fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
prime_decompn :=
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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | prime_decomp | |
primesn := unzip1 (prime_decomp n). | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | primes | |
primep := if prime_decomp p is [:: (_ , 1)] then true else false. | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | prime | |
nat_pred:= simpl_pred nat. | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | nat_pred | |
pi_arg:= nat. | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pi_arg | |
pi_arg_of_nat(n : nat) : pi_arg := n. | Coercion | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pi_arg_of_nat | |
pi_arg_of_fin_predT pT (A : @fin_pred_sort T pT) : pi_arg := #|A|.
Arguments pi_arg_of_nat n /.
Arguments pi_arg_of_fin_pred {T pT} A /. | Coercion | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pi_arg_of_fin_pred | |
pi_of(n : pi_arg) : nat_pred := [pred p in primes n]. | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pi_of | |
pdivn := head 1 (primes n). | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pdiv | |
max_pdivn := last 1 (primes n). | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | max_pdiv | |
divisorsn := foldr add_divisors [:: 1] (prime_decomp n). | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | divisors | |
totientn := foldr add_totient_factor (n > 0) (prime_decomp n). | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | totient | |
prime_decomp_correct:
let pd_val pd := \prod_(f <- pd) pfactor f.1 f.2 in
let lb_dvd q m := ~~ has [pred d | d %| m] (index_iota 2 q) in
let pf_ok f := lb_dvd f.1 f.1 && (0 < f.2) in
let pd_ord q pd := path ltn q (unzip1 pd) in
let pd_ok q n pd := [/\ n = pd_val pd, all pf_ok pd & pd_ord q pd] in
forall n, n > 0 -> pd_ok 1 n (prime_decomp n).
Proof.
rewrite unlock => pd_val lb_dvd pf_ok pd_ord pd_ok.
have leq_pd_ok m p q pd: q <= p -> pd_ok p m pd -> pd_ok q m pd.
rewrite /pd_ok /pd_ord; case: pd => [|[r _] pd] //= leqp [<- ->].
by case/andP=> /(leq_trans _)->.
have apd_ok m e q p pd: lb_dvd p p || (e == 0) -> q < p ->
pd_ok p m pd -> pd_ok q (p ^ e * m) (p ^? e :: pd).
- case: e => [|e]; rewrite orbC /= => pr_p ltqp.
by rewrite mul1n; apply: leq_pd_ok; apply: ltnW.
by rewrite /pd_ok /pd_ord /pf_ok /= pr_p ltqp => [[<- -> ->]].
case=> // n _; rewrite /prime_decomp.
case: elogn2P => e2 m2 -> {n}; case: m2 => [|[|abc]]; try exact: apd_ok.
rewrite [_.-2]/= !ltnS ltn0 natTrecE; case: edivnP => a bc ->{abc}.
case: edivnP => b c def_bc /= ltc2 ltbc3; apply: (apd_ok) => //.
move def_m: _.*2.+1 => m; set k := {2}1; rewrite -[2]/k.*2; set e := 0.
pose p := k.*2.+1; rewrite -{1}[m]mul1n -[1]/(p ^ e)%N.
have{def_m bc def_bc ltc2 ltbc3}:
let kb := (ifnz e k 1).*2 in
[&& k > 0, p < m, lb_dvd p m, c < kb & lb_dvd p p || (e == 0)]
/\ m + (b * kb + c).*2 = p ^ 2 + (a * p).*2.
- rewrite -def_m [in lb_dvd _ _]def_m; split=> //=; last first.
by rewrite -def_bc addSn -double
... | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | prime_decomp_correct | |
primePnn :
reflect (n < 2 \/ exists2 d, 1 < d < n & d %| n) (~~ prime n).
