n > 0 → p^'.-nat n = (p \notin \pi(n)).
Proof. by case: n ⇒ // n _; rewrite /pnat all_predC has_pred1. Qed.
Lemma p'natE p n : prime p → p^'.-nat n = ~~ (p %| n).
Proof.
case: n ⇒ [|n] p_pr; first by case: p p_pr.
by rewrite p'natEpi // mem_primes p_pr.
Qed.
Lemma pnatPpi pi n p : pi.-nat n → p \in \pi(n) → p \in pi.
Proof. by case/andP⇒ _ /allP; exact. Qed.
Lemma pnat_dvd m n pi : m %| n → pi.-nat n → pi.-nat m.
Proof. by case/dvdnP⇒ q ->; rewrite pnat_mul; case/andP. Qed.
Lemma pnat_div m n pi : m %| n → pi.-nat n → pi.-nat (n %/ m).
Proof.
case/dvdnP⇒ q ->; rewrite pnat_mul andbC ⇒ /andP[].
by case: m ⇒ // m _; rewrite mulnK.
Qed.
Lemma pnat_coprime pi m n : pi.-nat m → pi^'.-nat n → coprime m n.
Proof.
case/andP⇒ m_gt0 pi_m /andP[n_gt0 pi'_n].
rewrite coprime_has_primes //; apply/hasPn⇒ p /(allP pi'_n).
apply: contra; exact: allP.
Qed.
Lemma p'nat_coprime pi m n : pi^'.-nat m → pi.-nat n → coprime m n.
Proof. by move⇒ pi'm pi_n; rewrite (pnat_coprime pi'm) ?pnatNK. Qed.
Lemma sub_pnat_coprime pi rho m n :
{subset rho ≤ pi^'} → pi.-nat m → rho.-nat n → coprime m n.
Proof.
by move⇒ pi'rho pi_m; move/(sub_in_pnat (in1W pi'rho)); exact: pnat_coprime.
Qed.
Lemma coprime_partC pi m n : coprime m`_pi n`_pi^'.
Proof. by apply: (@pnat_coprime pi); exact: part_pnat. Qed.
Lemma pnat_1 pi n : pi.-nat n → pi^'.-nat n → n = 1.
Proof.
by move⇒ pi_n pi'_n; rewrite -(eqnP (pnat_coprime pi_n pi'_n)) gcdnn.
Qed.
Lemma part_pnat_id pi n : pi.-nat n → n`_pi = n.
Proof.
case/andP⇒ n_gt0 pi_n.
rewrite -{2}(partnT n_gt0) /partn big_mkcond; apply: eq_bigr⇒ p _.
case: (posnP (logn p n)) ⇒ [-> |]; first by rewrite if_same.
by rewrite logn_gt0 ⇒ /(allP pi_n)/= →.
Qed.
Lemma part_p'nat pi n : pi^'.-nat n → n`_pi = 1.
Proof.
case/andP⇒ n_gt0 pi'_n; apply: big1_seq ⇒ p /andP[pi_p _].
case: (posnP (logn p n)) ⇒ [-> //|].
by rewrite logn_gt0; move/(allP pi'_n); case/negP.
Qed.
Lemma partn_eq1 pi n : n > 0 → (n`_pi == 1) = pi^'.-nat n.
Proof.
move⇒ n_gt0; apply/eqP/idP⇒ [pi_n_1|]; last exact: part_p'nat.
by rewrite -(partnC pi n_gt0) pi_n_1 mul1n part_pnat.
Qed.
Lemma pnatP pi n :
n > 0 → reflect (∀ p, prime p → p %| n → p \in pi) (pi.-nat n).
Proof.
move⇒ n_gt0; rewrite /pnat n_gt0.
apply: (iffP allP) ⇒ /= pi_n p ⇒ [pr_p p_n|].
by rewrite pi_n // mem_primes pr_p n_gt0.
by rewrite mem_primes n_gt0 /=; case/andP; move: p.
Qed.
Lemma pi_pnat pi p n : p.-nat n → p \in pi → pi.-nat n.
Proof.
move⇒ p_n pi_p; have [n_gt0 _] := andP p_n.
by apply/pnatP⇒ // q q_pr /(pnatP _ n_gt0 p_n _ q_pr)/eqnP→.
Qed.
Lemma p_natP p n : p.-nat n → {k | n = p ^ k}.
Proof. by move⇒ p_n; ∃ (logn p n); rewrite -p_part part_pnat_id. Qed.
Lemma pi'_p'nat pi p n : pi^'.-nat n → p \in pi → p^'.-nat n.
Proof.
move⇒ pi'n pi_p; apply: sub_in_pnat pi'n ⇒ q _.
by apply: contraNneq ⇒ →.
Qed.
Lemma pi_p'nat p pi n : pi.-nat n → p \in pi^' → p^'.-nat n.
Proof. by move⇒ pi_n; apply: pi'_p'nat; rewrite pnatNK. Qed.
Lemma partn_part pi rho n : {subset pi ≤ rho} → n`_rho`_pi = n`_pi.
Proof.
move⇒ pi_sub_rho; have [->|n_gt0] := posnP n; first by rewrite !partn0 partn1.
rewrite -{2}(partnC rho n_gt0) partnM //.
suffices: pi^'.-nat n`_rho^' by move/part_p'nat->; rewrite muln1.
apply: sub_in_pnat (part_pnat _ _) ⇒ q _; apply: contra; exact: pi_sub_rho.
Qed.
Lemma partnI pi rho n : n`_[predI pi & rho] = n`_pi`_rho.
Proof.
rewrite -(@partnC [predI pi & rho] _`_rho) //.
symmetry; rewrite 2?partn_part; try by move⇒ p /andP [].
rewrite mulnC part_p'nat ?mul1n // pnatNK pnatI part_pnat andbT.
exact: pnat_dvd (dvdn_part _ _) (part_pnat _ _).
Qed.
Lemma odd_2'nat n : odd n = 2^'.-nat n.
Proof. by case: n ⇒ // n; rewrite p'natE // dvdn2 negbK. Qed.
End PnatTheory.
Hint Resolve part_gt0.
Lemma divisors_correct n : n > 0 →
[/\ uniq (divisors n), sorted leq (divisors n)
& ∀ d, (d \in divisors n) = (d %| n)].
Proof.
move/prod_prime_decomp⇒ def_n; rewrite {4}def_n {def_n}.
have: all prime (primes n) by apply/allP⇒ p; rewrite mem_primes; case/andP.
have:= primes_uniq n; rewrite /primes /divisors; move/prime_decomp: n.
elim⇒ [|[p e] pd] /=; first by split⇒ // d; rewrite big_nil dvdn1 mem_seq1.
rewrite big_cons /=; move: (foldr _ _ pd) ⇒ divs.
move⇒ IHpd /andP[npd_p Upd] /andP[pr_p pr_pd].
have lt0p: 0 < p by exact: prime_gt0.
have {IHpd Upd}[Udivs Odivs mem_divs] := IHpd Upd pr_pd.
have ndivs_p m: p × m \notin divs.
suffices: p \notin divs; rewrite !mem_divs.
by apply: contra ⇒ /dvdnP[n ->]; rewrite mulnCA dvdn_mulr.
have ndv_p_1: ~~(p %| 1) by rewrite dvdn1 neq_ltn orbC prime_gt1.
rewrite big_seq; elim/big_ind: _ ⇒ [//|u v npu npv|[q f] /= pd_qf].
by rewrite Euclid_dvdM //; apply/norP.
elim: (f) ⇒ // f'; rewrite expnS Euclid_dvdM // orbC negb_or ⇒ → {f'}/=.
have pd_q: q \in unzip1 pd by apply/mapP; ∃ (q, f).
by apply: contra npd_p; rewrite dvdn_prime2 // ?(allP pr_pd) // ⇒ /eqP→.
elim: e ⇒ [|e] /=; first by split⇒ // d; rewrite mul1n.
have Tmulp_inj: injective (NatTrec.mul p).
by move⇒ u v /eqP; rewrite !natTrecE eqn_pmul2l // ⇒ /eqP.
move: (iter e _ _) ⇒ divs' [Udivs' Odivs' mem_divs']; split⇒ [||d].
- rewrite merge_uniq cat_uniq map_inj_uniq // Udivs Udivs' andbT /=.
apply/hasP⇒ [[d dv_d /mapP[d' _ def_d]]].
by case/idPn: dv_d; rewrite def_d natTrecE.
- rewrite (merge_sorted leq_total) //; case: (divs') Odivs' ⇒ //= d ds.
rewrite (@map_path _ _ _ _ leq xpred0) ?has_pred0 // ⇒ u v _.
by rewrite !natTrecE leq_pmul2l.
rewrite mem_merge mem_cat; case dv_d_p: (p %| d).
case/dvdnP: dv_d_p ⇒ d' ->{d}; rewrite mulnC (negbTE (ndivs_p d')) orbF.
rewrite expnS -mulnA dvdn_pmul2l // -mem_divs'.
by rewrite -(mem_map Tmulp_inj divs') natTrecE.
case pdiv_d: (_ \in _).
by case/mapP: pdiv_d dv_d_p ⇒ d' _ ->; rewrite natTrecE dvdn_mulr.
rewrite mem_divs Gauss_dvdr // coprime_sym.
by rewrite coprime_expl ?prime_coprime ?dv_d_p.
Qed.
Lemma sorted_divisors n : sorted leq (divisors n).
Proof. by case: (posnP n) ⇒ [-> | /divisors_correct[]]. Qed.
Lemma divisors_uniq n : uniq (divisors n).
Proof. by case: (posnP n) ⇒ [-> | /divisors_correct[]]. Qed.
Lemma sorted_divisors_ltn n : sorted ltn (divisors n).
Proof. by rewrite ltn_sorted_uniq_leq divisors_uniq sorted_divisors. Qed.
Lemma dvdn_divisors d m : 0 < m → (d %| m) = (d \in divisors m).
Proof. by case/divisors_correct. Qed.
Lemma divisor1 n : 1 \in divisors n.
Proof. by case: n ⇒ // n; rewrite -dvdn_divisors // dvd1n. Qed.
Lemma divisors_id n : 0 < n → n \in divisors n.
Proof. by move/dvdn_divisors <-. Qed.
Lemma dvdn_sum d I r (K : pred I) F :
(∀ i, K i → d %| F i) → d %| \sum_(i <- r | K i) F i.
Proof. move⇒ dF; elim/big_ind: _ ⇒ //; exact: dvdn_add. Qed.
Lemma dvdn_partP n m : 0 < n →
reflect (∀ p, p \in \pi(n) → n`_p %| m) (n %| m).
Proof.
move⇒ n_gt0; apply: (iffP idP) ⇒ n_dvd_m ⇒ [p _|].
apply: dvdn_trans n_dvd_m; exact: dvdn_part.
have [-> // | m_gt0] := posnP m.
rewrite -(partnT n_gt0) -(partnT m_gt0).
rewrite !(@widen_partn (m + n)) ?leq_addl ?leq_addr // /in_mem /=.
elim/big_ind2: _ ⇒ // [* | q _]; first exact: dvdn_mul.
have [-> // | ] := posnP (logn q n); rewrite logn_gt0 ⇒ q_n.
have pr_q: prime q by move: q_n; rewrite mem_primes; case/andP.
by have:= n_dvd_m q q_n; rewrite p_part !pfactor_dvdn // pfactorK.
Qed.
Lemma modn_partP n a b : 0 < n →
reflect (∀ p : nat, p \in \pi(n) → a = b %[mod n`_p]) (a == b %[mod n]).
Proof.
move⇒ n_gt0; wlog le_b_a: a b / b ≤ a.
move⇒ IH; case: (leqP b a) ⇒ [|/ltnW] /IH {IH}// IH.
by rewrite eq_sym; apply: (iffP IH) ⇒ eqab p; move/eqab.
rewrite eqn_mod_dvd //; apply: (iffP (dvdn_partP _ n_gt0)) ⇒ eqab p /eqab;
by rewrite -eqn_mod_dvd // ⇒ /eqP.
Qed.
Lemma totientE n :
n > 0 → totient n = \prod_(p <- primes n) (p.-1 × p ^ (logn p n).-1).
Proof.
move⇒ n_gt0; rewrite /totient n_gt0 prime_decompE unlock.
by elim: (primes n) ⇒ //= [p pr ->]; rewrite !natTrecE.
