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 |
|---|---|---|---|---|---|---|
trunc_log2Sn : 1 < n -> trunc_log 2 n = (trunc_log 2 n./2).+1.
Proof.
move=> n_gt1.
rewrite -trunc_log2_double ?half_gt0//.
rewrite -[n in LHS]odd_double_half.
case: odd => //; rewrite add1n.
apply: trunc_log_eq => //.
rewrite leqW ?trunc_logP //= ?double_gt0 ?half_gt0//.
rewrite trunc_log2_double ?half_gt0// expnS.
by rewrite -doubleS mul2n leq_double 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_log2S | |
up_logp n :=
if (p <= 1) then 0 else
let v := trunc_log p n in if n <= p ^ v then v else v.+1. | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | up_log | |
up_log0p : up_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 | up_log0 | |
up_log1p : up_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 | up_log1 | |
up_log_eq0p n : (up_log p n == 0) = (p <= 1) || (n <= 1).
Proof.
case: p => // [] [] // p.
case: n => [|[|n]]; rewrite /up_log //=.
have /= := trunc_log_bounds (isT : 1 < p.+2) (isT : 0 < n.+2).
by case: (leqP _ n.+1); case: trunc_log.
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 | up_log_eq0 | |
up_log_gt0p n : (0 < up_log p n) = (1 < p) && (1 < n).
Proof. by rewrite ltnNge leqn0 up_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 | up_log_gt0 | |
up_log_boundsp n :
1 < p -> 1 < n -> let k := up_log p n in p ^ k.-1 < n <= p ^ k.
Proof.
move=> p_gt1 n_gt1.
have n_gt0 : 0 < n by apply: leq_trans n_gt1.
rewrite /up_log (leqNgt p 1) p_gt1 /=.
have /= /andP[tpLn nLtpS] := trunc_log_bounds p_gt1 n_gt0.
have [nLnp|npLn] := leqP n (p ^ trunc_log p n); last by rewrite npLn ltnW.
rewrite nLnp (leq_trans _ tpLn) // ltn_exp2l // prednK ?leqnn //.
by case: trunc_log (leq_trans n_gt1 nLnp).
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 | up_log_bounds | |
up_logPp n : 1 < p -> n <= p ^ up_log p n.
Proof.
case: n => [|[|n]] // p_gt1; first by rewrite up_log1.
by have /andP[] := up_log_bounds p_gt1 (isT: 1 < n.+2).
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 | up_logP | |
up_log_gtnp n : 1 < p -> 1 < n -> p ^ (up_log p n).-1 < n.
Proof.
by case: n => [|[|n]] p_gt1 n_gt1 //; have /andP[] := up_log_bounds p_gt1 n_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 | up_log_gtn | |
up_log_minp k j : 1 < p -> k <= p ^ j -> up_log p k <= j.
Proof.
case: k => [|[|k]] // p_gt1 kLj; rewrite ?(up_log0, up_log1) //.
rewrite -[up_log _ _]prednK ?up_log_gt0 ?p_gt1 // -(@ltn_exp2l p) //.
by apply: leq_trans (up_log_gtn p_gt1 (isT : 1 < k.+2)) _.
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 | up_log_min | |
leq_up_logp m n : m <= n -> up_log p m <= up_log p n.
Proof.
move=> mLn; case: p => [|[|p]] //.
by apply/up_log_min => //; apply: leq_trans mLn (up_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_up_log | |
up_log_eqp n k : 1 < p -> p ^ n < k <= p ^ n.+1 -> up_log p k = n.+1.
Proof.
move=> p_gt1 /andP[npLk kLpn]; apply/eqP; rewrite eqn_leq.
apply/andP; split; first by apply: up_log_min.
rewrite -(ltn_exp2l _ _ p_gt1) //.
by apply: leq_trans npLk (up_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 | up_log_eq | |
up_lognnp : 1 < p -> up_log p p = 1.
Proof. by move=> p_gt1; apply: up_log_eq; rewrite p_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 | up_lognn | |
up_expnKp n : 1 < p -> up_log p (p ^ n) = n.