Proof.
rewrite /prime; case: n => [|[|p2]]; try by do 2!left.
case: (@prime_decomp_correct p2.+2) => //; rewrite unlock.
case: prime_decomp => [|[q [|[|e]]] pd] //=; last first; last by rewrite andbF.
rewrite {1}/pfactor 2!expnS -!mulnA /=.
case: (_ ^ _ * _) => [|u -> _ /andP[lt1q _]]; first by rewrite !muln0.
left; right; exists q; last by rewrite dvdn_mulr.
have lt0q := ltnW lt1q; rewrite lt1q -[ltnLHS]muln1 ltn_pmul2l //.
by rewrite -[2]muln1 leq_mul.
rewrite {1}/pfactor expn1; case: pd => [|[r e] pd] /=; last first.
case: e => [|e] /=; first by rewrite !andbF.
rewrite {1}/pfactor expnS -mulnA.
case: (_ ^ _ * _) => [|u -> _ /and3P[lt1q ltqr _]]; first by rewrite !muln0.
left; right; exists q; last by rewrite dvdn_mulr.
by rewrite lt1q -[ltnLHS]mul1n ltn_mul // -[q.+1]muln1 leq_mul.
rewrite muln1 !andbT => def_q pr_q lt1q; right=> [[]] // [d].
by rewrite def_q -mem_index_iota => in_d_2q dv_d_q; case/hasP: pr_q; exists d.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | primePn | |
primeNsign : ~~ 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | primeNsig | |
primePp :
reflect (p > 1 /\ forall d, d %| p -> xpred2 1 p d) (prime p).
Proof.
rewrite -[prime p]negbK; have [npr_p | pr_p] := primePn p.
right=> [[lt1p pr_p]]; case: npr_p => [|[d n1pd]].
by rewrite ltnNge lt1p.
by move/pr_p=> /orP[] /eqP def_d; rewrite def_d ltnn ?andbF in n1pd.
have [lep1 | lt1p] := leqP; first by case: pr_p; left.
left; split=> // d dv_d_p; apply/norP=> [[nd1 ndp]]; case: pr_p; right.
exists d; rewrite // andbC 2!ltn_neqAle ndp eq_sym nd1.
by have lt0p := ltnW lt1p; rewrite dvdn_leq // (dvdn_gt0 lt0p).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | primeP | |
prime_nt_dvdPd 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.
Arguments primeP {p}.
Arguments primePn {n}. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | prime_nt_dvdP | |
prime_gt1p : prime p -> 1 < p.
Proof. by case/primeP. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | prime_gt1 | |
prime_gt0p : prime p -> 0 < p.
Proof. by move/prime_gt1; apply: ltnW. Qed.
#[global] Hint Resolve prime_gt1 prime_gt0 : core. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | prime_gt0 | |
prod_prime_decompn :
n > 0 -> n = \prod_(f <- prime_decomp n) f.1 ^ f.2.
Proof. by case/prime_decomp_correct. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | prod_prime_decomp | |
even_primep : 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | even_prime | |
prime_oddPnp : prime p -> reflect (p = 2) (~~ odd p).
Proof.
by move=> p_pr; apply: (iffP idP) => [|-> //]; case/even_prime: p_pr => ->.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | prime_oddPn | |
odd_prime_gt2p : odd p -> prime p -> p > 2.
Proof. by move=> odd_p /prime_gt1; apply: odd_gt2. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | odd_prime_gt2 | |
mem_prime_decompn p e :
(p, e) \in prime_decomp n -> [/\ prime p, e > 0 & p ^ e %| n].
Proof.
case: (posnP n) => [-> //| /prime_decomp_correct[def_n mem_pd ord_pd pd_pe]].
have /andP[pr_p ->] := allP mem_pd _ pd_pe; split=> //; last first.
case/splitPr: pd_pe def_n => pd1 pd2 ->.
by rewrite big_cat big_cons /= mulnCA dvdn_mulr.
have lt1p: 1 < p.
apply: (allP (order_path_min ltn_trans ord_pd)).
by apply/mapP; exists (p, e).
apply/primeP; split=> // d dv_d_p; apply/norP=> [[nd1 ndp]].
case/hasP: pr_p; exists d => //.
rewrite mem_index_iota andbC 2!ltn_neqAle ndp eq_sym nd1.
by have lt0p := ltnW lt1p; rewrite dvdn_leq // (dvdn_gt0 lt0p).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | mem_prime_decomp | |
prime_coprimep 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | prime_coprime | |
dvdn_prime2p 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | dvdn_prime2 | |
Euclid_dvd1p : prime p -> (p %| 1) = false.
Proof. by rewrite dvdn1; case: eqP => // ->. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | Euclid_dvd1 | |
Euclid_dvdMm 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | Euclid_dvdM | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | Euclid_dvd_prod | |
Euclid_dvdXm 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | Euclid_dvdX | |
mem_primesp 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].