Qed.
Lemma totient_gt0 n : (0 < totient n) = (0 < n).
Proof.
case: n ⇒ // n; rewrite totientE // big_seq_cond prodn_cond_gt0 // ⇒ p.
by rewrite mem_primes muln_gt0 expn_gt0; case: p ⇒ [|[|]].
Qed.
Lemma totient_pfactor p e :
prime p → e > 0 → totient (p ^ e) = p.-1 × p ^ e.-1.
Proof.
move⇒ p_pr e_gt0; rewrite totientE ?expn_gt0 ?prime_gt0 //.
by rewrite primes_exp // primes_prime // unlock /= muln1 pfactorK.
Qed.
Lemma totient_coprime m n :
coprime m n → totient (m × n) = totient m × totient n.
Proof.
move⇒ co_mn; have [-> //| m_gt0] := posnP m.
have [->|n_gt0] := posnP n; first by rewrite !muln0.
rewrite !totientE ?muln_gt0 ?m_gt0 //.
have /(eq_big_perm _)->: perm_eq (primes (m × n)) (primes m ++ primes n).
apply: uniq_perm_eq ⇒ [||p]; first exact: primes_uniq.
by rewrite cat_uniq !primes_uniq -coprime_has_primes // co_mn.
by rewrite mem_cat primes_mul.
rewrite big_cat /= !big_seq.
congr (_ × _); apply: eq_bigr ⇒ p; rewrite mem_primes ⇒ /and3P[_ _ dvp].
rewrite (mulnC m) logn_Gauss //; move: co_mn.
by rewrite -(divnK dvp) coprime_mull ⇒ /andP[].
rewrite logn_Gauss //; move: co_mn.
by rewrite coprime_sym -(divnK dvp) coprime_mull ⇒ /andP[].
Qed.
Lemma totient_count_coprime n : totient n = \sum_(0 ≤ d < n) coprime n d.
Proof.
elim: {n}_.+1 {-2}n (ltnSn n) ⇒ // m IHm n; rewrite ltnS ⇒ le_n_m.
case: (leqP n 1) ⇒ [|lt1n]; first by rewrite unlock; case: (n) ⇒ [|[]].
pose p := pdiv n; have p_pr: prime p by exact: pdiv_prime.
have p1 := prime_gt1 p_pr; have p0 := ltnW p1.
pose np := n`_p; pose np' := n`_p^'.
have co_npp': coprime np np' by rewrite coprime_partC.
have [n0 np0 np'0]: [/\ n > 0, np > 0 & np' > 0] by rewrite ltnW ?part_gt0.
have def_n: n = np × np' by rewrite partnC.
have lnp0: 0 < logn p n by rewrite lognE p_pr n0 pdiv_dvd.
pose in_mod k (k0 : k > 0) d := Ordinal (ltn_pmod d k0).
rewrite {1}def_n totient_coprime // {IHm}(IHm np') ?big_mkord; last first.
apply: leq_trans le_n_m; rewrite def_n ltn_Pmull //.
by rewrite /np p_part -(expn0 p) ltn_exp2l.
have ->: totient np = #|[pred d : 'I_np | coprime np d]|.
rewrite {1}[np]p_part totient_pfactor //=; set q := p ^ _.
apply: (@addnI (1 × q)); rewrite -mulnDl [1 + _]prednK // mul1n.
have def_np: np = p × q by rewrite -expnS prednK // -p_part.
pose mulp := [fun d : 'I_q ⇒ in_mod _ np0 (p × d)].
rewrite -def_np -{1}[np]card_ord -(cardC (mem (codom mulp))).
rewrite card_in_image ⇒ [|[d1 ltd1] [d2 ltd2] /= _ _ []]; last first.
move/eqP; rewrite def_np -!muln_modr ?modn_small //.
by rewrite eqn_pmul2l // ⇒ eq_op12; exact/eqP.
rewrite card_ord; congr (q + _); apply: eq_card ⇒ d /=.
rewrite !inE [np in coprime np _]p_part coprime_pexpl ?prime_coprime //.
congr (~~ _); apply/codomP/idP⇒ [[d' → /=] | /dvdnP[r def_d]].
by rewrite def_np -muln_modr // dvdn_mulr.
do [rewrite mulnC; case: d ⇒ d ltd /=] in def_d ×.
have ltr: r < q by rewrite -(ltn_pmul2l p0) -def_np -def_d.
by ∃ (Ordinal ltr); apply: val_inj; rewrite /= -def_d modn_small.
pose h (d : 'I_n) := (in_mod _ np0 d, in_mod _ np'0 d).
pose h' (d : 'I_np × 'I_np') := in_mod _ n0 (chinese np np' d.1 d.2).
rewrite -!big_mkcond -sum_nat_const pair_big (reindex_onto h h') ⇒ [|[d d'] _].
apply: eq_bigl ⇒ [[d ltd] /=]; rewrite !inE /= -val_eqE /= andbC.
rewrite !coprime_modr def_n -chinese_mod // -coprime_mull -def_n.
by rewrite modn_small ?eqxx.
apply/eqP; rewrite /eq_op /= /eq_op /= !modn_dvdm ?dvdn_part //.
by rewrite chinese_modl // chinese_modr // !modn_small ?eqxx ?ltn_ord.
Qed.
Require Import div bigop.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Fixpoint edivn2 q r := if r is r'.+2 then edivn2 q.+1 r' else (q, r).
Lemma edivn2P n : edivn_spec n 2 (edivn2 0 n).
Proof.
rewrite -[n]odd_double_half addnC -{1}[n./2]addn0 -{1}mul2n mulnC.
elim: n./2 {1 4}0 ⇒ [|r IHr] q; first by case (odd n) ⇒ /=.
rewrite addSnnS; exact: IHr.
Qed.
Fixpoint elogn2 e q r {struct q} :=
match q, r with
| 0, _ | _, 0 ⇒ (e, q)
| q'.+1, 1 ⇒ elogn2 e.+1 q' q'
| q'.+1, r'.+2 ⇒ elogn2 e q' r'
end.
CoInductive elogn2_spec n : nat × nat → Type :=
Elogn2Spec e m of n = 2 ^ e × m.*2.+1 : elogn2_spec n (e, m).
Lemma elogn2P n : elogn2_spec n.+1 (elogn2 0 n n).
Proof.
rewrite -{1}[n.+1]mul1n -[1]/(2 ^ 0) -{1}(addKn n n) addnn.
elim: n {1 4 6}n {2 3}0 (leqnn n) ⇒ [|q IHq] [|[|r]] e //=; last first.
by move/ltnW; exact: IHq.
clear 1; rewrite subn1 -[_.-1.+1]doubleS -mul2n mulnA -expnSr.
rewrite -{1}(addKn q q) addnn; exact: IHq.
Qed.
Definition ifnz T n (x y : T) := if n is 0 then y else x.
CoInductive ifnz_spec T n (x y : T) : T → Type :=
| IfnzPos of n > 0 : ifnz_spec n x y x
| IfnzZero of n = 0 : ifnz_spec n x y y.
Lemma ifnzP T n (x y : T) : ifnz_spec n x y (ifnz n x y).
Proof. by case: n ⇒ [|n]; [right | left]. Qed.
Definition NumFactor (f : nat × nat) := ([Num of f.1], f.2).
Definition pfactor p e := p ^ e.
Definition cons_pfactor (p e : nat) pd := ifnz e ((p, e) :: pd) pd.
Notation Local "p ^? e :: pd" := (cons_pfactor p e pd)
(at level 30, e at level 30, pd at level 60) : nat_scope.
Section prime_decomp.
Import NatTrec.
Fixpoint prime_decomp_rec m k a b c e :=
let p := k.*2.+1 in
if a is a'.+1 then
if b - (ifnz e 1 k - c) is b'.+1 then
[rec m, k, a', b', ifnz c c.-1 (ifnz e p.-2 1), e] else
if (b == 0) && (c == 0) then
let b' := k + a' in [rec b'.*2.+3, k, a', b', k.-1, e.+1] else
let bc' := ifnz e (ifnz b (k, 0) (edivn2 0 c)) (b, c) in
p ^? e :: ifnz a' [rec m, k.+1, a'.-1, bc'.1 + a', bc'.2, 0] [:: (m, 1)]
else if (b == 0) && (c == 0) then [:: (p, e.+2)] else p ^? e :: [:: (m, 1)]
where "[ 'rec' m , k , a , b , c , e ]" := (prime_decomp_rec m k a b c e).
Definition prime_decomp n :=
let: (e2, m2) := elogn2 0 n.-1 n.-1 in
if m2 < 2 then 2 ^? e2 :: 3 ^? m2 :: [::] else
let: (a, bc) := edivn m2.-2 3 in
let: (b, c) := edivn (2 - bc) 2 in
2 ^? e2 :: [rec m2.*2.+1, 1, a, b, c, 0].
Definition add_divisors f divs :=
let: (p, e) := f in
let add1 divs' := merge leq (map (NatTrec.mul p) divs') divs in
iter e add1 divs.
Definition add_totient_factor f m := let: (p, e) := f in p.-1 × p ^ e.-1 × m.
End prime_decomp.
Definition primes n := unzip1 (prime_decomp n).
Definition prime p := if prime_decomp p is [:: (_ , 1)] then true else false.
Definition nat_pred := simpl_pred nat.
Definition pi_unwrapped_arg := nat.
Definition pi_wrapped_arg := wrapped nat.
Coercion unwrap_pi_arg (wa : pi_wrapped_arg) : pi_unwrapped_arg := unwrap wa.
Coercion pi_arg_of_nat (n : nat) := Wrap n : pi_wrapped_arg.
Coercion pi_arg_of_fin_pred T pT (A : @fin_pred_sort T pT) : pi_wrapped_arg :=
Wrap #|A|.
Definition pi_of (n : pi_unwrapped_arg) : nat_pred := [pred p in primes n].
Notation "\pi ( n )" := (pi_of n)
(at level 2, format "\pi ( n )") : nat_scope.
Notation "\p 'i' ( A )" := \pi(#|A|)
(at level 2, format "\p 'i' ( A )") : nat_scope.
Definition pdiv n := head 1 (primes n).
Definition max_pdiv n := last 1 (primes n).
Definition divisors n := foldr add_divisors [:: 1] (prime_decomp n).
Definition totient n := foldr add_totient_factor (n > 0) (prime_decomp n).
Lemma prime_decomp_correct :
let pd_val pd := \prod_(f <- pd) pfactor f.1 f.2 in
let lb_dvd q m := ~~ has [pred d | d %| m] (index_iota 2 q) in
let pf_ok f := lb_dvd f.1 f.1 && (0 < f.2) in
let pd_ord q pd := path ltn q (unzip1 pd) in
let pd_ok q n pd := [/\ n = pd_val pd, all pf_ok pd & pd_ord q pd] in
∀ n, n > 0 → pd_ok 1 n (prime_decomp n).
Proof.
rewrite unlock ⇒ pd_val lb_dvd pf_ok pd_ord pd_ok.
have leq_pd_ok m p q pd: q ≤ p → pd_ok p m pd → pd_ok q m pd.
rewrite /pd_ok /pd_ord; case: pd ⇒ [|[r _] pd] //= leqp [<- ->].
by case/andP⇒ /(leq_trans _)->.
have apd_ok m e q p pd: lb_dvd p p || (e == 0) → q < p →
pd_ok p m pd → pd_ok q (p ^ e × m) (p ^? e :: pd).
- case: e ⇒ [|e]; rewrite orbC /= ⇒ pr_p ltqp.
rewrite mul1n; apply: leq_pd_ok; exact: ltnW.
by rewrite /pd_ok /pd_ord /pf_ok /= pr_p ltqp ⇒ [[<- → ->]].
case⇒ // n _; rewrite /prime_decomp.
case: elogn2P ⇒ e2 m2 → {n}; case: m2 ⇒ [|[|abc]]; try exact: apd_ok.
rewrite [_.-2]/= !ltnS ltn0 natTrecE; case: edivnP ⇒ a bc ->{abc}.
case: edivnP ⇒ b c def_bc /= ltc2 ltbc3; apply: (apd_ok) ⇒ //.
move def_m: _.*2.+1 ⇒ m; set k := {2}1; rewrite -[2]/k.*2; set e := 0.
pose p := k.*2.+1; rewrite -{1}[m]mul1n -[1]/(p ^ e)%N.
have{def_m bc def_bc ltc2 ltbc3}:
let kb := (ifnz e k 1).*2 in
[&& k > 0, p < m, lb_dvd p m, c < kb & lb_dvd p p || (e == 0)]
∧ m + (b × kb + c).*2 = p ^ 2 + (a × p).*2.