Proof.
case: n => [|n] p_gt1 /=; first by rewrite up_log1.
by apply: up_log_eq; rewrite // leqnn andbT 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 | up_expnK | |
up_logMpp n : 1 < p -> 0 < n -> up_log p (p * n) = (up_log p n).+1.
Proof.
case: p => [//|p] p_gt0.
case: n => [//|[|n]] _; first by rewrite muln1 up_lognn// up_log1.
apply: up_log_eq => //.
rewrite expnS leq_pmul2l// up_logP// andbT.
rewrite -[up_log _ _]prednK ?up_log_gt0 ?p_gt0 //.
by rewrite expnS ltn_pmul2l// up_log_gtn.
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 | up_logMp | |
up_log2_doublen : 0 < n -> up_log 2 n.*2 = (up_log 2 n).+1.
Proof. by move=> n_gt0; rewrite -mul2n up_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 | up_log2_double | |
up_log2Sn : 0 < n -> up_log 2 n.+1 = (up_log 2 (n./2.+1)).+1.
Proof.
case: n=> // [] [|n] // _.
apply: up_log_eq => //; apply/andP; split.
apply: leq_trans (_ : n./2.+1.*2 < n.+3); last first.
by rewrite doubleS !ltnS -[leqRHS]odd_double_half leq_addl.
have /= /andP[H1n _] := up_log_bounds (isT : 1 < 2) (isT : 1 < n./2.+2).
by rewrite ltnS -leq_double -mul2n -expnS prednK ?up_log_gt0 // in H1n.
rewrite -[_./2.+1]/(n./2.+2).
have /= /andP[_ H2n] := up_log_bounds (isT : 1 < 2) (isT : 1 < n./2.+2).
rewrite -leq_double -!mul2n -expnS in H2n.
apply: leq_trans H2n.
rewrite mul2n !doubleS !ltnS.
by rewrite -[leqLHS]odd_double_half -add1n leq_add2r; case: odd.
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 | up_log2S | |
up_log_trunc_logp n :
1 < p -> 1 < n -> up_log p n = (trunc_log p n.-1).+1.
Proof.
move=> p_gt1 n_gt1; apply: up_log_eq => //.
rewrite -[n]prednK ?ltnS -?pred_Sn ?[0 < n]ltnW//.
by rewrite trunc_logP ?ltn_predRL// 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 | up_log_trunc_log | |
trunc_log_up_logp n :
1 < p -> 0 < n -> trunc_log p n = (up_log p n.+1).-1.
Proof. by move=> ? ?; rewrite up_log_trunc_log.
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_up_log | |
nat_pred_pred:= Eval hnf in [predType of nat_pred]. | Canonical | 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_pred | |
nat_pred_of_nat(p : nat) : nat_pred := pred1 p. | Coercion | 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_of_nat | |
negn: nat_pred := [predC pi]. | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | negn | |
pnat: pred nat := fun m => (m > 0) && all [in pi] (primes m). | Definition | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pnat | |
partn:= \prod_(0 <= p < n.+1 | p \in pi) p ^ logn p 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 | partn | |
negnKpi : pi^'^' =i pi.
Proof. by move=> p; apply: negbK. 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 | negnK | |
eq_negnpi1 pi2 : pi1 =i pi2 -> pi1^' =i pi2^'.
Proof. by move=> eq_pi n; rewrite inE eq_pi. 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_negn | |
eq_piPm n : \pi(m) =i \pi(n) <-> \pi(m) = \pi(n).
Proof.
rewrite /pi_of; have eqs := irr_sorted_eq ltn_trans ltnn.
by split=> [|-> //] /(eqs _ _ (sorted_primes m) (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 | eq_piP | |
part_gt0pi n : 0 < n`_pi.
Proof. exact: prodn_gt0. Qed.
Hint Resolve part_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 | part_gt0 | |
sub_in_partnpi1 pi2 n :
{in \pi(n), {subset pi1 <= pi2}} -> n`_pi1 %| n`_pi2.