- by rewrite Euclid_dvd1.
- by rewrite Euclid_dvdM // => /orP[].
exists (q, e) => //=; case/mem_prime_decomp: mem_qe => pr_q _ _.
by rewrite Euclid_dvdX // dvdn_prime2 // in dv_p_qe; case: eqP dv_p_qe.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | mem_primes | |
sorted_primesn : sorted ltn (primes n).
Proof.
by case: (posnP n) => [-> // | /prime_decomp_correct[_ _]]; apply: path_sorted.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | sorted_primes | |
all_prime_primesn : all prime (primes n).
Proof. by apply/allP => p; rewrite mem_primes => /and3P[]. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | all_prime_primes | |
eq_primesm 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | eq_primes | |
primes_uniqn : uniq (primes n).
Proof. exact: (sorted_uniq ltn_trans ltnn (sorted_primes n)). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | primes_uniq | |
pi_pdivn : (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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pi_pdiv | |
pdiv_primen : 1 < n -> prime (pdiv n).
Proof. by rewrite -pi_pdiv mem_primes; case/and3P. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pdiv_prime | |
pdiv_dvdn : pdiv n %| n.
Proof. by case: n (pi_pdiv n) => [|[|n]] //; rewrite mem_primes=> /and3P[]. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pdiv_dvd | |
pi_max_pdivn : (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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pi_max_pdiv | |
max_pdiv_primen : n > 1 -> prime (max_pdiv n).
Proof. by rewrite -pi_max_pdiv mem_primes => /andP[]. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | max_pdiv_prime | |
max_pdiv_dvdn : max_pdiv n %| n.
Proof.
by case: n (pi_max_pdiv n) => [|[|n]] //; rewrite mem_primes => /andP[].
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | max_pdiv_dvd | |
pdiv_leqn : 0 < n -> pdiv n <= n.
Proof. by move=> n_gt0; rewrite dvdn_leq // pdiv_dvd. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pdiv_leq | |
max_pdiv_leqn : 0 < n -> max_pdiv n <= n.
Proof. by move=> n_gt0; rewrite dvdn_leq // max_pdiv_dvd. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | max_pdiv_leq | |
pdiv_gt0n : 0 < pdiv n.
Proof. by case: n => [|[|n]] //; rewrite prime_gt0 ?pdiv_prime. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pdiv_gt0 | |
max_pdiv_gt0n : 0 < max_pdiv n.
Proof. by case: n => [|[|n]] //; rewrite prime_gt0 ?max_pdiv_prime. Qed.
#[global] Hint Resolve pdiv_gt0 max_pdiv_gt0 : core. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | max_pdiv_gt0 | |
pdiv_min_dvdm 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 (order_path_min ltn_trans ord_pm)) /ltnW].
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pdiv_min_dvd | |
max_pdiv_maxn 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | max_pdiv_max | |
ltn_pdiv2_primen : 0 < n -> n < pdiv n ^ 2 -> prime n.
Proof.
case def_n: n => [|[|n']] // _; rewrite -def_n => lt_n_p2.
suffices ->: n = pdiv n by rewrite pdiv_prime ?def_n.
apply/eqP; rewrite eqn_leq leqNgt andbC pdiv_leq; last by rewrite def_n.
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_dvd //.
by rewrite -[pdiv n]mul1n [ltnRHS]def_q ltn_pmul2r in lt_pm_m.
by rewrite def_q dvdn_mulr.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | ltn_pdiv2_prime | |
primePnsn :
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); rewrite pdiv_dvd pdiv_prime //; split=> //.
by case: leqP npr_p => // /ltn_pdiv2_prime -> //; exact: ltnW.
Qed.
Arguments primePns {n}. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | primePns | |
pdivPn : n > 1 -> {p | prime p & p %| n}.
Proof. by move=> lt1n; exists (pdiv n); rewrite ?pdiv_dvd ?pdiv_prime. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pdivP | |
primes_eq0n : (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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | primes_eq0 | |
primesMm 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | primesM | |
primesXm 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | primesX | |
primes_primep : 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | primes_prime | |
coprime_has_primesm 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/negP.
move/pdiv_prime; set p := pdiv _ => pr_p.
move/implyP: (no_p_mn p); rewrite /= !mem_primes m_gt0 n_gt0 pr_p /=.
by rewrite !(dvdn_trans (pdiv_dvd _)) // (dvdn_gcdl, dvdn_gcdr).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | coprime_has_primes | |
pdiv_idp : prime p -> pdiv p = p.
