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+(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *)
+Require Import mathcomp.ssreflect.ssreflect.
+From mathcomp
+Require Import ssrfun ssrbool eqtype ssrnat seq path fintype.
+From mathcomp
+Require Import div bigop.
+
+(******************************************************************************)
+(* This file contains the definitions of:                                     *)
+(*        prime p <=> p is a prime.                                           *)
+(*       primes m == the sorted list of prime divisors of m > 1, else [::].   *)
+(*        pfactor == the type of prime factors, syntax (p ^ e)%pfactor.       *)
+(* prime_decomp m == the list of prime factors of m > 1, sorted by primes.    *)
+(*       logn p m == the e such that (p ^ e) \in prime_decomp n, else 0.      *)
+(*  trunc_log p m == the largest e such that p ^ e <= m, or 0 if p or m is 0. *)
+(*         pdiv n == the smallest prime divisor of n > 1, else 1.             *)
+(*     max_pdiv n == the largest prime divisor of n > 1, else 1.              *)
+(*     divisors m == the sorted list of divisors of m > 0, else [::].         *)
+(*      totient n == the Euler totient (#|{i < n | i and n coprime}|).        *)
+(*       nat_pred == the type of explicit collective nat predicates.          *)
+(*                := simpl_pred nat.                                          *)
+(*    -> We allow the coercion nat >-> nat_pred, interpreting p as pred1 p.   *)
+(*    -> We define a predType for nat_pred, enabling the notation p \in pi.   *)
+(*    -> We don't have nat_pred >-> pred, which would imply nat >-> Funclass. *)
+(*           pi^' == the complement of pi : nat_pred, i.e., the nat_pred such *)
+(*                   that (p \in pi^') = (p \notin pi).                       *)
+(*         \pi(n) == the set of prime divisors of n, i.e., the nat_pred such  *)
+(*                   that (p \in \pi(n)) = (p \in primes n).                  *)
+(*         \pi(A) == the set of primes of #|A|, with A a collective predicate *)
+(*                   over a finite Type.                                      *)
+(*     -> The notation \pi(A) is implemented with a collapsible Coercion, so  *)
+(*        the type of A must coerce to finpred_class (e.g., by coercing to    *)
+(*        {set T}), not merely implement the predType interface (as seq T     *)
+(*        does).                                                              *)
+(*     -> The expression #|A| will only appear in \pi(A) after simplification *)
+(*        collapses the coercion stack, so it is advisable to do so early on. *)
+(*     pi.-nat n <=> n > 0 and all prime divisors of n are in pi.             *)
+(*          n`_pi == the pi-part of n -- the largest pi.-nat divisor of n.    *)
+(*               := \prod_(0 <= p < n.+1 | p \in pi) p ^ logn p n.            *)
+(*     -> The nat >-> nat_pred coercion lets us write p.-nat n and n`_p.      *)
+(* In addition to the lemmas relevant to these definitions, this file also    *)
+(* contains the dvdn_sum lemma, so that bigop.v doesn't depend on div.v.      *)
+(******************************************************************************)
+
+Set Implicit Arguments.
+Unset Strict Implicit.
+Unset Printing Implicit Defensive.
+
+(* The complexity of any arithmetic operation with the Peano representation *)
+(* is pretty dreadful, so using algorithms for "harder" problems such as    *)
+(* factoring, that are geared for efficient artihmetic leads to dismal      *)
+(* performance -- it takes a significant time, for instance, to compute the *)
+(* divisors of just a two-digit number. On the other hand, for Peano        *)
+(* integers, prime factoring (and testing) is linear-time with a small      *)
+(* constant factor -- indeed, the same as converting in and out of a binary *)
+(* representation. This is implemented by the code below, which is then     *)
+(* used to give the "standard" definitions of prime, primes, and divisors,  *)
+(* which can then be used casually in proofs with moderately-sized numeric  *)
+(* values (indeed, the code here performs well for up to 6-digit numbers).  *)
+
+(* We start with faster mod-2 functions. *)
+
+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) => /=.
+by rewrite addSnnS; apply: 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; apply: IHq.
+clear 1; rewrite subn1 -[_.-1.+1]doubleS -mul2n mulnA -expnSr.
+by rewrite -{1}(addKn q q) addnn; apply: 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.
+
+(* For pretty-printing. *)
+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].
