phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
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.	Fixpoint	elogn2	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
elogn2_spec n : nat * nat -> Type	:= Elogn2Spec e m of n = 2 ^ e * m.*2.+1 : elogn2_spec n (e, m).	Variant	elogn2_spec	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
elogn2P n : elogn2_spec n.+1 (elogn2 0 n n).	Proof. rewrite -[n.+1]mul1n -[1]/(2 ^ 0) -[n in _ * n.+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. rewrite subn1 prednK // -mul2n mulnA -expnSr. by rewrite -[q in _ * q.+1](addKn q q) addnn => _; apply: IHq. Qed.	Lemma	elogn2P	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "addKn", "addnn", "apply", "elogn2", "elogn2_spec", "expnSr", "last", "leqnn", "ltnW", "mul1n", "mul2n", "mulnA", "prednK", "subn1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ifnz T n (x y : T)	:= if n is 0 then y else x.	Definition	ifnz	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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.	Variant	ifnz_spec	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ifnzP T n (x y : T) : ifnz_spec n x y (ifnz n x y).	Proof. by case: n => [\|n]; [right \| left]. Qed.	Lemma	ifnzP	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "ifnz", "ifnz_spec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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_divisors	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "iter", "leq", "map", "merge", "mul" ]	the decomposition, using a merge_sort variant sort of the divisor list.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_totient_factor f m	:= let: (p, e) := f in p.-1 * p ^ e.-1 * m.	Definition	add_totient_factor	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cons_pfactor (p e : nat) pd	:= ifnz e ((p, e) :: pd) pd.	Definition	cons_pfactor	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "ifnz", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"p ^? e :: pd"	:= (cons_pfactor p e pd) (at level 30, e at level 30, pd at level 60) : nat_scope.	Notation	p ^? e :: pd	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "cons_pfactor" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
NumFactor (f : nat * nat)	:= ([Num of f.1], f.2).	Definition	NumFactor	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "nat" ]	For pretty-printing.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pfactor p e	:= p ^ e.	Definition	pfactor	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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 ^...	Fixpoint	prime_decomp_rec	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "edivn2", "ifnz" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	prime_decomp	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "edivn", "elogn2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
primes n	:= unzip1 (prime_decomp n).	Definition	primes	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "prime_decomp", "unzip1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prime p	:= if prime_decomp p is [:: (_ , 1)] then true else false.	Definition	prime	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "prime_decomp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nat_pred	:= simpl_pred nat.	Definition	nat_pred	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_arg	:= nat.	Definition	pi_arg	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_arg_of_nat (n : nat) : pi_arg	:= n.	Coercion	pi_arg_of_nat	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "nat", "pi_arg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_arg_of_fin_pred T pT (A : @fin_pred_sort T pT) : pi_arg	:= #\|A\|.	Coercion	pi_arg_of_fin_pred	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "fin_pred_sort", "pi_arg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_of (n : pi_arg) : nat_pred	:= [pred p in primes n].	Definition	pi_of	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "nat_pred", "pi_arg", "primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\pi ( n )"	:= (pi_of n) (format "\pi ( n )") : nat_scope.	Notation	\pi ( n )	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "pi_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\p 'i' ( A )"	:= \pi(#\|A\|) (format "\p 'i' ( A )") : nat_scope.	Notation	\p 'i' ( A )	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pdiv n	:= head 1 (primes n).	Definition	pdiv	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "head", "primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_pdiv n	:= last 1 (primes n).	Definition	max_pdiv	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "last", "primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divisors n	:= foldr add_divisors [:: 1] (prime_decomp n).	Definition	divisors	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "add_divisors", "foldr", "prime_decomp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
totient n	:= foldr add_totient_factor (n > 0) (prime_decomp n).	Definition	totient	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "add_totient_factor", "foldr", "prime_decomp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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,...	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 ^ ...	Lemma	prime_decomp_correct	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "add2n", "addIn", "addKn", "addSn", "addSnnS", "addn0", "addn1", "addn2", "addnA", "addnACA", "addnC", "addnCA", "addnS", "addn_eq0", "addnn", "all", "apply", "coprime", "def_q", "def_r", "doubleB", "doubleD", "doubleS", "double_gt0", "dvdn", "dvdn2", "dvdnP", "...	Correctness of the decomposition algorithm.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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. le...	Lemma	primePn	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "def_q", "dvdn_mulr", "expn1", "expnS", "hasP", "last", "leq_mul", "ltnLHS", "ltnW", "ltn_mul", "ltn_pmul2l", "mem_index_iota", "mul1n", "muln0", "muln1", "mulnA", "pfactor", "prime", "prime_decomp", "prime_decomp_correct" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