Proof. by move=> p_pr; rewrite /pdiv primes_prime. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pdiv_id | |
pdiv_pfactorp k : prime p -> pdiv (p ^ k.+1) = p.
Proof. by move=> p_pr; rewrite /pdiv primesX ?primes_prime. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pdiv_pfactor | |
prime_abovem : {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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | prime_above | |
logn_recd m r :=
match r, edivn m d with
| r'.+1, (_.+1 as m', 0) => (logn_rec d m' r').+1
| _, _ => 0
end. | Fixpoint | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | logn_rec | |
lognp m := if prime p then logn_rec p m m else 0. | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | logn | |
lognEp 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: m' {q}q.+1 (ltn0Sn q) => -[_ []|r IHr m] //= m_gt0 le_mr.
rewrite -[m in logn_rec _ _ m]prednK //=.
case: edivnP => [[|q] [|_] def_q _] //; rewrite addn0 in def_q.
have{def_q} lt_qm1: q < m.-1.
by rewrite -[q.+1]muln1 -ltnS prednK // def_q ltn_pmul2l // prime_gt1.
have{le_mr} le_m1r: m.-1 <= r by rewrite -ltnS prednK.
by rewrite (IHr r) ?(IHr m.-1) // (leq_trans lt_qm1).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | lognE | |
logn_gt0p n : (0 < logn p n) = (p \in primes n).
Proof. by rewrite lognE -mem_primes; case: {+}(p \in _). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | logn_gt0 | |
ltn_log0p n : n < p -> logn p n = 0.
Proof. by case: n => [|n] ltnp; rewrite lognE ?andbF // gtnNdvd ?andbF. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | ltn_log0 | |
logn0p : logn p 0 = 0.
Proof. by rewrite /logn if_same. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | logn0 | |
logn1p : logn p 1 = 0.
Proof. by rewrite lognE dvdn1 /= andbC; case: eqP => // ->. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | logn1 | |
pfactor_gt0p n : 0 < p ^ logn p n.
Proof. by rewrite expn_gt0 lognE; case: (posnP p) => // ->. Qed.
#[global] Hint Resolve pfactor_gt0 : core. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pfactor_gt0 | |
pfactor_dvdnp n m : prime p -> m > 0 -> (p ^ n %| m) = (n <= logn p m).
Proof.
move=> p_pr; elim: n m => [|n IHn] m m_gt0; first exact: dvd1n.
rewrite lognE p_pr m_gt0 /=; case dv_pm: (p %| m); last first.
apply/dvdnP=> [] [/= q def_m].
by rewrite def_m expnS mulnCA dvdn_mulr in dv_pm.
case/dvdnP: dv_pm m_gt0 => q ->{m}; rewrite muln_gt0 => /andP[p_gt0 q_gt0].
by rewrite expnSr dvdn_pmul2r // mulnK // IHn.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pfactor_dvdn | |
pfactor_dvdnnp 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pfactor_dvdnn | |
logn_primep 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | logn_prime | |
pfactor_coprimep 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pfactor_coprime | |
pfactorKp 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pfactorK | |
pfactorKpdivp n : prime p -> logn (pdiv (p ^ n)) (p ^ n) = n.
Proof. by case: n => // n p_pr; rewrite pdiv_pfactor ?pfactorK. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pfactorKpdiv | |
dvdn_leq_logp 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | dvdn_leq_log | |
ltn_loglp 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | ltn_logl | |
logn_Gaussp m n : coprime p m -> logn p (m * n) = logn p n.
Proof.
move=> co_pm; case p_pr: (prime p); last by rewrite /logn p_pr.
have [-> | n_gt0] := posnP n; first by rewrite muln0.
have [m0 | m_gt0] := posnP m; first by rewrite m0 prime_coprime ?dvdn0 in co_pm.
have mn_gt0: m * n > 0 by rewrite muln_gt0 m_gt0.
apply/eqP; rewrite eqn_leq andbC dvdn_leq_log ?dvdn_mull //.
set k := logn p _; have: p ^ k %| m * n by rewrite pfactor_dvdn.
by rewrite Gauss_dvdr ?coprimeXl // -pfactor_dvdn.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | logn_Gauss | |
logn_coprimep m : coprime p m -> logn p m = 0.