+
+(* The list of divisors and the Euler function are computed directly from *)
+(* the decomposition, using a merge_sort variant sort the divisor list.   *)
+
+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).
+
+(* Correctness of the decomposition algorithm. *)
+
+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
+  forall 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.
+    by rewrite mul1n; apply: leq_pd_ok; apply: 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 exists r; rewrite // mem_index_iota lt1r.
+  case/dvdnP: dvrm => q def_q; exists 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->; apply: 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; exists 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 exists q; rewrite // mem_index_iota lt1q.
+  case/dvdnP: dvqm => r def_r; exists 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; exists 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; exists 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; exists d.
+Qed.
+
+Lemma primeP p :
+  reflect (p > 1 /\ forall 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.
+exists 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; apply: 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; exists (p, e).
+apply/primeP; split=> // d dv_d_p; apply/norP=> [[nd1 ndp]].
+case/hasP: pr_p; exists 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[].
+exists (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[_ _]]; apply: 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.
+
+(* The smallest prime divisor *)
+
+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[-> //|].
+by move/(allP (order_path_min ltn_trans ord_pm)); apply: 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 \/ exists 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].
+exists (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; exists (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.
+
+(* Primes are unbounded. *)
+
+Lemma dvdn_fact m n : 0 < m <= n -> m %| n`!.
+Proof.
+case: m => //= m; elim: n => //= n IHn; rewrite ltnS leq_eqVlt.
+by case/predU1P=> [-> | /IHn]; [apply: dvdn_mulr | apply: dvdn_mull].
+Qed.
+
+Lemma prime_above m : {p | m < p & prime p}.
+Proof.
+have /pdivP[p pr_p p_dv_m1]: 1 < m`! + 1 by rewrite addn1 ltnS fact_gt0.
+exists p => //; rewrite ltnNge; apply: contraL p_dv_m1 => p_le_m.
+by rewrite dvdn_addr ?dvdn_fact ?prime_gt0 // gtnNdvd ?prime_gt1.
+Qed.
+
+(* "prime" logarithms and p-parts. *)
+
+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.
+exists (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.
+exists (logn p d); first by rewrite -(pfactorK n p_pr) dvdn_leq_log.
+have d_gt0: d > 0 by apply: 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.
+
+(* Some combinatorial formulae. *)
+
+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.
+
+(* Truncated real log. *)
+
+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.
+
+(* pi- parts *)
+
+(* Testing for membership in set of prime factors. *)
+
+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. by move=> p; apply: 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; apply: 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 <-; apply: 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 => //.
+by case=> _ n_gt0 dv_p_n; apply: leq_ltn_trans lt_n_m; apply: 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 :
+    (forall 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 elim/big_ind: m => // 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 -> (forall 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].
+by apply/allP=> p pr_p; apply: subpi => //; apply: (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.
+by rewrite primes_mul // => /orP[]; [apply: (allP pi_m) | apply: (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. by rewrite /pnat => ->; apply/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. by move=> m_gt0 n_gt0 p; apply: 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; apply. 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 //.
+by apply/hasPn=> p /(allP pi'_n); apply/contra/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)); apply: pnat_coprime.
+Qed.
+
+Lemma coprime_partC pi m n : coprime m`_pi n`_pi^'.
+Proof. by apply: (@pnat_coprime pi); apply: 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 (forall 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; exists (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.
+by apply: sub_in_pnat (part_pnat _ _) => q _; apply/contra/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.
+
+(************************************)
+(* Properties of the divisors list. *)
+(************************************)
+
+Lemma divisors_correct n : n > 0 ->
+  [/\ uniq (divisors n), sorted leq (divisors n)
+    & forall 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 apply: 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; exists (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.
+
+(* Big sum / product lemmas*)
+
+Lemma dvdn_sum d I r (K : pred I) F :
+  (forall i, K i -> d %| F i) -> d %| \sum_(i <- r | K i) F i.
+Proof. by move=> dF; elim/big_ind: _ => //; apply: dvdn_add. Qed.
+
+Lemma dvdn_partP n m : 0 < n ->
+  reflect (forall p, p \in \pi(n) -> n`_p %| m) (n %| m).
+Proof.
+move=> n_gt0; apply: (iffP idP) => n_dvd_m => [p _|].
+  by apply: dvdn_trans n_dvd_m; apply: 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 (forall 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.
+
+(* The Euler totient function *)
+
+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 apply: 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; apply/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 exists (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.
+
+
+