primeNsig n : ~~ prime n -> 2 <= n -> { d : nat \| 1 < d < n & d %\| n }.	Proof. by move=> /primePn; case: ltnP => // lt1n nP _; apply/sig2W; case: nP. Qed.	Lemma	primeNsig	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "ltnP", "nat", "prime", "primePn", "sig2W" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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 n...	Lemma	primeP	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "dvdn_gt0", "dvdn_leq", "eq_sym", "leqP", "ltnNge", "ltnW", "ltn_neqAle", "ltnn", "pr_p", "prime", "primePn", "split", "xpred2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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.	Lemma	prime_nt_dvdP	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "prime", "primeP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prime_gt1 p : prime p -> 1 < p.	Proof. by case/primeP. Qed.	Lemma	prime_gt1	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "prime", "primeP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prime_gt0 p : prime p -> 0 < p.	Proof. by move/prime_gt1; apply: ltnW. Qed.	Lemma	prime_gt0	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "ltnW", "prime", "prime_gt1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_prime_decomp n : n > 0 -> n = \prod_(f <- prime_decomp n) f.1 ^ f.2.	Proof. by case/prime_decomp_correct. Qed.	Lemma	prod_prime_decomp	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "prime_decomp", "prime_decomp_correct" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	even_prime	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "dvdn2", "odd", "pr_p", "prime", "primeP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	prime_oddPn	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "even_prime", "odd", "p_pr", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
odd_prime_gt2 p : odd p -> prime p -> p > 2.	Proof. by move=> odd_p /prime_gt1; apply: odd_gt2. Qed.	Lemma	odd_prime_gt2	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "odd", "odd_gt2", "prime", "prime_gt1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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...	Lemma	mem_prime_decomp	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "allP", "apply", "big_cat", "big_cons", "def_n", "dvdn_gt0", "dvdn_leq", "dvdn_mulr", "eq_sym", "hasP", "last", "ltnW", "ltn_neqAle", "ltn_trans", "mapP", "mem_index_iota", "mulnCA", "order_path_min", "posnP", "pr_p", "prime", "primeP", "prime_decomp", "prime_decomp_cor...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	prime_coprime	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "addn0", "apply", "coprime", "dvdnP", "gcdn0", "gcdnMDl", "gcdn_def", "p_gt1", "p_pr", "prime", "primeP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	dvdn_prime2	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "coprime_sym", "eqn_dvd", "pr_p", "prime", "prime_coprime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Euclid_dvd1 p : prime p -> (p %\| 1) = false.	Proof. by rewrite dvdn1; case: eqP => // ->. Qed.	Lemma	Euclid_dvd1	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "dvdn1", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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_dvdM	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "Gauss_dvdr", "dvdn_mulr", "pr_p", "prime", "prime_coprime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Euclid_dvd_prod (I : Type) (r : seq I) (P : pred I) (f : I -> nat) p : prime p -> (p %\| \prod_(i <- r \| P i) f i) = \big[orb/false]_(i <- r \| P i) (p %\| f i).	Proof. move=> pP; apply: big_morph=> [x y\|]; [exact: Euclid_dvdM \| exact: Euclid_dvd1]. Qed.	Lemma	Euclid_dvd_prod	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "Euclid_dvd1", "Euclid_dvdM", "apply", "big_morph", "nat", "pP", "prime", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	Euclid_dvdX	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "Euclid_dvd1", "apply", "coprime_pexpr", "pr_p", "prime", "prime_coprime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_primes p n : (p \in primes n) = [&& prime p, n > 0 & p %\| n].	Proof. rewrite andbCA; have [-> // \| /= n_gt0] := posnP. apply/mapP/andP=> [[[q e]]\|[pr_p]] /=. case/mem_prime_decomp=> pr_q e_gt0 /dvdnP [u ->] -> {p}. by rewrite -(prednK e_gt0) expnS mulnCA dvdn_mulr. rewrite [n in _ %\| n]prod_prime_decomp // big_seq. apply big_ind => [\| u v IHu IHv \| [q e] /= mem_qe dv_p_qe]. -...	Lemma	mem_primes	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "Euclid_dvd1", "Euclid_dvdM", "Euclid_dvdX", "apply", "big_ind", "big_seq", "dvdnP", "dvdn_mulr", "dvdn_prime2", "expnS", "mapP", "mem_prime_decomp", "mulnCA", "n_gt0", "posnP", "pr_p", "prednK", "prime", "primes", "prod_prime_decomp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sorted_primes n : sorted ltn (primes n).	Proof. by case: (posnP n) => [-> // \| /prime_decomp_correct[_ _]]; apply: path_sorted. Qed.	Lemma	sorted_primes	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "ltn", "path_sorted", "posnP", "prime_decomp_correct", "primes", "sorted" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
all_prime_primes n : all prime (primes n).	Proof. by apply/allP => p; rewrite mem_primes => /and3P[]. Qed.	Lemma	all_prime_primes	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "all", "allP", "apply", "mem_primes", "prime", "primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_primes m n : (primes m =i primes n) <-> (primes m = primes n).	Proof. split=> [eqpr\| -> //]. by apply: (irr_sorted_eq ltn_trans ltnn); rewrite ?sorted_primes. Qed.	Lemma	eq_primes	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "irr_sorted_eq", "ltn_trans", "ltnn", "primes", "sorted_primes", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
primes_uniq n : uniq (primes n).	Proof. exact: (sorted_uniq ltn_trans ltnn (sorted_primes n)). Qed.	Lemma	primes_uniq	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "ltn_trans", "ltnn", "primes", "sorted_primes", "sorted_uniq", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	pi_pdiv	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "inE", "ltn0Sn", "mem_head", "pdiv", "pi", "prime_decomp", "primes", "prod_prime_decomp" ]	The smallest prime divisor	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pdiv_prime n : 1 < n -> prime (pdiv n).	Proof. by rewrite -pi_pdiv mem_primes; case/and3P. Qed.	Lemma	pdiv_prime	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "mem_primes", "pdiv", "pi_pdiv", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pdiv_dvd n : pdiv n %\| n.	Proof. by case: n (pi_pdiv n) => [\|[\|n]] //; rewrite mem_primes=> /and3P[]. Qed.	Lemma	pdiv_dvd	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "mem_primes", "pdiv", "pi_pdiv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	pi_max_pdiv	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "inE", "max_pdiv", "mem_head", "mem_last", "pdiv", "pi", "pi_pdiv", "primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_pdiv_prime n : n > 1 -> prime (max_pdiv n).	Proof. by rewrite -pi_max_pdiv mem_primes => /andP[]. Qed.	Lemma	max_pdiv_prime	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "max_pdiv", "mem_primes", "pi_max_pdiv", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_pdiv_dvd n : max_pdiv n %\| n.	Proof. by case: n (pi_max_pdiv n) => [\|[\|n]] //; rewrite mem_primes => /andP[]. Qed.	Lemma	max_pdiv_dvd	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "max_pdiv", "mem_primes", "pi_max_pdiv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pdiv_leq n : 0 < n -> pdiv n <= n.	Proof. by move=> n_gt0; rewrite dvdn_leq // pdiv_dvd. Qed.	Lemma	pdiv_leq	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "dvdn_leq", "n_gt0", "pdiv", "pdiv_dvd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_pdiv_leq n : 0 < n -> max_pdiv n <= n.	Proof. by move=> n_gt0; rewrite dvdn_leq // max_pdiv_dvd. Qed.	Lemma	max_pdiv_leq	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "dvdn_leq", "max_pdiv", "max_pdiv_dvd", "n_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pdiv_gt0 n : 0 < pdiv n.	Proof. by case: n => [\|[\|n]] //; rewrite prime_gt0 ?pdiv_prime. Qed.	Lemma	pdiv_gt0	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "pdiv", "pdiv_prime", "prime_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