Proof. by move=> coprime_pm; rewrite -[m]muln1 logn_Gauss// logn1. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | logn_coprime | |
lognMp 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | lognM | |
lognXp 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | lognX | |
logn_divp 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | logn_div | |
dvdn_pfactorp d n : prime p ->
reflect (exists2 m, m <= n & d = p ^ m) (d %| p ^ n).
Proof.
move=> p_pr; have pn_gt0: p ^ n > 0 by rewrite expn_gt0 prime_gt0.
apply: (iffP idP) => [dv_d_pn|[m le_m_n ->]]; last first.
by rewrite -(subnK le_m_n) expnD dvdn_mull.
exists (logn p d); first by rewrite -(pfactorK n p_pr) dvdn_leq_log.
have d_gt0: d > 0 by apply: dvdn_gt0 dv_d_pn.
case: (pfactor_coprime p_pr d_gt0) => q co_p_q def_d.
rewrite [LHS]def_d ((q =P 1) _) ?mul1n // -dvdn1.
suff: q %| p ^ n * 1 by rewrite Gauss_dvdr // coprime_sym coprimeXl.
by rewrite muln1 (dvdn_trans _ dv_d_pn) // def_d dvdn_mulr.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | dvdn_pfactor | |
prime_decompEn : prime_decomp n = [seq (p, logn p n) | p <- primes n].
Proof.
case: n => // n; pose f0 := (0, 0); rewrite -map_comp.
apply: (@eq_from_nth _ f0) => [|i lt_i_n]; first by rewrite size_map.
rewrite (nth_map f0) //; case def_f: (nth _ _ i) => [p e] /=.
congr (_, _); rewrite [n.+1]prod_prime_decomp //.
have: (p, e) \in prime_decomp n.+1 by rewrite -def_f mem_nth.
case/mem_prime_decomp=> pr_p _ _.
rewrite (big_nth f0) big_mkord (bigD1 (Ordinal lt_i_n)) //=.
rewrite def_f mulnC logn_Gauss ?pfactorK //.
apply big_ind => [|m1 m2 com1 com2| [j ltj] /=]; first exact: coprimen1.
by rewrite coprimeMr com1.
rewrite -val_eqE /= => nji; case def_j: (nth _ _ j) => [q e1] /=.
have: (q, e1) \in prime_decomp n.+1 by rewrite -def_j mem_nth.
case/mem_prime_decomp=> pr_q e1_gt0 _; rewrite coprime_pexpr //.
rewrite prime_coprime // dvdn_prime2 //; apply: contra nji => eq_pq.
rewrite -(nth_uniq 0 _ _ (primes_uniq n.+1)) ?size_map //=.
by rewrite !(nth_map f0) // def_f def_j /= eq_sym.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | prime_decompE | |
divn_count_dvdd 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 ?] _].
rewrite (bigD1 (inord (q * d))) /eq_op /= !inordK ?ltnS -?leq_divRL ?mulnK //.
rewrite dvdn_mull ?big1 // => [[i /= ?] /andP[/eqP <- /negPf]].
by rewrite eq_sym dvdn_eq inordK ?ltnS ?leq_div2r // => ->.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | divn_count_dvd | |
logn_count_dvdp 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | logn_count_dvd | |
trunc_logp 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 | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log | |
trunc_log0p : trunc_log p 0 = 0.
Proof. by case: p => [] // []. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log0 | |
trunc_log1p : trunc_log p 1 = 0.
Proof. by case: p => [|[]]. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log1 | |
trunc_log_boundsp n :
1 < p -> 0 < n -> let k := trunc_log p n in p ^ k <= n < p ^ k.+1.