- rewrite -{-2}def_m; split⇒ //=; last first.
by rewrite -def_bc addSn -doubleD 2!addSn -addnA subnKC // addnC.
rewrite ltc2 /lb_dvd /index_iota /= dvdn2 -def_m.
by rewrite [_.+2]lock /= odd_double.
move: {2}a.+1 (ltnSn a) ⇒ n; clearbody k e.
elim: n ⇒ // n IHn in a k p m b c e *; rewrite ltnS ⇒ le_a_n [].
set kb := _.*2; set d := _ + c ⇒ /and5P[lt0k ltpm leppm ltc pr_p def_m].
have def_k1: k.-1.+1 = k := ltn_predK lt0k.
have def_kb1: kb.-1.+1 = kb by rewrite /kb -def_k1; case e.
have eq_bc_0: (b == 0) && (c == 0) = (d == 0).
by rewrite addn_eq0 muln_eq0 orbC -def_kb1.
have lt1p: 1 < p by rewrite ltnS double_gt0.
have co_p_2: coprime p 2 by rewrite /coprime gcdnC gcdnE modn2 /= odd_double.
have if_d0: d = 0 → [/\ m = (p + a.*2) × p, lb_dvd p p & lb_dvd p (p + a.*2)].
move⇒ d0; have{d0 def_m} def_m: m = (p + a.*2) × p.
by rewrite d0 addn0 -mulnn -!mul2n mulnA -mulnDl in def_m ×.
split⇒ //; apply/hasPn⇒ r /(hasPn leppm); apply: contra ⇒ /= dv_r.
by rewrite def_m dvdn_mull.
by rewrite def_m dvdn_mulr.
case def_a: a ⇒ [|a'] /= in le_a_n *; rewrite !natTrecE -/p {}eq_bc_0.
case: d if_d0 def_m ⇒ [[//| def_m {pr_p}pr_p pr_m'] _ | d _ def_m] /=.
rewrite def_m def_a addn0 mulnA -2!expnSr.
by split; rewrite /pd_ord /pf_ok /= ?muln1 ?pr_p ?leqnn.
apply: apd_ok; rewrite // /pd_ok /= /pfactor expn1 muln1 /pd_ord /= ltpm.
rewrite /pf_ok !andbT /=; split⇒ //; apply: contra leppm.
case/hasP⇒ r /=; rewrite mem_index_iota ⇒ /andP[lt1r ltrm] dvrm; apply/hasP.
have [ltrp | lepr] := ltnP r p.
by ∃ r; rewrite // mem_index_iota lt1r.
case/dvdnP: dvrm ⇒ q def_q; ∃ q; last by rewrite def_q /= dvdn_mulr.
rewrite mem_index_iota -(ltn_pmul2r (ltnW lt1r)) -def_q mul1n ltrm.
move: def_m; rewrite def_a addn0 -(@ltn_pmul2r p) // mulnn ⇒ <-.
apply: (@leq_ltn_trans m); first by rewrite def_q leq_mul.
by rewrite -addn1 leq_add2l.
have def_k2: k.*2 = ifnz e 1 k × kb.
by rewrite /kb; case: (e) ⇒ [|e']; rewrite (mul1n, muln2).
case def_b': (b - _) ⇒ [|b']; last first.
have ->: ifnz e k.*2.-1 1 = kb.-1 by rewrite /kb; case e.
apply: IHn ⇒ {n le_a_n}//; rewrite -/p -/kb; split⇒ //.
rewrite lt0k ltpm leppm pr_p andbT /=.
by case: ifnzP; [move/ltn_predK->; exact: ltnW | rewrite def_kb1].
apply: (@addIn p.*2).
rewrite -2!addnA -!doubleD -addnA -mulSnr -def_a -def_m /d.
have ->: b × kb = b' × kb + (k.*2 - c × kb + kb).
rewrite addnCA addnC -mulSnr -def_b' def_k2 -mulnBl -mulnDl subnK //.
by rewrite ltnW // -subn_gt0 def_b'.
rewrite -addnA; congr (_ + (_ + _).*2).
case: (c) ltc; first by rewrite -addSnnS def_kb1 subn0 addn0 addnC.
rewrite /kb; case e ⇒ [[] // _ | e' c' _] /=; last first.
by rewrite subnDA subnn addnC addSnnS.
by rewrite mul1n -doubleB -doubleD subn1 !addn1 def_k1.
have ltdp: d < p.
move/eqP: def_b'; rewrite subn_eq0 -(@leq_pmul2r kb); last first.
by rewrite -def_kb1.
rewrite mulnBl -def_k2 ltnS -(leq_add2r c); move/leq_trans; apply.
have{ltc} ltc: c < k.*2.
by apply: (leq_trans ltc); rewrite leq_double /kb; case e.
rewrite -{2}(subnK (ltnW ltc)) leq_add2r leq_sub2l //.
by rewrite -def_kb1 mulnS leq_addr.
case def_d: d if_d0 ⇒ [|d'] ⇒ [[//|{def_m ltdp pr_p} def_m pr_p pr_m'] | _].
rewrite eqxx -doubleS -addnS -def_a doubleD -addSn -/p def_m.
rewrite mulnCA mulnC -expnSr.
apply: IHn ⇒ {n le_a_n}//; rewrite -/p -/kb; split.
rewrite lt0k -addn1 leq_add2l {1}def_a pr_m' pr_p /= def_k1 -addnn.
by rewrite leq_addr.
rewrite -addnA -doubleD addnCA def_a addSnnS def_k1 -(addnC k) -mulnSr.
rewrite -[_.*2.+1]/p mulnDl doubleD addnA -mul2n mulnA mul2n -mulSn.
by rewrite -/p mulnn.
have next_pm: lb_dvd p.+2 m.
rewrite /lb_dvd /index_iota 2!subSS subn0 -(subnK lt1p) iota_add.
rewrite has_cat; apply/norP; split⇒ //=; rewrite orbF subnKC // orbC.
apply/norP; split; apply/dvdnP⇒ [[q def_q]].
case/hasP: leppm; ∃ 2; first by rewrite /p -(subnKC lt0k).
by rewrite /= def_q dvdn_mull // dvdn2 /= odd_double.
move/(congr1 (dvdn p)): def_m; rewrite -mulnn -!mul2n mulnA -mulnDl.
rewrite dvdn_mull // dvdn_addr; last by rewrite def_q dvdn_mull.
case/dvdnP⇒ r; rewrite mul2n ⇒ def_r; move: ltdp (congr1 odd def_r).
rewrite odd_double -ltn_double {1}def_r -mul2n ltn_pmul2r //.
by case: r def_r ⇒ [|[|[]]] //; rewrite def_d // mul1n /= odd_double.
apply: apd_ok ⇒ //; case: a' def_a le_a_n ⇒ [|a'] def_a ⇒ [_ | lta] /=.
rewrite /pd_ok /= /pfactor expn1 muln1 /pd_ord /= ltpm /pf_ok !andbT /=.
split⇒ //; apply: contra next_pm.
case/hasP⇒ q; rewrite mem_index_iota ⇒ /andP[lt1q ltqm] dvqm; apply/hasP.
have [ltqp | lepq] := ltnP q p.+2.
by ∃ q; rewrite // mem_index_iota lt1q.
case/dvdnP: dvqm ⇒ r def_r; ∃ r; last by rewrite def_r /= dvdn_mulr.
rewrite mem_index_iota -(ltn_pmul2r (ltnW lt1q)) -def_r mul1n ltqm /=.
rewrite -(@ltn_pmul2l p.+2) //; apply: (@leq_ltn_trans m).
by rewrite def_r mulnC leq_mul.
rewrite -addn2 mulnn sqrnD mul2n muln2 -addnn addnCA -addnA addnCA addnA.
by rewrite def_a mul1n in def_m; rewrite -def_m addnS -addnA ltnS leq_addr.
set bc := ifnz _ _ _; apply: leq_pd_ok (leqnSn _) _.
rewrite -doubleS -{1}[m]mul1n -[1]/(k.+1.*2.+1 ^ 0)%N.
apply: IHn; first exact: ltnW.
rewrite doubleS -/p [ifnz 0 _ _]/=; do 2?split ⇒ //.
rewrite orbT next_pm /= -(leq_add2r d.*2) def_m 2!addSnnS -doubleS leq_add.
- move: ltc; rewrite /kb {}/bc andbT; case e ⇒ //= e' _; case: ifnzP ⇒ //.
by case: edivn2P.
- by rewrite -{1}[p]muln1 -mulnn ltn_pmul2l.
by rewrite leq_double def_a mulSn (leq_trans ltdp) ?leq_addr.
rewrite mulnDl !muln2 -addnA addnCA doubleD addnCA.
rewrite (_ : _ + bc.2 = d); last first.
rewrite /d {}/bc /kb -muln2.
case: (e) (b) def_b' ⇒ //= _ []; first by case: edivn2P.
by case c; do 2?case; rewrite // mul1n /= muln2.
rewrite def_m 3!doubleS addnC -(addn2 p) sqrnD mul2n muln2 -3!addnA.
congr (_ + _); rewrite 4!addnS -!doubleD; congr _.*2.+2.+2.
by rewrite def_a -add2n mulnDl -addnA -muln2 -mulnDr mul2n.
Qed.
Lemma primePn n :
reflect (n < 2 ∨ exists2 d, 1 < d < n & d %| n) (~~ prime n).
Proof.
rewrite /prime; case: n ⇒ [|[|p2]]; try by do 2!left.
case: (@prime_decomp_correct p2.+2) ⇒ //; rewrite unlock.
case: prime_decomp ⇒ [|[q [|[|e]]] pd] //=; last first; last by rewrite andbF.
rewrite {1}/pfactor 2!expnS -!mulnA /=.
case: (_ ^ _ × _) ⇒ [|u → _ /andP[lt1q _]]; first by rewrite !muln0.
left; right; ∃ q; last by rewrite dvdn_mulr.
have lt0q := ltnW lt1q; rewrite lt1q -{1}[q]muln1 ltn_pmul2l //.
by rewrite -[2]muln1 leq_mul.
rewrite {1}/pfactor expn1; case: pd ⇒ [|[r e] pd] /=; last first.
case: e ⇒ [|e] /=; first by rewrite !andbF.
rewrite {1}/pfactor expnS -mulnA.
case: (_ ^ _ × _) ⇒ [|u → _ /and3P[lt1q ltqr _]]; first by rewrite !muln0.
left; right; ∃ q; last by rewrite dvdn_mulr.
by rewrite lt1q -{1}[q]mul1n ltn_mul // -[q.+1]muln1 leq_mul.
rewrite muln1 !andbT ⇒ def_q pr_q lt1q; right⇒ [[]] // [d].
by rewrite def_q -mem_index_iota ⇒ in_d_2q dv_d_q; case/hasP: pr_q; ∃ d.
Qed.
Lemma primeP p :
reflect (p > 1 ∧ ∀ d, d %| p → xpred2 1 p d) (prime p).
Proof.
rewrite -[prime p]negbK; have [npr_p | pr_p] := primePn p.
right⇒ [[lt1p pr_p]]; case: npr_p ⇒ [|[d n1pd]].
by rewrite ltnNge lt1p.
by move/pr_p⇒ /orP[] /eqP def_d; rewrite def_d ltnn ?andbF in n1pd.
have [lep1 | lt1p] := leqP; first by case: pr_p; left.
left; split⇒ // d dv_d_p; apply/norP⇒ [[nd1 ndp]]; case: pr_p; right.
∃ d; rewrite // andbC 2!ltn_neqAle ndp eq_sym nd1.
by have lt0p := ltnW lt1p; rewrite dvdn_leq // (dvdn_gt0 lt0p).
Qed.
Lemma prime_nt_dvdP d p : prime p → d != 1 → reflect (d = p) (d %| p).
Proof.
case/primeP⇒ _ min_p d_neq1; apply: (iffP idP) ⇒ [/min_p|-> //].
by rewrite (negPf d_neq1) /= ⇒ /eqP.
Qed.
Implicit Arguments primeP [p].