max_pdiv_gt0 n : 0 < max_pdiv n.	Proof. by case: n => [\|[\|n]] //; rewrite prime_gt0 ?max_pdiv_prime. Qed.	Lemma	max_pdiv_gt0	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "max_pdiv", "max_pdiv_prime", "prime_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pdiv_min_dvd m d : 1 < d -> d %\| m -> pdiv m <= d.	Proof. case: (posnP m) => [->\|mpos] lt1d dv_d_m; 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. by case/predU1P => [-> \| /(allP...	Lemma	pdiv_min_dvd	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "allP", "apply", "dvdn_trans", "inE", "leq_trans", "ltnW", "ltn_trans", "mem_primes", "order_path_min", "pdiv", "pdiv_dvd", "pdiv_leq", "pdiv_prime", "posnP", "predU1P", "primes", "sorted_primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	max_pdiv_max	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "all_rcons", "cat_path", "cat_rcons", "headI", "inE", "lastP", "last_cat", "last_cons", "last_rcons", "ltn_neqAle", "ltn_trans", "max_pdiv", "order_path_min", "pi", "sorted_primes", "splitPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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; first by rewrite def_n. apply/contraL: lt_n_p2 => lt_pm_m; case/dvdnP: (pdiv_dvd n) => q def_q. rewrite -leqNgt [leqRHS]def_q leq_pmul2r // pdiv_min_d...	Lemma	ltn_pdiv2_prime	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "def_n", "def_q", "dvdnP", "dvdn_mulr", "eqn_leq", "leqNgt", "leqRHS", "leq_pmul2r", "ltnRHS", "ltn_pmul2r", "mul1n", "n'", "pdiv", "pdiv_dvd", "pdiv_leq", "pdiv_min_dvd", "pdiv_prime", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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 -[ltnLHS]muln1 ltn_pmul2l ?prime_gt1 ?prime_gt0. have [lt1p\|] := leqP; [right \| by left]. exists (pdiv n)...	Lemma	primePns	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "dvdn_prime2", "last", "leqNgt", "leqP", "ltnLHS", "ltnW", "ltn_pdiv2_prime", "ltn_pmul2l", "muln1", "pdiv", "pdiv_dvd", "pdiv_prime", "pr_p", "prime", "prime_gt0", "prime_gt1", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pdivP n : n > 1 -> {p \| prime p & p %\| n}.	Proof. by move=> lt1n; exists (pdiv n); rewrite ?pdiv_dvd ?pdiv_prime. Qed.	Lemma	pdivP	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "pdiv", "pdiv_dvd", "pdiv_prime", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
primes_eq0 n : (primes n == [::]) = (n < 2).	Proof. case: n => [\|[\|n']]//=; have [//\|p pp pn] := @pdivP (n'.+2). suff: p \in primes n'.+2 by case: primes. by rewrite mem_primes pp pn. Qed.	Lemma	primes_eq0	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "mem_primes", "n'", "pdivP", "primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
primesM 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	primesM	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "Euclid_dvdM", "mem_primes", "muln_gt0", "n_gt0", "pr_p", "prime", "primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
primesX m n : n > 0 -> primes (m ^ n) = primes m.	Proof. case: n => // n _; rewrite expnS; have [-> // \| m_gt0] := posnP m. apply/eq_primes => /= p; elim: n => [\|n IHn]; first by rewrite muln1. by rewrite primesM ?(expn_gt0, expnS, IHn, orbb, m_gt0). Qed.	Lemma	primesX	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "eq_primes", "expnS", "expn_gt0", "muln1", "posnP", "primes", "primesM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
primes_prime p : prime p -> primes p = [:: p].	Proof. move=> pr_p; apply: (irr_sorted_eq 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	primes_prime	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "dvdn_prime2", "irr_sorted_eq", "ltn_trans", "ltnn", "mem_primes", "mem_seq1", "pr_p", "prime", "prime_gt0", "primes", "sorted_primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprime_has_primes m n : 0 < m -> 0 < n -> coprime m n = ~~ has [in primes m] (primes n).	Proof. move=> m_gt0 n_gt0; apply/eqP/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. apply/eqP; rewrite eqn_leq gcdn_gt0 m_gt0 andbT leqNgt; apply/n...	Lemma	coprime_has_primes	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "coprime", "dvdn_gcd", "dvdn_gcdl", "dvdn_gcdr", "dvdn_leq", "dvdn_trans", "eqn_leq", "gcdn_gt0", "has", "hasPn", "leqNgt", "ltnNge", "mem_primes", "n_gt0", "pdiv", "pdiv_dvd", "pdiv_prime", "pr_p", "prime_gt1", "primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pdiv_id p : prime p -> pdiv p = p.	Proof. by move=> p_pr; rewrite /pdiv primes_prime. Qed.	Lemma	pdiv_id	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "p_pr", "pdiv", "prime", "primes_prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pdiv_pfactor p k : prime p -> pdiv (p ^ k.+1) = p.	Proof. by move=> p_pr; rewrite /pdiv primesX ?primes_prime. Qed.	Lemma	pdiv_pfactor	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "p_pr", "pdiv", "prime", "primesX", "primes_prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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.	Lemma	prime_above	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "addn1", "apply", "dvdn_addr", "dvdn_fact", "fact_gt0", "gtnNdvd", "ltnNge", "ltnS", "pdivP", "pr_p", "prime", "prime_gt0", "prime_gt1" ]	Primes are unbounded.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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.	Fixpoint	logn_rec	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "edivn" ]	"prime" logarithms and p-parts.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
logn p m	:= if prime p then logn_rec p m m else 0.	Definition	logn	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "logn_rec", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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) => //. case def_m: m => // [m']; rewrite !andTb [LHS]/= -def_m /divn modn_def. 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/ltn_ind...	Lemma	lognE	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "addn0", "def_q", "divn", "dvdn", "edivnP", "leq_trans", "logn", "logn_rec", "ltn0Sn", "ltnS", "ltn_ind", "ltn_pmul2l", "ltn_pmul2r", "modn_def", "mul1n", "muln1", "mulnC", "p_pr", "prednK", "prime", "prime_gt1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