Proof.
rewrite {+}/trunc_log => p_gt1; have p_gt0 := ltnW p_gt1.
rewrite [p <= 1]leqNgt p_gt1 /=.
set loop := (loop in loop n n); set m := n; rewrite [in n in loop m n]/m.
have: m <= n by []; elim: n m => [|n IHn] [|m] //= /ltnSE-le_m_n _.
have [le_p_n | // ] := leqP p _; rewrite 2!expnSr -leq_divRL -?ltn_divLR //.
by apply: IHn; rewrite ?divn_gt0 // -ltnS (leq_trans (ltn_Pdiv _ _)).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log_bounds | |
trunc_logPp n : 1 < p -> 0 < n -> p ^ trunc_log p n <= n.
Proof. by move=> p_gt1 /(trunc_log_bounds p_gt1)/andP[]. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_logP | |
trunc_log_ltnp n : 1 < p -> n < p ^ (trunc_log p n).+1.
Proof.
have [-> | n_gt0] := posnP n; first by rewrite trunc_log0 => /ltnW.
by case/trunc_log_bounds/(_ n_gt0)/andP.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log_ltn | |
trunc_log_maxp k j : 1 < p -> p ^ j <= k -> j <= trunc_log p k.
Proof.
move=> p_gt1 le_pj_k; rewrite -ltnS -(@ltn_exp2l p) //.
exact: leq_ltn_trans (trunc_log_ltn _ _).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log_max | |
trunc_log_eq0p n : (trunc_log p n == 0) = (p <= 1) || (n <= p.-1).
Proof.
case: p => [|[|p]]; case: n => // n; rewrite /= ltnS.
have /= /andP[] := trunc_log_bounds (isT : 1 < p.+2) (isT : 0 < n.+1).
case: trunc_log => [//|k] b1 b2.
apply/idP/idP => [/eqP sk0 | nlep]; first by move: b2; rewrite sk0.
symmetry; rewrite -[_ == _]/false /is_true -b1; apply/negbTE; rewrite -ltnNge.
move: nlep; rewrite -ltnS => nlep; apply: (leq_ltn_trans nlep).
by rewrite -[leqLHS]expn1; apply: leq_pexp2l.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log_eq0 | |
trunc_log_gt0p n : (0 < trunc_log p n) = (1 < p) && (p.-1 < n).
Proof. by rewrite ltnNge leqn0 trunc_log_eq0 negb_or -!ltnNge. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log_gt0 | |
trunc_log0nn : trunc_log 0 n = 0.
Proof. by []. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log0n | |
trunc_log1nn : trunc_log 1 n = 0.
Proof. by []. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log1n | |
leq_trunc_logp m n : m <= n -> trunc_log p m <= trunc_log p n.
Proof.
move=> mlen; case: p => [|[|p]]; rewrite ?trunc_log0n ?trunc_log1n //.
case: m mlen => [|m] mlen; first by rewrite trunc_log0.
apply/trunc_log_max => //; apply: leq_trans mlen; exact: trunc_logP.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | leq_trunc_log | |
trunc_log_eqp n k : 1 < p -> p ^ n <= k < p ^ n.+1 -> trunc_log p k = n.
Proof.
move=> p_gt1 /andP[npLk kLpn]; apply/anti_leq.
rewrite trunc_log_max// andbT -ltnS -(ltn_exp2l _ _ p_gt1).
apply: leq_ltn_trans kLpn; apply: trunc_logP => //.
by apply: leq_trans npLk; rewrite expn_gt0 ltnW.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log_eq | |
trunc_lognnp : 1 < p -> trunc_log p p = 1.
Proof. by case: p => [|[|p]] // _; rewrite /trunc_log ltnSn divnn. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_lognn | |
trunc_expnKp n : 1 < p -> trunc_log p (p ^ n) = n.
Proof. by move=> ?; apply: trunc_log_eq; rewrite // leqnn ltn_exp2l /=. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_expnK | |
trunc_logMpp n : 1 < p -> 0 < n ->
trunc_log p (p * n) = (trunc_log p n).+1.
Proof.
case: p => [//|p] => p_gt0 n_gt0; apply: trunc_log_eq => //.
rewrite expnS leq_pmul2l// trunc_logP//=.
by rewrite expnS ltn_pmul2l// trunc_log_ltn.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_logMp | |
trunc_log2_doublen : 0 < n -> trunc_log 2 n.*2 = (trunc_log 2 n).+1.
Proof. by move=> n_gt0; rewrite -mul2n trunc_logMp. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | trunc_log2_double |
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