Implicit Arguments primePn [n].
Prenex Implicits primePn primeP.
Lemma prime_gt1 p : prime p → 1 < p.
Proof. by case/primeP. Qed.
Lemma prime_gt0 p : prime p → 0 < p.
Proof. by move/prime_gt1; exact: ltnW. Qed.
Hint Resolve prime_gt1 prime_gt0.
Lemma prod_prime_decomp n :
n > 0 → n = \prod_(f <- prime_decomp n) f.1 ^ f.2.
Proof. by case/prime_decomp_correct. Qed.
Lemma even_prime p : prime p → p = 2 ∨ odd p.
Proof.
move⇒ pr_p; case odd_p: (odd p); [by right | left].
have: 2 %| p by rewrite dvdn2 odd_p.
by case/primeP: pr_p ⇒ _ dv_p /dv_p/(2 =P p).
Qed.
Lemma prime_oddPn p : prime p → reflect (p = 2) (~~ odd p).
Proof.
by move⇒ p_pr; apply: (iffP idP) ⇒ [|-> //]; case/even_prime: p_pr ⇒ →.
Qed.
Lemma odd_prime_gt2 p : odd p → prime p → p > 2.
Proof. by move⇒ odd_p /prime_gt1; apply: odd_gt2. Qed.
Lemma mem_prime_decomp n p e :
(p, e) \in prime_decomp n → [/\ prime p, e > 0 & p ^ e %| n].
Proof.
case: (posnP n) ⇒ [-> //| /prime_decomp_correct[def_n mem_pd ord_pd pd_pe]].
have /andP[pr_p ->] := allP mem_pd _ pd_pe; split⇒ //; last first.
case/splitPr: pd_pe def_n ⇒ pd1 pd2 →.
by rewrite big_cat big_cons /= mulnCA dvdn_mulr.
have lt1p: 1 < p.
apply: (allP (order_path_min ltn_trans ord_pd)).
by apply/mapP; ∃ (p, e).
apply/primeP; split⇒ // d dv_d_p; apply/norP⇒ [[nd1 ndp]].
case/hasP: pr_p; ∃ d ⇒ //.
rewrite mem_index_iota andbC 2!ltn_neqAle ndp eq_sym nd1.
by have lt0p := ltnW lt1p; rewrite dvdn_leq // (dvdn_gt0 lt0p).
Qed.
Lemma prime_coprime p m : prime p → coprime p m = ~~ (p %| m).
Proof.
case/primeP⇒ p_gt1 p_pr; apply/eqP/negP⇒ [d1 | ndv_pm].
case/dvdnP⇒ k def_m; rewrite -(addn0 m) def_m gcdnMDl gcdn0 in d1.
by rewrite d1 in p_gt1.
by apply: gcdn_def ⇒ // d /p_pr /orP[] /eqP→.
Qed.
Lemma dvdn_prime2 p q : prime p → prime q → (p %| q) = (p == q).
Proof.
move⇒ pr_p pr_q; apply: negb_inj.
by rewrite eqn_dvd negb_and -!prime_coprime // coprime_sym orbb.
Qed.
Lemma Euclid_dvdM m n p : prime p → (p %| m × n) = (p %| m) || (p %| n).
Proof.
move⇒ pr_p; case dv_pm: (p %| m); first exact: dvdn_mulr.
by rewrite Gauss_dvdr // prime_coprime // dv_pm.
Qed.
Lemma Euclid_dvd1 p : prime p → (p %| 1) = false.
Proof. by rewrite dvdn1; case: eqP ⇒ // →. Qed.
Lemma Euclid_dvdX m n p : prime p → (p %| m ^ n) = (p %| m) && (n > 0).
Proof.
case: n ⇒ [|n] pr_p; first by rewrite andbF Euclid_dvd1.
by apply: (inv_inj negbK); rewrite !andbT -!prime_coprime // coprime_pexpr.
Qed.
Lemma mem_primes p n : (p \in primes n) = [&& prime p, n > 0 & p %| n].
Proof.
rewrite andbCA; case: posnP ⇒ [-> // | /= n_gt0].
apply/mapP/andP⇒ [[[q e]]|[pr_p]] /=.
case/mem_prime_decomp⇒ pr_q e_gt0; case/dvdnP⇒ u → → {p}.
by rewrite -(prednK e_gt0) expnS mulnCA dvdn_mulr.
rewrite {1}(prod_prime_decomp n_gt0) big_seq.
apply big_ind ⇒ [| u v IHu IHv | [q e] /= mem_qe dv_p_qe].
- by rewrite Euclid_dvd1.
- by rewrite Euclid_dvdM // ⇒ /orP[].
∃ (q, e) ⇒ //=; case/mem_prime_decomp: mem_qe ⇒ pr_q _ _.
by rewrite Euclid_dvdX // dvdn_prime2 // in dv_p_qe; case: eqP dv_p_qe.
Qed.
Lemma sorted_primes n : sorted ltn (primes n).
Proof.
by case: (posnP n) ⇒ [-> // | /prime_decomp_correct[_ _]]; exact: path_sorted.
Qed.
Lemma eq_primes m n : (primes m =i primes n) ↔ (primes m = primes n).
Proof.
split⇒ [eqpr| → //].
by apply: (eq_sorted_irr ltn_trans ltnn); rewrite ?sorted_primes.
Qed.
Lemma primes_uniq n : uniq (primes n).
Proof. exact: (sorted_uniq ltn_trans ltnn (sorted_primes n)). Qed.
Lemma pi_pdiv n : (pdiv n \in \pi(n)) = (n > 1).
Proof.
case: n ⇒ [|[|n]] //; rewrite /pdiv !inE /primes.
have:= prod_prime_decomp (ltn0Sn n.+1); rewrite unlock.
by case: prime_decomp ⇒ //= pf pd _; rewrite mem_head.
Qed.
Lemma pdiv_prime n : 1 < n → prime (pdiv n).
Proof. by rewrite -pi_pdiv mem_primes; case/and3P. Qed.
Lemma pdiv_dvd n : pdiv n %| n.
Proof.
by case: n (pi_pdiv n) ⇒ [|[|n]] //; rewrite mem_primes⇒ /and3P[].
Qed.
Lemma pi_max_pdiv n : (max_pdiv n \in \pi(n)) = (n > 1).
Proof.
rewrite !inE -pi_pdiv /max_pdiv /pdiv !inE.
by case: (primes n) ⇒ //= p ps; rewrite mem_head mem_last.
Qed.
Lemma max_pdiv_prime n : n > 1 → prime (max_pdiv n).
Proof. by rewrite -pi_max_pdiv mem_primes ⇒ /andP[]. Qed.
Lemma max_pdiv_dvd n : max_pdiv n %| n.
Proof.
by case: n (pi_max_pdiv n) ⇒ [|[|n]] //; rewrite mem_primes ⇒ /andP[].
Qed.
Lemma pdiv_leq n : 0 < n → pdiv n ≤ n.
Proof. by move⇒ n_gt0; rewrite dvdn_leq // pdiv_dvd. Qed.
Lemma max_pdiv_leq n : 0 < n → max_pdiv n ≤ n.
Proof. by move⇒ n_gt0; rewrite dvdn_leq // max_pdiv_dvd. Qed.
Lemma pdiv_gt0 n : 0 < pdiv n.
Proof. by case: n ⇒ [|[|n]] //; rewrite prime_gt0 ?pdiv_prime. Qed.
Lemma max_pdiv_gt0 n : 0 < max_pdiv n.
Proof. by case: n ⇒ [|[|n]] //; rewrite prime_gt0 ?max_pdiv_prime. Qed.
Hint Resolve pdiv_gt0 max_pdiv_gt0.
Lemma pdiv_min_dvd m d : 1 < d → d %| m → pdiv m ≤ d.
Proof.
move⇒ lt1d dv_d_m; case: (posnP m) ⇒ [->|mpos]; first exact: ltnW.
rewrite /pdiv; apply: leq_trans (pdiv_leq (ltnW lt1d)).
have: pdiv d \in primes m.
by rewrite mem_primes mpos pdiv_prime // (dvdn_trans (pdiv_dvd d)).
case: (primes m) (sorted_primes m) ⇒ //= p pm ord_pm.
rewrite inE ⇒ /predU1P[-> //|].
move/(allP (order_path_min ltn_trans ord_pm)); exact: ltnW.
Qed.
Lemma max_pdiv_max n p : p \in \pi(n) → p ≤ max_pdiv n.
Proof.
rewrite /max_pdiv !inE ⇒ n_p.
case/splitPr: n_p (sorted_primes n) ⇒ p1 p2; rewrite last_cat -cat_rcons /=.
rewrite headI /= cat_path -(last_cons 0) -headI last_rcons; case/andP⇒ _.
move/(order_path_min ltn_trans); case/lastP: p2 ⇒ //= p2 q.
by rewrite all_rcons last_rcons ltn_neqAle -andbA ⇒ /and3P[].
Qed.
Lemma ltn_pdiv2_prime n : 0 < n → n < pdiv n ^ 2 → prime n.
Proof.
case def_n: n ⇒ [|[|n']] // _; rewrite -def_n ⇒ lt_n_p2.
suffices ->: n = pdiv n by rewrite pdiv_prime ?def_n.
apply/eqP; rewrite eqn_leq leqNgt andbC pdiv_leq; last by rewrite def_n.
move: lt_n_p2; rewrite ltnNge; apply: contra ⇒ lt_pm_m.
case/dvdnP: (pdiv_dvd n) ⇒ q def_q.
rewrite {2}def_q -mulnn leq_pmul2r // pdiv_min_dvd //.
by rewrite -[pdiv n]mul1n {2}def_q ltn_pmul2r in lt_pm_m.
by rewrite def_q dvdn_mulr.
Qed.
Lemma primePns n :
reflect (n < 2 ∨ ∃ p, [/\ prime p, p ^ 2 ≤ n & p %| n]) (~~ prime n).
Proof.
apply: (iffP idP) ⇒ [npr_p|]; last first.
case⇒ [|[p [pr_p le_p2_n dv_p_n]]]; first by case: n ⇒ [|[]].
apply/negP⇒ pr_n; move: dv_p_n le_p2_n; rewrite dvdn_prime2 //; move/eqP→.
by rewrite leqNgt -{1}[n]muln1 -mulnn ltn_pmul2l ?prime_gt1 ?prime_gt0.
case: leqP ⇒ [lt1p|]; [right | by left].
∃ (pdiv n); rewrite pdiv_dvd pdiv_prime //; split⇒ //.
by case: leqP npr_p ⇒ //; move/ltn_pdiv2_prime->; auto.
Qed.
Implicit Arguments primePns [n].
Prenex Implicits primePns.
Lemma pdivP n : n > 1 → {p | prime p & p %| n}.
Proof. by move⇒ lt1n; ∃ (pdiv n); rewrite ?pdiv_dvd ?pdiv_prime. Qed.
Lemma primes_mul m n p : m > 0 → n > 0 →
(p \in primes (m × n)) = (p \in primes m) || (p \in primes n).
Proof.
move⇒ m_gt0 n_gt0; rewrite !mem_primes muln_gt0 m_gt0 n_gt0.
by case pr_p: (prime p); rewrite // Euclid_dvdM.
Qed.
Lemma primes_exp m n : n > 0 → primes (m ^ n) = primes m.
Proof.
case: n ⇒ // n _; rewrite expnS; case: (posnP m) ⇒ [-> //| m_gt0].
apply/eq_primes ⇒ /= p; elim: n ⇒ [|n IHn]; first by rewrite muln1.
by rewrite primes_mul ?(expn_gt0, expnS, IHn, orbb, m_gt0).
Qed.
Lemma primes_prime p : prime p → primes p = [::p].
Proof.
move⇒ pr_p; apply: (eq_sorted_irr ltn_trans ltnn) ⇒ // [|q].
exact: sorted_primes.
rewrite mem_seq1 mem_primes prime_gt0 //=.
by apply/andP/idP⇒ [[pr_q q_p] | /eqP→ //]; rewrite -dvdn_prime2.
Qed.
Lemma coprime_has_primes m n : m > 0 → n > 0 →
coprime m n = ~~ has (mem (primes m)) (primes n).