logn_gt0 p n : (0 < logn p n) = (p \in primes n).	Proof. by rewrite lognE -mem_primes; case: {+}(p \in _). Qed.	Lemma	logn_gt0	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "logn", "lognE", "mem_primes", "primes" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltn_log0 p n : n < p -> logn p n = 0.	Proof. by case: n => [\|n] ltnp; rewrite lognE ?andbF // gtnNdvd ?andbF. Qed.	Lemma	ltn_log0	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "gtnNdvd", "logn", "lognE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
logn0 p : logn p 0 = 0.	Proof. by rewrite /logn if_same. Qed.	Lemma	logn0	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "logn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
logn1 p : logn p 1 = 0.	Proof. by rewrite lognE dvdn1 /= andbC; case: eqP => // ->. Qed.	Lemma	logn1	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "dvdn1", "logn", "lognE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pfactor_gt0 p n : 0 < p ^ logn p n.	Proof. by rewrite expn_gt0 lognE; case: (posnP p) => // ->. Qed.	Lemma	pfactor_gt0	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "expn_gt0", "logn", "lognE", "posnP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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_pm...	Lemma	pfactor_dvdn	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "dvd1n", "dvdnP", "dvdn_mulr", "dvdn_pmul2r", "expnS", "expnSr", "last", "logn", "lognE", "mulnCA", "mulnK", "muln_gt0", "p_gt0", "p_pr", "prime", "q_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	pfactor_dvdnn	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "dvd1n", "logn", "lognE", "pfactor_dvdn", "pr_p", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	logn_prime	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "divnn", "dvdn_prime2", "last", "logn", "logn1", "lognE", "pr_p", "prime", "prime_gt0", "q_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	pfactor_coprime	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "coprime", "divnK", "dvdn_pmul2r", "expnS", "expn_gt0", "last", "logn", "ltnn", "n_gt0", "p_pr", "pfactor_dvdn", "prime", "prime_coprime", "prime_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	pfactorK	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "dvdn_leq", "dvdnn", "eqn_leq", "expn_gt0", "leq_exp2l", "logn", "p_pr", "pfactor_dvdn", "prime", "prime_gt0", "prime_gt1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	pfactorKpdiv	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "logn", "p_pr", "pdiv", "pdiv_pfactor", "pfactorK", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	dvdn_leq_log	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "dvdn_gt0", "dvdn_trans", "last", "logn", "lognE", "n_gt0", "p_pr", "pfactor_dvdn", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	ltn_logl	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "dvdn_leq", "leq_trans", "logn", "lognE", "ltn_expl", "n_gt0", "p_gt1", "pfactor_dvdnn", "prime_gt1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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...	Lemma	logn_Gauss	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "Gauss_dvdr", "apply", "coprime", "coprimeXl", "dvdn0", "dvdn_leq_log", "dvdn_mull", "eqn_leq", "last", "logn", "muln0", "muln_gt0", "n_gt0", "p_pr", "pfactor_dvdn", "posnP", "prime", "prime_coprime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
logn_coprime p m : coprime p m -> logn p m = 0.	Proof. by move=> coprime_pm; rewrite -[m]muln1 logn_Gauss// logn1. Qed.	Lemma	logn_coprime	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "coprime", "logn", "logn1", "logn_Gauss", "muln1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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}. rewrite [in LHS]def_m [in LHS]def_n mulnCA -mulnA -expnD !logn_Gauss //. exact: pfactorK. Qed.	Lemma	lognM	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "def_n", "expnD", "last", "logn", "logn_Gauss", "mulnA", "mulnCA", "n'", "p_pr", "pfactorK", "pfactor_coprime", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	lognX	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "exp0n", "expnS", "expn_gt0", "last", "logn", "logn1", "lognE", "lognM", "muln0", "p_pr", "posnP", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	logn_div	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "addnK", "def_n", "div0n", "dvdn_eq", "logn", "logn0", "lognM", "muln_gt0", "posnP", "q_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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...	Lemma	dvdn_pfactor	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "Gauss_dvdr", "apply", "coprimeXl", "coprime_sym", "d_gt0", "dvdn1", "dvdn_gt0", "dvdn_leq_log", "dvdn_mull", "dvdn_mulr", "dvdn_trans", "expnD", "expn_gt0", "last", "logn", "mul1n", "muln1", "p_pr", "pfactorK", "pfactor_coprime", "prime", "prime_gt0", "subnK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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...	Lemma	prime_decompE	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "apply", "bigD1", "big_ind", "big_mkord", "big_nth", "coprimeMr", "coprime_pexpr", "coprimen1", "dvdn_prime2", "eq_from_nth", "eq_sym", "logn", "logn_Gauss", "map_comp", "mem_nth", "mem_prime_decomp", "mulnC", "nth", "nth_map", "nth_uniq", "pfactorK", "pr_p", "prime_copri...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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 in LHS]card_ord -sum1_card. apply: eq_bigr => [[q ?] _]. re...	Lemma	divn_count_dvd	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "add1n", "addnI", "apply", "big1", "bigD1", "big_add1", "big_ltn", "big_mkord", "card_ord", "d_gt0", "divn0", "dvdn", "dvdn0", "dvdn_eq", "dvdn_mull", "eq_bigr", "eq_sym", "inord", "inordK", "leq_div2r", "leq_divRL", "ltnS", "mulnK", "partition_big", "posnP", "sum1_...	Some combinatorial formulae.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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.	Lemma	logn_count_dvd	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "big_add1", "big_geq", "big_mkcond", "big_mkord", "big_ord_narrow", "card_ord", "eq_bigl", "logn", "logn0", "ltnS", "ltn_logl", "pfactor_dvdn", "prime", "sum1_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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 if p <= 1 then 0 else loop n n.	Definition	trunc_log	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[]	Truncated real log.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trunc_log0 p : trunc_log p 0 = 0.	Proof. by case: p => [] // []. Qed.	Lemma	trunc_log0	boot	boot/prime.v	[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]	[ "trunc_log" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

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.

Fixpoint

elogn2

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

elogn2_spec n : nat * nat -> Type

:= Elogn2Spec e m of n = 2 ^ e * m.*2.+1 : elogn2_spec n (e, m).

Variant

elogn2_spec

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

elogn2P n : elogn2_spec n.+1 (elogn2 0 n n).

Proof. rewrite -[n.+1]mul1n -[1]/(2 ^ 0) -[n in _ * n.+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. rewrite subn1 prednK // -mul2n mulnA -expnSr. by rewrite -[q in _ * q.+1](addKn q q) addnn => _; apply: IHq. Qed.

Lemma

elogn2P

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "addKn", "addnn", "apply", "elogn2", "elogn2_spec", "expnSr", "last", "leqnn", "ltnW", "mul1n", "mul2n", "mulnA", "prednK", "subn1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ifnz T n (x y : T)

:= if n is 0 then y else x.

Definition

ifnz

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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.

Variant

ifnz_spec

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ifnzP T n (x y : T) : ifnz_spec n x y (ifnz n x y).

Proof. by case: n => [|n]; [right | left]. Qed.

Lemma

ifnzP

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "ifnz", "ifnz_spec" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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_divisors

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "iter", "leq", "map", "merge", "mul" ]

the decomposition, using a merge_sort variant sort of the divisor list.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

add_totient_factor f m

:= let: (p, e) := f in p.-1 * p ^ e.-1 * m.

Definition

add_totient_factor

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cons_pfactor (p e : nat) pd

:= ifnz e ((p, e) :: pd) pd.

Definition

cons_pfactor

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "ifnz", "nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"p ^? e :: pd"

:= (cons_pfactor p e pd) (at level 30, e at level 30, pd at level 60) : nat_scope.

Notation

p ^? e :: pd

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "cons_pfactor" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

NumFactor (f : nat * nat)

:= ([Num of f.1], f.2).

Definition

NumFactor

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "nat" ]

For pretty-printing.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pfactor p e

:= p ^ e.

Definition

pfactor

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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 ^...

Fixpoint

prime_decomp_rec

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "edivn2", "ifnz" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

prime_decomp

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "edivn", "elogn2" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

primes n

:= unzip1 (prime_decomp n).

Definition

primes

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "prime_decomp", "unzip1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prime p

:= if prime_decomp p is [:: (_ , 1)] then true else false.

Definition

prime

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "prime_decomp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nat_pred

:= simpl_pred nat.

Definition

nat_pred

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_arg

:= nat.

Definition

pi_arg

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_arg_of_nat (n : nat) : pi_arg

:= n.

Coercion

pi_arg_of_nat

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "nat", "pi_arg" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_arg_of_fin_pred T pT (A : @fin_pred_sort T pT) : pi_arg

:= #|A|.

Coercion

pi_arg_of_fin_pred

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "fin_pred_sort", "pi_arg" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_of (n : pi_arg) : nat_pred

:= [pred p in primes n].

Definition

pi_of

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "nat_pred", "pi_arg", "primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\pi ( n )"

:= (pi_of n) (format "\pi ( n )") : nat_scope.

Notation

\pi ( n )

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "pi_of" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\p 'i' ( A )"

:= \pi(#|A|) (format "\p 'i' ( A )") : nat_scope.

Notation

\p 'i' ( A )

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "pi" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pdiv n

:= head 1 (primes n).

Definition

pdiv

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "head", "primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

max_pdiv n

:= last 1 (primes n).

Definition

max_pdiv

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "last", "primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divisors n

:= foldr add_divisors [:: 1] (prime_decomp n).

Definition

divisors

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "add_divisors", "foldr", "prime_decomp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

totient n

:= foldr add_totient_factor (n > 0) (prime_decomp n).

Definition

totient

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "add_totient_factor", "foldr", "prime_decomp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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,...

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 ^ ...

Lemma

prime_decomp_correct

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "add2n", "addIn", "addKn", "addSn", "addSnnS", "addn0", "addn1", "addn2", "addnA", "addnACA", "addnC", "addnCA", "addnS", "addn_eq0", "addnn", "all", "apply", "coprime", "def_q", "def_r", "doubleB", "doubleD", "doubleS", "double_gt0", "dvdn", "dvdn2", "dvdnP", "...

Correctness of the decomposition algorithm.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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. le...

Lemma

primePn

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "def_q", "dvdn_mulr", "expn1", "expnS", "hasP", "last", "leq_mul", "ltnLHS", "ltnW", "ltn_mul", "ltn_pmul2l", "mem_index_iota", "mul1n", "muln0", "muln1", "mulnA", "pfactor", "prime", "prime_decomp", "prime_decomp_correct" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

primeNsig n : ~~ prime n -> 2 <= n -> { d : nat | 1 < d < n & d %| n }.

Proof. by move=> /primePn; case: ltnP => // lt1n nP _; apply/sig2W; case: nP. Qed.

Lemma

primeNsig

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "ltnP", "nat", "prime", "primePn", "sig2W" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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 n...

Lemma

primeP

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "dvdn_gt0", "dvdn_leq", "eq_sym", "leqP", "ltnNge", "ltnW", "ltn_neqAle", "ltnn", "pr_p", "prime", "primePn", "split", "xpred2" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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.

Lemma

prime_nt_dvdP

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "prime", "primeP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prime_gt1 p : prime p -> 1 < p.

Proof. by case/primeP. Qed.

Lemma

prime_gt1

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "prime", "primeP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prime_gt0 p : prime p -> 0 < p.

Proof. by move/prime_gt1; apply: ltnW. Qed.

Lemma

prime_gt0

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "ltnW", "prime", "prime_gt1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod_prime_decomp n : n > 0 -> n = \prod_(f <- prime_decomp n) f.1 ^ f.2.

Proof. by case/prime_decomp_correct. Qed.

Lemma

prod_prime_decomp

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "prime_decomp", "prime_decomp_correct" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

even_prime

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "dvdn2", "odd", "pr_p", "prime", "primeP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

prime_oddPn

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "even_prime", "odd", "p_pr", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

odd_prime_gt2 p : odd p -> prime p -> p > 2.

Proof. by move=> odd_p /prime_gt1; apply: odd_gt2. Qed.