Proof.
move⇒ m_gt0 n_gt0; apply/eqnP/hasPn⇒ [mn1 p | no_p_mn].
rewrite /= !mem_primes m_gt0 n_gt0 /= ⇒ /andP[pr_p p_n].
have:= prime_gt1 pr_p; rewrite pr_p ltnNge -mn1 /=; apply: contra ⇒ p_m.
by rewrite dvdn_leq ?gcdn_gt0 ?m_gt0 // dvdn_gcd ?p_m.
case: (ltngtP (gcdn m n) 1) ⇒ //; first by rewrite ltnNge gcdn_gt0 ?m_gt0.
move/pdiv_prime; set p := pdiv _ ⇒ pr_p.
move/implyP: (no_p_mn p); rewrite /= !mem_primes m_gt0 n_gt0 pr_p /=.
by rewrite !(dvdn_trans (pdiv_dvd _)) // (dvdn_gcdl, dvdn_gcdr).
Qed.
Lemma pdiv_id p : prime p → pdiv p = p.
Proof. by move⇒ p_pr; rewrite /pdiv primes_prime. Qed.
Lemma pdiv_pfactor p k : prime p → pdiv (p ^ k.+1) = p.
Proof. by move⇒ p_pr; rewrite /pdiv primes_exp ?primes_prime. Qed.
Fixpoint logn_rec d m r :=
match r, edivn m d with
| r'.+1, (_.+1 as m', 0) ⇒ (logn_rec d m' r').+1
| _, _ ⇒ 0
end.
Definition logn p m := if prime p then logn_rec p m m else 0.
Lemma lognE p m :
logn p m = if [&& prime p, 0 < m & p %| m] then (logn p (m %/ p)).+1 else 0.
Proof.
rewrite /logn /dvdn; case p_pr: (prime p) ⇒ //.
rewrite /divn modn_def; case def_m: {2 3}m ⇒ [|m'] //=.
case: edivnP def_m ⇒ [[|q] [|r] → _] // def_m; congr _.+1; rewrite [_.1]/=.
have{m def_m}: q < m'.
by rewrite -ltnS -def_m addn0 mulnC -{1}[q.+1]mul1n ltn_pmul2r // prime_gt1.
elim: {m' q}_.+1 {-2}m' q.+1 (ltnSn m') (ltn0Sn q) ⇒ // s IHs.
case⇒ [[]|r] //= m; rewrite ltnS ⇒ lt_rs m_gt0 le_mr.
rewrite -{3}[m]prednK //=; case: edivnP ⇒ [[|q] [|_] def_q _] //.
have{def_q} lt_qm': q < m.-1.
by rewrite -[q.+1]muln1 -ltnS prednK // def_q addn0 ltn_pmul2l // prime_gt1.
have{le_mr} le_m'r: m.-1 ≤ r by rewrite -ltnS prednK.
by rewrite (IHs r) ?(IHs m.-1) // ?(leq_trans lt_qm', leq_trans _ lt_rs).
Qed.
Lemma logn_gt0 p n : (0 < logn p n) = (p \in primes n).
Proof. by rewrite lognE -mem_primes; case: {+}(p \in _). Qed.
Lemma ltn_log0 p n : n < p → logn p n = 0.
Proof. by case: n ⇒ [|n] ltnp; rewrite lognE ?andbF // gtnNdvd ?andbF. Qed.
Lemma logn0 p : logn p 0 = 0.
Proof. by rewrite /logn if_same. Qed.
Lemma logn1 p : logn p 1 = 0.
Proof. by rewrite lognE dvdn1 /= andbC; case: eqP ⇒ // →. Qed.
Lemma pfactor_gt0 p n : 0 < p ^ logn p n.
Proof. by rewrite expn_gt0 lognE; case: (posnP p) ⇒ // →. Qed.
Hint Resolve pfactor_gt0.
Lemma pfactor_dvdn p n m : prime p → m > 0 → (p ^ n %| m) = (n ≤ logn p m).
Proof.
move⇒ p_pr; elim: n m ⇒ [|n IHn] m m_gt0; first exact: dvd1n.
rewrite lognE p_pr m_gt0 /=; case dv_pm: (p %| m); last first.
apply/dvdnP⇒ [] [/= q def_m].
by rewrite def_m expnS mulnCA dvdn_mulr in dv_pm.
case/dvdnP: dv_pm m_gt0 ⇒ q ->{m}; rewrite muln_gt0 ⇒ /andP[p_gt0 q_gt0].
by rewrite expnSr dvdn_pmul2r // mulnK // IHn.
Qed.
Lemma pfactor_dvdnn p n : p ^ logn p n %| n.
Proof.
case: n ⇒ // n; case pr_p: (prime p); first by rewrite pfactor_dvdn.
by rewrite lognE pr_p dvd1n.
Qed.
Lemma logn_prime p q : prime q → logn p q = (p == q).
Proof.
move⇒ pr_q; have q_gt0 := prime_gt0 pr_q; rewrite lognE q_gt0 /=.
case pr_p: (prime p); last by case: eqP pr_p pr_q ⇒ // → →.
by rewrite dvdn_prime2 //; case: eqP ⇒ // ->; rewrite divnn q_gt0 logn1.
Qed.
Lemma pfactor_coprime p n :
prime p → n > 0 → {m | coprime p m & n = m × p ^ logn p n}.
Proof.
move⇒ p_pr n_gt0; set k := logn p n.
have dv_pk_n: p ^ k %| n by rewrite pfactor_dvdn.
∃ (n %/ p ^ k); last by rewrite divnK.
rewrite prime_coprime // -(@dvdn_pmul2r (p ^ k)) ?expn_gt0 ?prime_gt0 //.
by rewrite -expnS divnK // pfactor_dvdn // ltnn.
Qed.
Lemma pfactorK p n : prime p → logn p (p ^ n) = n.
Proof.
move⇒ p_pr; have pn_gt0: p ^ n > 0 by rewrite expn_gt0 prime_gt0.
apply/eqP; rewrite eqn_leq -pfactor_dvdn // dvdnn andbT.
by rewrite -(leq_exp2l _ _ (prime_gt1 p_pr)) dvdn_leq // pfactor_dvdn.
Qed.
Lemma pfactorKpdiv p n : prime p → logn (pdiv (p ^ n)) (p ^ n) = n.
Proof. by case: n ⇒ // n p_pr; rewrite pdiv_pfactor ?pfactorK. Qed.
Lemma dvdn_leq_log p m n : 0 < n → m %| n → logn p m ≤ logn p n.
Proof.
move⇒ n_gt0 dv_m_n; have m_gt0 := dvdn_gt0 n_gt0 dv_m_n.
case p_pr: (prime p); last by do 2!rewrite lognE p_pr /=.
by rewrite -pfactor_dvdn //; apply: dvdn_trans dv_m_n; rewrite pfactor_dvdn.
Qed.
Lemma ltn_logl p n : 0 < n → logn p n < n.
Proof.
move⇒ n_gt0; have [p_gt1 | p_le1] := boolP (1 < p).
by rewrite (leq_trans (ltn_expl _ p_gt1)) // dvdn_leq ?pfactor_dvdnn.
by rewrite lognE (contraNF (@prime_gt1 _)).
Qed.
Lemma logn_Gauss p m n : coprime p m → logn p (m × n) = logn p n.
Proof.
move⇒ co_pm; case p_pr: (prime p); last by rewrite /logn p_pr.
have [-> | n_gt0] := posnP n; first by rewrite muln0.
have [m0 | m_gt0] := posnP m; first by rewrite m0 prime_coprime ?dvdn0 in co_pm.
have mn_gt0: m × n > 0 by rewrite muln_gt0 m_gt0.
apply/eqP; rewrite eqn_leq andbC dvdn_leq_log ?dvdn_mull //.
set k := logn p _; have: p ^ k %| m × n by rewrite pfactor_dvdn.
by rewrite Gauss_dvdr ?coprime_expl // -pfactor_dvdn.
Qed.
Lemma lognM p m n : 0 < m → 0 < n → logn p (m × n) = logn p m + logn p n.
Proof.
case p_pr: (prime p); last by rewrite /logn p_pr.
have xlp := pfactor_coprime p_pr.
case/xlp⇒ m' co_m' def_m /xlp[n' co_n' def_n] {xlp}.
by rewrite {1}def_m {1}def_n mulnCA -mulnA -expnD !logn_Gauss // pfactorK.
Qed.
Lemma lognX p m n : logn p (m ^ n) = n × logn p m.
Proof.
case p_pr: (prime p); last by rewrite /logn p_pr muln0.
elim: n ⇒ [|n IHn]; first by rewrite logn1.
have [->|m_gt0] := posnP m; first by rewrite exp0n // lognE andbF muln0.
by rewrite expnS lognM ?IHn // expn_gt0 m_gt0.
Qed.
Lemma logn_div p m n : m %| n → logn p (n %/ m) = logn p n - logn p m.
Proof.
rewrite dvdn_eq ⇒ /eqP def_n.
case: (posnP n) ⇒ [-> |]; first by rewrite div0n logn0.
by rewrite -{1 3}def_n muln_gt0 ⇒ /andP[q_gt0 m_gt0]; rewrite lognM ?addnK.
Qed.
Lemma dvdn_pfactor p d n : prime p →
reflect (exists2 m, m ≤ n & d = p ^ m) (d %| p ^ n).
Proof.
move⇒ p_pr; have pn_gt0: p ^ n > 0 by rewrite expn_gt0 prime_gt0.
apply: (iffP idP) ⇒ [dv_d_pn|[m le_m_n ->]]; last first.
by rewrite -(subnK le_m_n) expnD dvdn_mull.
∃ (logn p d); first by rewrite -(pfactorK n p_pr) dvdn_leq_log.
have d_gt0: d > 0 by exact: dvdn_gt0 dv_d_pn.
case: (pfactor_coprime p_pr d_gt0) ⇒ q co_p_q def_d.
rewrite {1}def_d ((q =P 1) _) ?mul1n // -dvdn1.
suff: q %| p ^ n × 1 by rewrite Gauss_dvdr // coprime_sym coprime_expl.
by rewrite muln1 (dvdn_trans _ dv_d_pn) // def_d dvdn_mulr.
Qed.
Lemma prime_decompE n : prime_decomp n = [seq (p, logn p n) | p <- primes n].
Proof.
case: n ⇒ // n; pose f0 := (0, 0); rewrite -map_comp.
apply: (@eq_from_nth _ f0) ⇒ [|i lt_i_n]; first by rewrite size_map.
rewrite (nth_map f0) //; case def_f: (nth _ _ i) ⇒ [p e] /=.
congr (_, _); rewrite [n.+1]prod_prime_decomp //.
have: (p, e) \in prime_decomp n.+1 by rewrite -def_f mem_nth.
case/mem_prime_decomp⇒ pr_p _ _.
rewrite (big_nth f0) big_mkord (bigD1 (Ordinal lt_i_n)) //=.
rewrite def_f mulnC logn_Gauss ?pfactorK //.
apply big_ind ⇒ [|m1 m2 com1 com2| [j ltj] /=]; first exact: coprimen1.
by rewrite coprime_mulr com1.
rewrite -val_eqE /= ⇒ nji; case def_j: (nth _ _ j) ⇒ [q e1] /=.
have: (q, e1) \in prime_decomp n.+1 by rewrite -def_j mem_nth.
case/mem_prime_decomp⇒ pr_q e1_gt0 _; rewrite coprime_pexpr //.
rewrite prime_coprime // dvdn_prime2 //; apply: contra nji ⇒ eq_pq.
rewrite -(nth_uniq 0 _ _ (primes_uniq n.+1)) ?size_map //=.
by rewrite !(nth_map f0) // def_f def_j /= eq_sym.
Qed.
Lemma divn_count_dvd d n : n %/ d = \sum_(1 ≤ i < n.+1) (d %| i).
Proof.
have [-> | d_gt0] := posnP d; first by rewrite big_add1 divn0 big1.
apply: (@addnI (d %| 0)); rewrite -(@big_ltn _ 0 _ 0 _ (dvdn d)) // big_mkord.
rewrite (partition_big (fun i : 'I_n.+1 ⇒ inord (i %/ d)) 'I_(n %/ d).+1) //=.
rewrite dvdn0 add1n -{1}[_.+1]card_ord -sum1_card; apply: eq_bigr ⇒ [[q ?] _].
rewrite (bigD1 (inord (q × d))) /eq_op /= !inordK ?ltnS -?leq_divRL ?mulnK //.
rewrite dvdn_mull ?big1 // ⇒ [[i /= ?] /andP[/eqP <- /negPf]].
by rewrite eq_sym dvdn_eq inordK ?ltnS ?leq_div2r // ⇒ →.