Lemma

odd_prime_gt2

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "odd", "odd_gt2", "prime", "prime_gt1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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...

Lemma

mem_prime_decomp

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "allP", "apply", "big_cat", "big_cons", "def_n", "dvdn_gt0", "dvdn_leq", "dvdn_mulr", "eq_sym", "hasP", "last", "ltnW", "ltn_neqAle", "ltn_trans", "mapP", "mem_index_iota", "mulnCA", "order_path_min", "posnP", "pr_p", "prime", "primeP", "prime_decomp", "prime_decomp_cor...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

prime_coprime

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "addn0", "apply", "coprime", "dvdnP", "gcdn0", "gcdnMDl", "gcdn_def", "p_gt1", "p_pr", "prime", "primeP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

dvdn_prime2

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "coprime_sym", "eqn_dvd", "pr_p", "prime", "prime_coprime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Euclid_dvd1 p : prime p -> (p %| 1) = false.

Proof. by rewrite dvdn1; case: eqP => // ->. Qed.

Lemma

Euclid_dvd1

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "dvdn1", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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_dvdM

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "Gauss_dvdr", "dvdn_mulr", "pr_p", "prime", "prime_coprime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Euclid_dvd_prod (I : Type) (r : seq I) (P : pred I) (f : I -> nat) p : prime p -> (p %| \prod_(i <- r | P i) f i) = \big[orb/false]_(i <- r | P i) (p %| f i).

Proof. move=> pP; apply: big_morph=> [x y|]; [exact: Euclid_dvdM | exact: Euclid_dvd1]. Qed.

Lemma

Euclid_dvd_prod

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "Euclid_dvd1", "Euclid_dvdM", "apply", "big_morph", "nat", "pP", "prime", "seq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Euclid_dvdX

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "Euclid_dvd1", "apply", "coprime_pexpr", "pr_p", "prime", "prime_coprime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_primes p n : (p \in primes n) = [&& prime p, n > 0 & p %| n].

Proof. rewrite andbCA; have [-> // | /= n_gt0] := posnP. apply/mapP/andP=> [[[q e]]|[pr_p]] /=. case/mem_prime_decomp=> pr_q e_gt0 /dvdnP [u ->] -> {p}. by rewrite -(prednK e_gt0) expnS mulnCA dvdn_mulr. rewrite [n in _ %| n]prod_prime_decomp // big_seq. apply big_ind => [| u v IHu IHv | [q e] /= mem_qe dv_p_qe]. -...

Lemma

mem_primes

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "Euclid_dvd1", "Euclid_dvdM", "Euclid_dvdX", "apply", "big_ind", "big_seq", "dvdnP", "dvdn_mulr", "dvdn_prime2", "expnS", "mapP", "mem_prime_decomp", "mulnCA", "n_gt0", "posnP", "pr_p", "prednK", "prime", "primes", "prod_prime_decomp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sorted_primes n : sorted ltn (primes n).

Proof. by case: (posnP n) => [-> // | /prime_decomp_correct[_ _]]; apply: path_sorted. Qed.

Lemma

sorted_primes

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "ltn", "path_sorted", "posnP", "prime_decomp_correct", "primes", "sorted" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

all_prime_primes n : all prime (primes n).

Proof. by apply/allP => p; rewrite mem_primes => /and3P[]. Qed.

Lemma

all_prime_primes

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "all", "allP", "apply", "mem_primes", "prime", "primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_primes m n : (primes m =i primes n) <-> (primes m = primes n).

Proof. split=> [eqpr| -> //]. by apply: (irr_sorted_eq ltn_trans ltnn); rewrite ?sorted_primes. Qed.

Lemma

eq_primes

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "irr_sorted_eq", "ltn_trans", "ltnn", "primes", "sorted_primes", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

primes_uniq n : uniq (primes n).

Proof. exact: (sorted_uniq ltn_trans ltnn (sorted_primes n)). Qed.

Lemma

primes_uniq

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "ltn_trans", "ltnn", "primes", "sorted_primes", "sorted_uniq", "uniq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

pi_pdiv

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "inE", "ltn0Sn", "mem_head", "pdiv", "pi", "prime_decomp", "primes", "prod_prime_decomp" ]

The smallest prime divisor

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pdiv_prime n : 1 < n -> prime (pdiv n).

Proof. by rewrite -pi_pdiv mem_primes; case/and3P. Qed.

Lemma

pdiv_prime

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "mem_primes", "pdiv", "pi_pdiv", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pdiv_dvd n : pdiv n %| n.

Proof. by case: n (pi_pdiv n) => [|[|n]] //; rewrite mem_primes=> /and3P[]. Qed.

Lemma

pdiv_dvd

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "mem_primes", "pdiv", "pi_pdiv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

pi_max_pdiv

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "inE", "max_pdiv", "mem_head", "mem_last", "pdiv", "pi", "pi_pdiv", "primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

max_pdiv_prime n : n > 1 -> prime (max_pdiv n).

Proof. by rewrite -pi_max_pdiv mem_primes => /andP[]. Qed.

Lemma

max_pdiv_prime

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "max_pdiv", "mem_primes", "pi_max_pdiv", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

max_pdiv_dvd n : max_pdiv n %| n.

Proof. by case: n (pi_max_pdiv n) => [|[|n]] //; rewrite mem_primes => /andP[]. Qed.

Lemma

max_pdiv_dvd

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "max_pdiv", "mem_primes", "pi_max_pdiv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pdiv_leq n : 0 < n -> pdiv n <= n.

Proof. by move=> n_gt0; rewrite dvdn_leq // pdiv_dvd. Qed.

Lemma

pdiv_leq

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "dvdn_leq", "n_gt0", "pdiv", "pdiv_dvd" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

max_pdiv_leq n : 0 < n -> max_pdiv n <= n.

Proof. by move=> n_gt0; rewrite dvdn_leq // max_pdiv_dvd. Qed.

Lemma

max_pdiv_leq

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "dvdn_leq", "max_pdiv", "max_pdiv_dvd", "n_gt0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pdiv_gt0 n : 0 < pdiv n.

Proof. by case: n => [|[|n]] //; rewrite prime_gt0 ?pdiv_prime. Qed.

Lemma

pdiv_gt0

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "pdiv", "pdiv_prime", "prime_gt0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

max_pdiv_gt0 n : 0 < max_pdiv n.

Proof. by case: n => [|[|n]] //; rewrite prime_gt0 ?max_pdiv_prime. Qed.

Lemma

max_pdiv_gt0

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "max_pdiv", "max_pdiv_prime", "prime_gt0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pdiv_min_dvd m d : 1 < d -> d %| m -> pdiv m <= d.

Proof. case: (posnP m) => [->|mpos] lt1d dv_d_m; 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. by case/predU1P => [-> | /(allP...

Lemma

pdiv_min_dvd

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "allP", "apply", "dvdn_trans", "inE", "leq_trans", "ltnW", "ltn_trans", "mem_primes", "order_path_min", "pdiv", "pdiv_dvd", "pdiv_leq", "pdiv_prime", "posnP", "predU1P", "primes", "sorted_primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

max_pdiv_max

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "all_rcons", "cat_path", "cat_rcons", "headI", "inE", "lastP", "last_cat", "last_cons", "last_rcons", "ltn_neqAle", "ltn_trans", "max_pdiv", "order_path_min", "pi", "sorted_primes", "splitPr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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; first by rewrite def_n. apply/contraL: lt_n_p2 => lt_pm_m; case/dvdnP: (pdiv_dvd n) => q def_q. rewrite -leqNgt [leqRHS]def_q leq_pmul2r // pdiv_min_d...

Lemma

ltn_pdiv2_prime

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "def_n", "def_q", "dvdnP", "dvdn_mulr", "eqn_leq", "leqNgt", "leqRHS", "leq_pmul2r", "ltnRHS", "ltn_pmul2r", "mul1n", "n'", "pdiv", "pdiv_dvd", "pdiv_leq", "pdiv_min_dvd", "pdiv_prime", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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 -[ltnLHS]muln1 ltn_pmul2l ?prime_gt1 ?prime_gt0. have [lt1p|] := leqP; [right | by left]. exists (pdiv n)...

Lemma

primePns

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "dvdn_prime2", "last", "leqNgt", "leqP", "ltnLHS", "ltnW", "ltn_pdiv2_prime", "ltn_pmul2l", "muln1", "pdiv", "pdiv_dvd", "pdiv_prime", "pr_p", "prime", "prime_gt0", "prime_gt1", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pdivP n : n > 1 -> {p | prime p & p %| n}.

Proof. by move=> lt1n; exists (pdiv n); rewrite ?pdiv_dvd ?pdiv_prime. Qed.

Lemma

pdivP

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "pdiv", "pdiv_dvd", "pdiv_prime", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

primes_eq0 n : (primes n == [::]) = (n < 2).

Proof. case: n => [|[|n']]//=; have [//|p pp pn] := @pdivP (n'.+2). suff: p \in primes n'.+2 by case: primes. by rewrite mem_primes pp pn. Qed.

Lemma

primes_eq0

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "mem_primes", "n'", "pdivP", "primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

primesM 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

primesM

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "Euclid_dvdM", "mem_primes", "muln_gt0", "n_gt0", "pr_p", "prime", "primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

primesX m n : n > 0 -> primes (m ^ n) = primes m.

Proof. case: n => // n _; rewrite expnS; have [-> // | m_gt0] := posnP m. apply/eq_primes => /= p; elim: n => [|n IHn]; first by rewrite muln1. by rewrite primesM ?(expn_gt0, expnS, IHn, orbb, m_gt0). Qed.

Lemma

primesX

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "eq_primes", "expnS", "expn_gt0", "muln1", "posnP", "primes", "primesM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

primes_prime p : prime p -> primes p = [:: p].

Proof. move=> pr_p; apply: (irr_sorted_eq 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

primes_prime

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "dvdn_prime2", "irr_sorted_eq", "ltn_trans", "ltnn", "mem_primes", "mem_seq1", "pr_p", "prime", "prime_gt0", "primes", "sorted_primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprime_has_primes m n : 0 < m -> 0 < n -> coprime m n = ~~ has [in primes m] (primes n).

Proof. move=> m_gt0 n_gt0; apply/eqP/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. apply/eqP; rewrite eqn_leq gcdn_gt0 m_gt0 andbT leqNgt; apply/n...

Lemma

coprime_has_primes

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "coprime", "dvdn_gcd", "dvdn_gcdl", "dvdn_gcdr", "dvdn_leq", "dvdn_trans", "eqn_leq", "gcdn_gt0", "has", "hasPn", "leqNgt", "ltnNge", "mem_primes", "n_gt0", "pdiv", "pdiv_dvd", "pdiv_prime", "pr_p", "prime_gt1", "primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pdiv_id p : prime p -> pdiv p = p.

Proof. by move=> p_pr; rewrite /pdiv primes_prime. Qed.

Lemma

pdiv_id

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "p_pr", "pdiv", "prime", "primes_prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pdiv_pfactor p k : prime p -> pdiv (p ^ k.+1) = p.

Proof. by move=> p_pr; rewrite /pdiv primesX ?primes_prime. Qed.

Lemma

pdiv_pfactor

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "p_pr", "pdiv", "prime", "primesX", "primes_prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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.

Lemma

prime_above

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "addn1", "apply", "dvdn_addr", "dvdn_fact", "fact_gt0", "gtnNdvd", "ltnNge", "ltnS", "pdivP", "pr_p", "prime", "prime_gt0", "prime_gt1" ]

Primes are unbounded.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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.

Fixpoint

logn_rec

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "edivn" ]

"prime" logarithms and p-parts.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

logn p m

:= if prime p then logn_rec p m m else 0.

Definition

logn

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "logn_rec", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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) => //. case def_m: m => // [m']; rewrite !andTb [LHS]/= -def_m /divn modn_def. 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/ltn_ind...

Lemma

lognE

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "addn0", "def_q", "divn", "dvdn", "edivnP", "leq_trans", "logn", "logn_rec", "ltn0Sn", "ltnS", "ltn_ind", "ltn_pmul2l", "ltn_pmul2r", "modn_def", "mul1n", "muln1", "mulnC", "p_pr", "prednK", "prime", "prime_gt1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

logn_gt0 p n : (0 < logn p n) = (p \in primes n).