Qed.
Lemma logn_count_dvd p n : prime p → logn p n = \sum_(1 ≤ k < n) (p ^ k %| n).
Proof.
rewrite big_add1 ⇒ p_prime; case: n ⇒ [|n]; first by rewrite logn0 big_geq.
rewrite big_mkord -big_mkcond (eq_bigl _ _ (fun _ ⇒ pfactor_dvdn _ _ _)) //=.
by rewrite big_ord_narrow ?sum1_card ?card_ord // -ltnS ltn_logl.
Qed.
Definition trunc_log p n :=
let fix loop n k :=
if k is k'.+1 then if p ≤ n then (loop (n %/ p) k').+1 else 0 else 0
in loop n n.
Lemma trunc_log_bounds p n :
1 < p → 0 < n → let k := trunc_log p n in p ^ k ≤ n < p ^ k.+1.
Proof.
rewrite {+}/trunc_log ⇒ p_gt1; have p_gt0 := ltnW p_gt1.
elim: n {-2 5}n (leqnn n) ⇒ [|m IHm] [|n] //=; rewrite ltnS ⇒ le_n_m _.
have [le_p_n | // ] := leqP p _; rewrite 2!expnSr -leq_divRL -?ltn_divLR //.
by apply: IHm; rewrite ?divn_gt0 // -ltnS (leq_trans (ltn_Pdiv _ _)).
Qed.
Lemma trunc_log_ltn p n : 1 < p → n < p ^ (trunc_log p n).+1.
Proof.
have [-> | n_gt0] := posnP n; first by move⇒ /ltnW; rewrite expn_gt0.
by case/trunc_log_bounds/(_ n_gt0)/andP.
Qed.
Lemma trunc_logP p n : 1 < p → 0 < n → p ^ trunc_log p n ≤ n.
Proof. by move⇒ p_gt1 /(trunc_log_bounds p_gt1)/andP[]. Qed.
Lemma trunc_log_max p k j : 1 < p → p ^ j ≤ k → j ≤ trunc_log p k.
Proof.
move⇒ p_gt1 le_pj_k; rewrite -ltnS -(@ltn_exp2l p) //.
exact: leq_ltn_trans (trunc_log_ltn _ _).
Qed.
Canonical nat_pred_pred := Eval hnf in [predType of nat_pred].
Coercion nat_pred_of_nat (p : nat) : nat_pred := pred1 p.
Section NatPreds.
Variables (n : nat) (pi : nat_pred).
Definition negn : nat_pred := [predC pi].
Definition pnat : pred nat := fun m ⇒ (m > 0) && all (mem pi) (primes m).
Definition partn := \prod_(0 ≤ p < n.+1 | p \in pi) p ^ logn p n.
End NatPreds.
Notation "pi ^'" := (negn pi) (at level 2, format "pi ^'") : nat_scope.
Notation "pi .-nat" := (pnat pi) (at level 2, format "pi .-nat") : nat_scope.
Notation "n `_ pi" := (partn n pi) : nat_scope.
Section PnatTheory.
Implicit Types (n p : nat) (pi rho : nat_pred).
Lemma negnK pi : pi^'^' =i pi.
Proof. move⇒ p; exact: negbK. Qed.
Lemma eq_negn pi1 pi2 : pi1 =i pi2 → pi1^' =i pi2^'.
Proof. by move⇒ eq_pi n; rewrite 3!inE /= eq_pi. Qed.
Lemma eq_piP m n : \pi(m) =i \pi(n) ↔ \pi(m) = \pi(n).
Proof.
rewrite /pi_of; have eqs := eq_sorted_irr ltn_trans ltnn.
by split⇒ [|-> //]; move/(eqs _ _ (sorted_primes m) (sorted_primes n)) →.
Qed.
Lemma part_gt0 pi n : 0 < n`_pi.
Proof. exact: prodn_gt0. Qed.
Hint Resolve part_gt0.
Lemma sub_in_partn pi1 pi2 n :
{in \pi(n), {subset pi1 ≤ pi2}} → n`_pi1 %| n`_pi2.
Proof.
move⇒ pi12; rewrite ![n`__]big_mkcond /=.
apply (big_ind2 (fun m1 m2 ⇒ m1 %| m2)) ⇒ // [*|p _]; first exact: dvdn_mul.
rewrite lognE -mem_primes; case: ifP ⇒ pi1p; last exact: dvd1n.
by case: ifP ⇒ pr_p; [rewrite pi12 | rewrite if_same].
Qed.
Lemma eq_in_partn pi1 pi2 n : {in \pi(n), pi1 =i pi2} → n`_pi1 = n`_pi2.
Proof.
by move⇒ pi12; apply/eqP; rewrite eqn_dvd ?sub_in_partn // ⇒ p /pi12→.
Qed.
Lemma eq_partn pi1 pi2 n : pi1 =i pi2 → n`_pi1 = n`_pi2.
Proof. by move⇒ pi12; apply: eq_in_partn ⇒ p _. Qed.
Lemma partnNK pi n : n`_pi^'^' = n`_pi.
Proof. by apply: eq_partn; exact: negnK. Qed.
Lemma widen_partn m pi n :
n ≤ m → n`_pi = \prod_(0 ≤ p < m.+1 | p \in pi) p ^ logn p n.
Proof.
move⇒ le_n_m; rewrite big_mkcond /=.
rewrite [n`_pi](big_nat_widen _ _ m.+1) // big_mkcond /=.
apply: eq_bigr ⇒ p _; rewrite ltnS lognE.
by case: and3P ⇒ [[_ n_gt0 p_dv_n]|]; rewrite ?if_same // andbC dvdn_leq.
Qed.
Lemma partn0 pi : 0`_pi = 1.
Proof. by apply: big1_seq ⇒ [] [|n]; rewrite andbC. Qed.
Lemma partn1 pi : 1`_pi = 1.
Proof. by apply: big1_seq ⇒ [] [|[|n]]; rewrite andbC. Qed.
Lemma partnM pi m n : m > 0 → n > 0 → (m × n)`_pi = m`_pi × n`_pi.
Proof.
have le_pmul m' n': m' > 0 → n' ≤ m' × n' by move/prednK <-; exact: leq_addr.
move⇒ mpos npos; rewrite !(@widen_partn (n × m)) 3?(le_pmul, mulnC) //.
rewrite !big_mkord -big_split; apply: eq_bigr ⇒ p _ /=.
by rewrite lognM // expnD.
Qed.
Lemma partnX pi m n : (m ^ n)`_pi = m`_pi ^ n.
Proof.
elim: n ⇒ [|n IHn]; first exact: partn1.
rewrite expnS; case: (posnP m) ⇒ [->|m_gt0]; first by rewrite partn0 exp1n.
by rewrite expnS partnM ?IHn // expn_gt0 m_gt0.
Qed.
Lemma partn_dvd pi m n : n > 0 → m %| n → m`_pi %| n`_pi.
Proof.
move⇒ n_gt0 dvmn; case/dvdnP: dvmn n_gt0 ⇒ q ->{n}.
by rewrite muln_gt0 ⇒ /andP[q_gt0 m_gt0]; rewrite partnM ?dvdn_mull.
Qed.
Lemma p_part p n : n`_p = p ^ logn p n.
Proof.
case (posnP (logn p n)) ⇒ [log0 |].
by rewrite log0 [n`_p]big1_seq // ⇒ q; case/andP; move/eqnP->; rewrite log0.
rewrite logn_gt0 mem_primes; case/and3P⇒ _ n_gt0 dv_p_n.
have le_p_n: p < n.+1 by rewrite ltnS dvdn_leq.
by rewrite [n`_p]big_mkord (big_pred1 (Ordinal le_p_n)).
Qed.
Lemma p_part_eq1 p n : (n`_p == 1) = (p \notin \pi(n)).
Proof.
rewrite mem_primes p_part lognE; case: and3P ⇒ // [[p_pr _ _]].
by rewrite -dvdn1 pfactor_dvdn // logn1.
Qed.
Lemma p_part_gt1 p n : (n`_p > 1) = (p \in \pi(n)).
Proof. by rewrite ltn_neqAle part_gt0 andbT eq_sym p_part_eq1 negbK. Qed.
Lemma primes_part pi n : primes n`_pi = filter (mem pi) (primes n).
Proof.
have ltnT := ltn_trans.
case: (posnP n) ⇒ [-> | n_gt0]; first by rewrite partn0.
apply: (eq_sorted_irr ltnT ltnn); rewrite ?(sorted_primes, sorted_filter) //.
move⇒ p; rewrite mem_filter /= !mem_primes n_gt0 part_gt0 /=.
apply/andP/and3P⇒ [[p_pr] | [pi_p p_pr dv_p_n]].
rewrite /partn; apply big_ind ⇒ [|n1 n2 IHn1 IHn2|q pi_q].
- by rewrite dvdn1; case: eqP p_pr ⇒ // →.
- by rewrite Euclid_dvdM //; case/orP.
rewrite -{1}(expn1 p) pfactor_dvdn // lognX muln_gt0.
rewrite logn_gt0 mem_primes n_gt0 - andbA /=; case/and3P⇒ pr_q dv_q_n.
by rewrite logn_prime //; case: eqP ⇒ // →.
have le_p_n: p < n.+1 by rewrite ltnS dvdn_leq.
rewrite [n`_pi]big_mkord (bigD1 (Ordinal le_p_n)) //= dvdn_mulr //.
by rewrite lognE p_pr n_gt0 dv_p_n expnS dvdn_mulr.
Qed.
Lemma filter_pi_of n m : n < m → filter \pi(n) (index_iota 0 m) = primes n.
Proof.
move⇒ lt_n_m; have ltnT := ltn_trans; apply: (eq_sorted_irr ltnT ltnn).
- by rewrite sorted_filter // iota_ltn_sorted.
- exact: sorted_primes.
move⇒ p; rewrite mem_filter mem_index_iota /= mem_primes; case: and3P ⇒ //.
case⇒ _ n_gt0 dv_p_n; apply: leq_ltn_trans lt_n_m; exact: dvdn_leq.
Qed.
Lemma partn_pi n : n > 0 → n`_\pi(n) = n.
Proof.
move⇒ n_gt0; rewrite {3}(prod_prime_decomp n_gt0) prime_decompE big_map.
by rewrite -[n`__]big_filter filter_pi_of.
Qed.
Lemma partnT n : n > 0 → n`_predT = n.
Proof.
move⇒ n_gt0; rewrite -{2}(partn_pi n_gt0) {2}/partn big_mkcond /=.
by apply: eq_bigr ⇒ p _; rewrite -logn_gt0; case: (logn p _).
Qed.
Lemma partnC pi n : n > 0 → n`_pi × n`_pi^' = n.
Proof.
move⇒ n_gt0; rewrite -{3}(partnT n_gt0) /partn.
do 2!rewrite mulnC big_mkcond /=; rewrite -big_split; apply: eq_bigr ⇒ p _ /=.
by rewrite mulnC inE /=; case: (p \in pi); rewrite /= (muln1, mul1n).
Qed.
Lemma dvdn_part pi n : n`_pi %| n.
Proof. by case: n ⇒ // n; rewrite -{2}[n.+1](@partnC pi) // dvdn_mulr. Qed.
Lemma logn_part p m : logn p m`_p = logn p m.
Proof.
case p_pr: (prime p); first by rewrite p_part pfactorK.
by rewrite lognE (lognE p m) p_pr.
Qed.
Lemma partn_lcm pi m n : m > 0 → n > 0 → (lcmn m n)`_pi = lcmn m`_pi n`_pi.
Proof.
move⇒ m_gt0 n_gt0; have p_gt0: lcmn m n > 0 by rewrite lcmn_gt0 m_gt0.
apply/eqP; rewrite eqn_dvd dvdn_lcm !partn_dvd ?dvdn_lcml ?dvdn_lcmr //.
rewrite -(dvdn_pmul2r (part_gt0 pi^' (lcmn m n))) partnC // dvdn_lcm !andbT.
rewrite -{1}(partnC pi m_gt0) andbC -{1}(partnC pi n_gt0).
by rewrite !dvdn_mul ?partn_dvd ?dvdn_lcml ?dvdn_lcmr.
Qed.
Lemma partn_gcd pi m n : m > 0 → n > 0 → (gcdn m n)`_pi = gcdn m`_pi n`_pi.
Proof.
move⇒ m_gt0 n_gt0; have p_gt0: gcdn m n > 0 by rewrite gcdn_gt0 m_gt0.
apply/eqP; rewrite eqn_dvd dvdn_gcd !partn_dvd ?dvdn_gcdl ?dvdn_gcdr //=.
rewrite -(dvdn_pmul2r (part_gt0 pi^' (gcdn m n))) partnC // dvdn_gcd.
rewrite -{3}(partnC pi m_gt0) andbC -{3}(partnC pi n_gt0).
by rewrite !dvdn_mul ?partn_dvd ?dvdn_gcdl ?dvdn_gcdr.
Qed.
Lemma partn_biglcm (I : finType) (P : pred I) F pi :
(∀ i, P i → F i > 0) →
(\big[lcmn/1%N]_(i | P i) F i)`_pi = \big[lcmn/1%N]_(i | P i) (F i)`_pi.
Proof.
move⇒ F_gt0; set m := \big[lcmn/1%N]_(i | P i) F i.
have m_gt0: 0 < m by apply big_ind ⇒ // p q p_gt0; rewrite lcmn_gt0 p_gt0.
apply/eqP; rewrite eqn_dvd andbC; apply/andP; split.
by apply/dvdn_biglcmP⇒ i Pi; rewrite partn_dvd // (@biglcmn_sup _ i).
rewrite -(dvdn_pmul2r (part_gt0 pi^' m)) partnC //.
apply/dvdn_biglcmP⇒ i Pi; rewrite -(partnC pi (F_gt0 i Pi)) dvdn_mul //.
by rewrite (@biglcmn_sup _ i).
by rewrite partn_dvd // (@biglcmn_sup _ i).
Qed.
Lemma partn_biggcd (I : finType) (P : pred I) F pi :
#|SimplPred P| > 0 → (∀ i, P i → F i > 0) →
(\big[gcdn/0]_(i | P i) F i)`_pi = \big[gcdn/0]_(i | P i) (F i)`_pi.
Proof.
move⇒ ntP F_gt0; set d := \big[gcdn/0]_(i | P i) F i.
have d_gt0: 0 < d.
case/card_gt0P: ntP ⇒ i /= Pi; have:= F_gt0 i Pi.
rewrite !lt0n -!dvd0n; apply: contra ⇒ dv0d.
by rewrite (dvdn_trans dv0d) // (@biggcdn_inf _ i).
apply/eqP; rewrite eqn_dvd; apply/andP; split.
by apply/dvdn_biggcdP⇒ i Pi; rewrite partn_dvd ?F_gt0 // (@biggcdn_inf _ i).
rewrite -(dvdn_pmul2r (part_gt0 pi^' d)) partnC //.
apply/dvdn_biggcdP⇒ i Pi; rewrite -(partnC pi (F_gt0 i Pi)) dvdn_mul //.
by rewrite (@biggcdn_inf _ i).
by rewrite partn_dvd ?F_gt0 // (@biggcdn_inf _ i).
Qed.
Lemma sub_in_pnat pi rho n :
{in \pi(n), {subset pi ≤ rho}} → pi.-nat n → rho.-nat n.
Proof.
rewrite /pnat ⇒ subpi /andP[-> pi_n].
apply/allP⇒ p pr_p; apply: subpi ⇒ //; exact: (allP pi_n).
Qed.
Lemma eq_in_pnat pi rho n : {in \pi(n), pi =i rho} → pi.-nat n = rho.-nat n.
Proof. by move⇒ eqpi; apply/idP/idP; apply: sub_in_pnat ⇒ p /eqpi→. Qed.
Lemma eq_pnat pi rho n : pi =i rho → pi.-nat n = rho.-nat n.
Proof. by move⇒ eqpi; apply: eq_in_pnat ⇒ p _. Qed.
Lemma pnatNK pi n : pi^'^'.-nat n = pi.-nat n.
Proof. exact: eq_pnat (negnK pi). Qed.
Lemma pnatI pi rho n : [predI pi & rho].-nat n = pi.-nat n && rho.-nat n.
Proof. by rewrite /pnat andbCA all_predI !andbA andbb. Qed.
Lemma pnat_mul pi m n : pi.-nat (m × n) = pi.-nat m && pi.-nat n.
Proof.
rewrite /pnat muln_gt0 andbCA -andbA andbCA.
case: posnP ⇒ // n_gt0; case: posnP ⇒ //= m_gt0.
apply/allP/andP⇒ [pi_mn | [pi_m pi_n] p].
by split; apply/allP⇒ p m_p; apply: pi_mn; rewrite primes_mul // m_p ?orbT.
rewrite primes_mul // ⇒ /orP[]; [exact: (allP pi_m) | exact: (allP pi_n)].
Qed.
Lemma pnat_exp pi m n : pi.-nat (m ^ n) = pi.-nat m || (n == 0).
Proof. by case: n ⇒ [|n]; rewrite orbC // /pnat expn_gt0 orbC primes_exp. Qed.
Lemma part_pnat pi n : pi.-nat n`_pi.
Proof.
rewrite /pnat primes_part part_gt0.
by apply/allP⇒ p; rewrite mem_filter ⇒ /andP[].
Qed.
Lemma pnatE pi p : prime p → pi.-nat p = (p \in pi).
Proof. by move⇒ pr_p; rewrite /pnat prime_gt0 ?primes_prime //= andbT. Qed.
Lemma pnat_id p : prime p → p.-nat p.
Proof. by move⇒ pr_p; rewrite pnatE ?inE /=. Qed.
Lemma coprime_pi' m n : m > 0 → n > 0 → coprime m n = \pi(m)^'.-nat n.
Proof.
by move⇒ m_gt0 n_gt0; rewrite /pnat n_gt0 all_predC coprime_has_primes.
Qed.
Lemma pnat_pi n : n > 0 → \pi(n).-nat n.
Proof. rewrite /pnat ⇒ ->; exact/allP. Qed.
Lemma pi_of_dvd m n : m %| n → n > 0 → {subset \pi(m) ≤ \pi(n)}.
Proof.
move⇒ m_dv_n n_gt0 p; rewrite !mem_primes n_gt0 ⇒ /and3P[-> _ p_dv_m].
exact: dvdn_trans p_dv_m m_dv_n.
Qed.
Lemma pi_ofM m n : m > 0 → n > 0 → \pi(m × n) =i [predU \pi(m) & \pi(n)].
Proof. move⇒ m_gt0 n_gt0 p; exact: primes_mul. Qed.
Lemma pi_of_part pi n : n > 0 → \pi(n`_pi) =i [predI \pi(n) & pi].
Proof. by move⇒ n_gt0 p; rewrite /pi_of primes_part mem_filter andbC. Qed.
Lemma pi_of_exp p n : n > 0 → \pi(p ^ n) = \pi(p).
Proof. by move⇒ n_gt0; rewrite /pi_of primes_exp. Qed.
Lemma pi_of_prime p : prime p → \pi(p) =i (p : nat_pred).
Proof. by move⇒ pr_p q; rewrite /pi_of primes_prime // mem_seq1. Qed.
Lemma p'natEpi p n : n > 0 → p^'.-nat n = (p \notin \pi(n)).
Proof. by case: n ⇒ // n _; rewrite /pnat all_predC has_pred1. Qed.
Lemma p'natE p n : prime p → p^'.-nat n = ~~ (p %| n).
Proof.
case: n ⇒ [|n] p_pr; first by case: p p_pr.
by rewrite p'natEpi // mem_primes p_pr.
Qed.
Lemma pnatPpi pi n p : pi.-nat n → p \in \pi(n) → p \in pi.
Proof. by case/andP⇒ _ /allP; exact. Qed.
Lemma pnat_dvd m n pi : m %| n → pi.-nat n → pi.-nat m.
Proof. by case/dvdnP⇒ q ->; rewrite pnat_mul; case/andP. Qed.
Lemma pnat_div m n pi : m %| n → pi.-nat n → pi.-nat (n %/ m).
Proof.
case/dvdnP⇒ q ->; rewrite pnat_mul andbC ⇒ /andP[].
by case: m ⇒ // m _; rewrite mulnK.
Qed.
Lemma pnat_coprime pi m n : pi.-nat m → pi^'.-nat n → coprime m n.
Proof.
case/andP⇒ m_gt0 pi_m /andP[n_gt0 pi'_n].
rewrite coprime_has_primes //; apply/hasPn⇒ p /(allP pi'_n).
apply: contra; exact: allP.
Qed.
Lemma p'nat_coprime pi m n : pi^'.-nat m → pi.-nat n → coprime m n.
Proof. by move⇒ pi'm pi_n; rewrite (pnat_coprime pi'm) ?pnatNK. Qed.
Lemma sub_pnat_coprime pi rho m n :
{subset rho ≤ pi^'} → pi.-nat m → rho.-nat n → coprime m n.
Proof.
by move⇒ pi'rho pi_m; move/(sub_in_pnat (in1W pi'rho)); exact: pnat_coprime.
Qed.
Lemma coprime_partC pi m n : coprime m`_pi n`_pi^'.
Proof. by apply: (@pnat_coprime pi); exact: part_pnat. Qed.
Lemma pnat_1 pi n : pi.-nat n → pi^'.-nat n → n = 1.
Proof.
by move⇒ pi_n pi'_n; rewrite -(eqnP (pnat_coprime pi_n pi'_n)) gcdnn.
Qed.
Lemma part_pnat_id pi n : pi.-nat n → n`_pi = n.
Proof.
case/andP⇒ n_gt0 pi_n.
rewrite -{2}(partnT n_gt0) /partn big_mkcond; apply: eq_bigr⇒ p _.
case: (posnP (logn p n)) ⇒ [-> |]; first by rewrite if_same.
by rewrite logn_gt0 ⇒ /(allP pi_n)/= →.
Qed.
Lemma part_p'nat pi n : pi^'.-nat n → n`_pi = 1.
Proof.
case/andP⇒ n_gt0 pi'_n; apply: big1_seq ⇒ p /andP[pi_p _].
case: (posnP (logn p n)) ⇒ [-> //|].
by rewrite logn_gt0; move/(allP pi'_n); case/negP.
Qed.
Lemma partn_eq1 pi n : n > 0 → (n`_pi == 1) = pi^'.-nat n.
Proof.
move⇒ n_gt0; apply/eqP/idP⇒ [pi_n_1|]; last exact: part_p'nat.
by rewrite -(partnC pi n_gt0) pi_n_1 mul1n part_pnat.
Qed.
Lemma pnatP pi n :
n > 0 → reflect (∀ p, prime p → p %| n → p \in pi) (pi.-nat n).
Proof.
move⇒ n_gt0; rewrite /pnat n_gt0.
apply: (iffP allP) ⇒ /= pi_n p ⇒ [pr_p p_n|].
by rewrite pi_n // mem_primes pr_p n_gt0.
by rewrite mem_primes n_gt0 /=; case/andP; move: p.
Qed.
Lemma pi_pnat pi p n : p.-nat n → p \in pi → pi.-nat n.
Proof.
move⇒ p_n pi_p; have [n_gt0 _] := andP p_n.
by apply/pnatP⇒ // q q_pr /(pnatP _ n_gt0 p_n _ q_pr)/eqnP→.
Qed.
Lemma p_natP p n : p.-nat n → {k | n = p ^ k}.
Proof. by move⇒ p_n; ∃ (logn p n); rewrite -p_part part_pnat_id. Qed.
Lemma pi'_p'nat pi p n : pi^'.-nat n → p \in pi → p^'.-nat n.
Proof.
move⇒ pi'n pi_p; apply: sub_in_pnat pi'n ⇒ q _.
by apply: contraNneq ⇒ →.
Qed.
Lemma pi_p'nat p pi n : pi.-nat n → p \in pi^' → p^'.-nat n.
Proof. by move⇒ pi_n; apply: pi'_p'nat; rewrite pnatNK. Qed.
Lemma partn_part pi rho n : {subset pi ≤ rho} → n`_rho`_pi = n`_pi.
Proof.
move⇒ pi_sub_rho; have [->|n_gt0] := posnP n; first by rewrite !partn0 partn1.
rewrite -{2}(partnC rho n_gt0) partnM //.
suffices: pi^'.-nat n`_rho^' by move/part_p'nat->; rewrite muln1.
apply: sub_in_pnat (part_pnat _ _) ⇒ q _; apply: contra; exact: pi_sub_rho.