Proof. by rewrite lognE -mem_primes; case: {+}(p \in _). Qed.

Lemma

logn_gt0

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "logn", "lognE", "mem_primes", "primes" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltn_log0 p n : n < p -> logn p n = 0.

Proof. by case: n => [|n] ltnp; rewrite lognE ?andbF // gtnNdvd ?andbF. Qed.

Lemma

ltn_log0

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "gtnNdvd", "logn", "lognE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

logn0 p : logn p 0 = 0.

Proof. by rewrite /logn if_same. Qed.

Lemma

logn0

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "logn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

logn1 p : logn p 1 = 0.

Proof. by rewrite lognE dvdn1 /= andbC; case: eqP => // ->. Qed.

Lemma

logn1

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "dvdn1", "logn", "lognE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pfactor_gt0 p n : 0 < p ^ logn p n.

Proof. by rewrite expn_gt0 lognE; case: (posnP p) => // ->. Qed.

Lemma

pfactor_gt0

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "expn_gt0", "logn", "lognE", "posnP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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_pm...

Lemma

pfactor_dvdn

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "dvd1n", "dvdnP", "dvdn_mulr", "dvdn_pmul2r", "expnS", "expnSr", "last", "logn", "lognE", "mulnCA", "mulnK", "muln_gt0", "p_gt0", "p_pr", "prime", "q_gt0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

pfactor_dvdnn

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "dvd1n", "logn", "lognE", "pfactor_dvdn", "pr_p", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

logn_prime

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "divnn", "dvdn_prime2", "last", "logn", "logn1", "lognE", "pr_p", "prime", "prime_gt0", "q_gt0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

pfactor_coprime

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "coprime", "divnK", "dvdn_pmul2r", "expnS", "expn_gt0", "last", "logn", "ltnn", "n_gt0", "p_pr", "pfactor_dvdn", "prime", "prime_coprime", "prime_gt0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

pfactorK

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "dvdn_leq", "dvdnn", "eqn_leq", "expn_gt0", "leq_exp2l", "logn", "p_pr", "pfactor_dvdn", "prime", "prime_gt0", "prime_gt1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

pfactorKpdiv

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "logn", "p_pr", "pdiv", "pdiv_pfactor", "pfactorK", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

dvdn_leq_log

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "dvdn_gt0", "dvdn_trans", "last", "logn", "lognE", "n_gt0", "p_pr", "pfactor_dvdn", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

ltn_logl

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "dvdn_leq", "leq_trans", "logn", "lognE", "ltn_expl", "n_gt0", "p_gt1", "pfactor_dvdnn", "prime_gt1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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...

Lemma

logn_Gauss

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "Gauss_dvdr", "apply", "coprime", "coprimeXl", "dvdn0", "dvdn_leq_log", "dvdn_mull", "eqn_leq", "last", "logn", "muln0", "muln_gt0", "n_gt0", "p_pr", "pfactor_dvdn", "posnP", "prime", "prime_coprime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

logn_coprime p m : coprime p m -> logn p m = 0.

Proof. by move=> coprime_pm; rewrite -[m]muln1 logn_Gauss// logn1. Qed.

Lemma

logn_coprime

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "coprime", "logn", "logn1", "logn_Gauss", "muln1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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}. rewrite [in LHS]def_m [in LHS]def_n mulnCA -mulnA -expnD !logn_Gauss //. exact: pfactorK. Qed.

Lemma

lognM

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "def_n", "expnD", "last", "logn", "logn_Gauss", "mulnA", "mulnCA", "n'", "p_pr", "pfactorK", "pfactor_coprime", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

lognX

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "exp0n", "expnS", "expn_gt0", "last", "logn", "logn1", "lognE", "lognM", "muln0", "p_pr", "posnP", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

logn_div

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "addnK", "def_n", "div0n", "dvdn_eq", "logn", "logn0", "lognM", "muln_gt0", "posnP", "q_gt0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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...

Lemma

dvdn_pfactor

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "Gauss_dvdr", "apply", "coprimeXl", "coprime_sym", "d_gt0", "dvdn1", "dvdn_gt0", "dvdn_leq_log", "dvdn_mull", "dvdn_mulr", "dvdn_trans", "expnD", "expn_gt0", "last", "logn", "mul1n", "muln1", "p_pr", "pfactorK", "pfactor_coprime", "prime", "prime_gt0", "subnK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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...

Lemma

prime_decompE

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "apply", "bigD1", "big_ind", "big_mkord", "big_nth", "coprimeMr", "coprime_pexpr", "coprimen1", "dvdn_prime2", "eq_from_nth", "eq_sym", "logn", "logn_Gauss", "map_comp", "mem_nth", "mem_prime_decomp", "mulnC", "nth", "nth_map", "nth_uniq", "pfactorK", "pr_p", "prime_copri...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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 in LHS]card_ord -sum1_card. apply: eq_bigr => [[q ?] _]. re...

Lemma

divn_count_dvd

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "add1n", "addnI", "apply", "big1", "bigD1", "big_add1", "big_ltn", "big_mkord", "card_ord", "d_gt0", "divn0", "dvdn", "dvdn0", "dvdn_eq", "dvdn_mull", "eq_bigr", "eq_sym", "inord", "inordK", "leq_div2r", "leq_divRL", "ltnS", "mulnK", "partition_big", "posnP", "sum1_...

Some combinatorial formulae.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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.

Lemma

logn_count_dvd

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "big_add1", "big_geq", "big_mkcond", "big_mkord", "big_ord_narrow", "card_ord", "eq_bigl", "logn", "logn0", "ltnS", "ltn_logl", "pfactor_dvdn", "prime", "sum1_card" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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 if p <= 1 then 0 else loop n n.

Definition

trunc_log

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[]

Truncated real log.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trunc_log0 p : trunc_log p 0 = 0.

Proof. by case: p => [] // []. Qed.

Lemma

trunc_log0

boot

boot/prime.v

[ "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "bigop", "NatTrec" ]

[ "trunc_log" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d