Qed.
Lemma partnI pi rho n : n`_[predI pi & rho] = n`_pi`_rho.
Proof.
rewrite -(@partnC [predI pi & rho] _`_rho) //.
symmetry; rewrite 2?partn_part; try by move⇒ p /andP [].
rewrite mulnC part_p'nat ?mul1n // pnatNK pnatI part_pnat andbT.
exact: pnat_dvd (dvdn_part _ _) (part_pnat _ _).
Qed.
Lemma odd_2'nat n : odd n = 2^'.-nat n.
Proof. by case: n ⇒ // n; rewrite p'natE // dvdn2 negbK. Qed.
End PnatTheory.
Hint Resolve part_gt0.
Lemma divisors_correct n : n > 0 →
[/\ uniq (divisors n), sorted leq (divisors n)
& ∀ d, (d \in divisors n) = (d %| n)].
Proof.
move/prod_prime_decomp⇒ def_n; rewrite {4}def_n {def_n}.
have: all prime (primes n) by apply/allP⇒ p; rewrite mem_primes; case/andP.
have:= primes_uniq n; rewrite /primes /divisors; move/prime_decomp: n.
elim⇒ [|[p e] pd] /=; first by split⇒ // d; rewrite big_nil dvdn1 mem_seq1.
rewrite big_cons /=; move: (foldr _ _ pd) ⇒ divs.
move⇒ IHpd /andP[npd_p Upd] /andP[pr_p pr_pd].
have lt0p: 0 < p by exact: prime_gt0.
have {IHpd Upd}[Udivs Odivs mem_divs] := IHpd Upd pr_pd.
have ndivs_p m: p × m \notin divs.
suffices: p \notin divs; rewrite !mem_divs.
by apply: contra ⇒ /dvdnP[n ->]; rewrite mulnCA dvdn_mulr.
have ndv_p_1: ~~(p %| 1) by rewrite dvdn1 neq_ltn orbC prime_gt1.
rewrite big_seq; elim/big_ind: _ ⇒ [//|u v npu npv|[q f] /= pd_qf].
by rewrite Euclid_dvdM //; apply/norP.
elim: (f) ⇒ // f'; rewrite expnS Euclid_dvdM // orbC negb_or ⇒ → {f'}/=.
have pd_q: q \in unzip1 pd by apply/mapP; ∃ (q, f).
by apply: contra npd_p; rewrite dvdn_prime2 // ?(allP pr_pd) // ⇒ /eqP→.
elim: e ⇒ [|e] /=; first by split⇒ // d; rewrite mul1n.
have Tmulp_inj: injective (NatTrec.mul p).
by move⇒ u v /eqP; rewrite !natTrecE eqn_pmul2l // ⇒ /eqP.
move: (iter e _ _) ⇒ divs' [Udivs' Odivs' mem_divs']; split⇒ [||d].
- rewrite merge_uniq cat_uniq map_inj_uniq // Udivs Udivs' andbT /=.
apply/hasP⇒ [[d dv_d /mapP[d' _ def_d]]].
by case/idPn: dv_d; rewrite def_d natTrecE.
- rewrite (merge_sorted leq_total) //; case: (divs') Odivs' ⇒ //= d ds.
rewrite (@map_path _ _ _ _ leq xpred0) ?has_pred0 // ⇒ u v _.
by rewrite !natTrecE leq_pmul2l.
rewrite mem_merge mem_cat; case dv_d_p: (p %| d).
case/dvdnP: dv_d_p ⇒ d' ->{d}; rewrite mulnC (negbTE (ndivs_p d')) orbF.
rewrite expnS -mulnA dvdn_pmul2l // -mem_divs'.
by rewrite -(mem_map Tmulp_inj divs') natTrecE.
case pdiv_d: (_ \in _).
by case/mapP: pdiv_d dv_d_p ⇒ d' _ ->; rewrite natTrecE dvdn_mulr.
rewrite mem_divs Gauss_dvdr // coprime_sym.
by rewrite coprime_expl ?prime_coprime ?dv_d_p.
Qed.
Lemma sorted_divisors n : sorted leq (divisors n).
Proof. by case: (posnP n) ⇒ [-> | /divisors_correct[]]. Qed.
Lemma divisors_uniq n : uniq (divisors n).
Proof. by case: (posnP n) ⇒ [-> | /divisors_correct[]]. Qed.
Lemma sorted_divisors_ltn n : sorted ltn (divisors n).
Proof. by rewrite ltn_sorted_uniq_leq divisors_uniq sorted_divisors. Qed.
Lemma dvdn_divisors d m : 0 < m → (d %| m) = (d \in divisors m).
Proof. by case/divisors_correct. Qed.
Lemma divisor1 n : 1 \in divisors n.
Proof. by case: n ⇒ // n; rewrite -dvdn_divisors // dvd1n. Qed.
Lemma divisors_id n : 0 < n → n \in divisors n.
Proof. by move/dvdn_divisors <-. Qed.
Lemma dvdn_sum d I r (K : pred I) F :
(∀ i, K i → d %| F i) → d %| \sum_(i <- r | K i) F i.
Proof. move⇒ dF; elim/big_ind: _ ⇒ //; exact: dvdn_add. Qed.
Lemma dvdn_partP n m : 0 < n →
reflect (∀ p, p \in \pi(n) → n`_p %| m) (n %| m).
Proof.
move⇒ n_gt0; apply: (iffP idP) ⇒ n_dvd_m ⇒ [p _|].
apply: dvdn_trans n_dvd_m; exact: dvdn_part.
have [-> // | m_gt0] := posnP m.
rewrite -(partnT n_gt0) -(partnT m_gt0).
rewrite !(@widen_partn (m + n)) ?leq_addl ?leq_addr // /in_mem /=.
elim/big_ind2: _ ⇒ // [* | q _]; first exact: dvdn_mul.
have [-> // | ] := posnP (logn q n); rewrite logn_gt0 ⇒ q_n.
have pr_q: prime q by move: q_n; rewrite mem_primes; case/andP.
by have:= n_dvd_m q q_n; rewrite p_part !pfactor_dvdn // pfactorK.
Qed.
Lemma modn_partP n a b : 0 < n →
reflect (∀ p : nat, p \in \pi(n) → a = b %[mod n`_p]) (a == b %[mod n]).
Proof.
move⇒ n_gt0; wlog le_b_a: a b / b ≤ a.
move⇒ IH; case: (leqP b a) ⇒ [|/ltnW] /IH {IH}// IH.
by rewrite eq_sym; apply: (iffP IH) ⇒ eqab p; move/eqab.
rewrite eqn_mod_dvd //; apply: (iffP (dvdn_partP _ n_gt0)) ⇒ eqab p /eqab;
by rewrite -eqn_mod_dvd // ⇒ /eqP.
Qed.
Lemma totientE n :
n > 0 → totient n = \prod_(p <- primes n) (p.-1 × p ^ (logn p n).-1).
Proof.
move⇒ n_gt0; rewrite /totient n_gt0 prime_decompE unlock.
by elim: (primes n) ⇒ //= [p pr ->]; rewrite !natTrecE.
Qed.
Lemma totient_gt0 n : (0 < totient n) = (0 < n).
Proof.
case: n ⇒ // n; rewrite totientE // big_seq_cond prodn_cond_gt0 // ⇒ p.
by rewrite mem_primes muln_gt0 expn_gt0; case: p ⇒ [|[|]].
Qed.
Lemma totient_pfactor p e :
prime p → e > 0 → totient (p ^ e) = p.-1 × p ^ e.-1.
Proof.
move⇒ p_pr e_gt0; rewrite totientE ?expn_gt0 ?prime_gt0 //.
by rewrite primes_exp // primes_prime // unlock /= muln1 pfactorK.
Qed.
Lemma totient_coprime m n :
coprime m n → totient (m × n) = totient m × totient n.
Proof.
move⇒ co_mn; have [-> //| m_gt0] := posnP m.
have [->|n_gt0] := posnP n; first by rewrite !muln0.
rewrite !totientE ?muln_gt0 ?m_gt0 //.
have /(eq_big_perm _)->: perm_eq (primes (m × n)) (primes m ++ primes n).
apply: uniq_perm_eq ⇒ [||p]; first exact: primes_uniq.
by rewrite cat_uniq !primes_uniq -coprime_has_primes // co_mn.
by rewrite mem_cat primes_mul.
rewrite big_cat /= !big_seq.
congr (_ × _); apply: eq_bigr ⇒ p; rewrite mem_primes ⇒ /and3P[_ _ dvp].
rewrite (mulnC m) logn_Gauss //; move: co_mn.
by rewrite -(divnK dvp) coprime_mull ⇒ /andP[].
rewrite logn_Gauss //; move: co_mn.
by rewrite coprime_sym -(divnK dvp) coprime_mull ⇒ /andP[].
Qed.
Lemma totient_count_coprime n : totient n = \sum_(0 ≤ d < n) coprime n d.
Proof.
elim: {n}_.+1 {-2}n (ltnSn n) ⇒ // m IHm n; rewrite ltnS ⇒ le_n_m.
case: (leqP n 1) ⇒ [|lt1n]; first by rewrite unlock; case: (n) ⇒ [|[]].
pose p := pdiv n; have p_pr: prime p by exact: pdiv_prime.
have p1 := prime_gt1 p_pr; have p0 := ltnW p1.
pose np := n`_p; pose np' := n`_p^'.
have co_npp': coprime np np' by rewrite coprime_partC.
have [n0 np0 np'0]: [/\ n > 0, np > 0 & np' > 0] by rewrite ltnW ?part_gt0.
have def_n: n = np × np' by rewrite partnC.
have lnp0: 0 < logn p n by rewrite lognE p_pr n0 pdiv_dvd.
pose in_mod k (k0 : k > 0) d := Ordinal (ltn_pmod d k0).
rewrite {1}def_n totient_coprime // {IHm}(IHm np') ?big_mkord; last first.
apply: leq_trans le_n_m; rewrite def_n ltn_Pmull //.
by rewrite /np p_part -(expn0 p) ltn_exp2l.
have ->: totient np = #|[pred d : 'I_np | coprime np d]|.
rewrite {1}[np]p_part totient_pfactor //=; set q := p ^ _.
apply: (@addnI (1 × q)); rewrite -mulnDl [1 + _]prednK // mul1n.
have def_np: np = p × q by rewrite -expnS prednK // -p_part.
pose mulp := [fun d : 'I_q ⇒ in_mod _ np0 (p × d)].
rewrite -def_np -{1}[np]card_ord -(cardC (mem (codom mulp))).
rewrite card_in_image ⇒ [|[d1 ltd1] [d2 ltd2] /= _ _ []]; last first.
move/eqP; rewrite def_np -!muln_modr ?modn_small //.
by rewrite eqn_pmul2l // ⇒ eq_op12; exact/eqP.
rewrite card_ord; congr (q + _); apply: eq_card ⇒ d /=.
rewrite !inE [np in coprime np _]p_part coprime_pexpl ?prime_coprime //.
congr (~~ _); apply/codomP/idP⇒ [[d' → /=] | /dvdnP[r def_d]].
by rewrite def_np -muln_modr // dvdn_mulr.
do [rewrite mulnC; case: d ⇒ d ltd /=] in def_d ×.
have ltr: r < q by rewrite -(ltn_pmul2l p0) -def_np -def_d.
by ∃ (Ordinal ltr); apply: val_inj; rewrite /= -def_d modn_small.
pose h (d : 'I_n) := (in_mod _ np0 d, in_mod _ np'0 d).
pose h' (d : 'I_np × 'I_np') := in_mod _ n0 (chinese np np' d.1 d.2).
rewrite -!big_mkcond -sum_nat_const pair_big (reindex_onto h h') ⇒ [|[d d'] _].
apply: eq_bigl ⇒ [[d ltd] /=]; rewrite !inE /= -val_eqE /= andbC.
rewrite !coprime_modr def_n -chinese_mod // -coprime_mull -def_n.
by rewrite modn_small ?eqxx.
apply/eqP; rewrite /eq_op /= /eq_op /= !modn_dvdm ?dvdn_part //.
by rewrite chinese_modl // chinese_modr // !modn_small ?eqxx ?ltn_ord.
Qed.