Proof.
move=> pi12; rewrite ![n`__]big_mkcond /=.
apply (big_ind2 (fun m1 m2 => m1 %| m2)) => // [*|p _]; first exact: dvdn_mul.
rewrite lognE -mem_primes; case: ifP => pi1p; last exact: dvd1n.
by case: ifP => pr_p; [rewrite pi12 | rewrite if_same].
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | sub_in_partn | |
eq_in_partnpi1 pi2 n : {in \pi(n), pi1 =i pi2} -> n`_pi1 = n`_pi2.
Proof.
by move=> pi12; apply/eqP; rewrite eqn_dvd ?sub_in_partn // => p /pi12->.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | eq_in_partn | |
eq_partnpi1 pi2 n : pi1 =i pi2 -> n`_pi1 = n`_pi2.
Proof. by move=> pi12; apply: eq_in_partn => 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 | eq_partn | |
partnNKpi n : n`_pi^'^' = n`_pi.
Proof. by apply: eq_partn; apply: negnK. 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 | partnNK | |
widen_partnm pi n :
n <= m -> n`_pi = \prod_(0 <= p < m.+1 | p \in pi) p ^ logn p n.
Proof.
move=> le_n_m; rewrite big_mkcond /=.
rewrite [n`_pi](big_nat_widen _ _ m.+1) // big_mkcond /=.
apply: eq_bigr => p _; rewrite ltnS lognE.
by case: and3P => [[_ n_gt0 p_dv_n]|]; rewrite ?if_same // andbC dvdn_leq.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | widen_partn | |
eq_partn_from_logm n (pi : nat_pred) : 0 < m -> 0 < n ->
{in pi, logn^~ m =1 logn^~ n} -> m`_pi = n`_pi.
Proof.
move=> m0 n0 eq_log; rewrite !(@widen_partn (maxn m n)) ?leq_maxl ?leq_maxr//.
by apply: eq_bigr => p /eq_log ->.
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_partn_from_log | |
partn0pi : 0`_pi = 1.
Proof. by apply: big1_seq => [] [|n]; rewrite andbC. 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 | partn0 | |
partn1pi : 1`_pi = 1.
Proof. by apply: big1_seq => [] [|[|n]]; rewrite andbC. 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 | partn1 | |
partnMpi m n : m > 0 -> n > 0 -> (m * n)`_pi = m`_pi * n`_pi.
Proof.
have le_pmul m' n': m' > 0 -> n' <= m' * n' by move/prednK <-; apply: leq_addr.
move=> mpos npos; rewrite !(@widen_partn (n * m)) 3?(le_pmul, mulnC) //.
rewrite !big_mkord -big_split; apply: eq_bigr => p _ /=.
by rewrite lognM // expnD.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | partnM | |
partnXpi m n : (m ^ n)`_pi = m`_pi ^ n.
Proof.
elim: n => [|n IHn]; first exact: partn1.
rewrite expnS; have [->|m_gt0] := posnP m; first by rewrite partn0 exp1n.
by rewrite expnS partnM ?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 | partnX | |
partn_dvdpi m n : n > 0 -> m %| n -> m`_pi %| n`_pi.
Proof.
move=> n_gt0 dvmn; case/dvdnP: dvmn n_gt0 => q ->{n}.
by rewrite muln_gt0 => /andP[q_gt0 m_gt0]; rewrite partnM ?dvdn_mull.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | partn_dvd | |
p_partp n : n`_p = p ^ logn p n.
Proof.
case (posnP (logn p n)) => [log0 |].
by rewrite log0 [n`_p]big1_seq // => q /andP [/eqP ->]; rewrite log0.
rewrite logn_gt0 mem_primes; case/and3P=> _ n_gt0 dv_p_n.
have le_p_n: p < n.+1 by rewrite ltnS dvdn_leq.
by rewrite [n`_p]big_mkord (big_pred1 (Ordinal le_p_n)).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | p_part | |
p_part_eq1p n : (n`_p == 1) = (p \notin \pi(n)).
Proof.
rewrite mem_primes p_part lognE; case: and3P => // [[p_pr _ _]].
by rewrite -dvdn1 pfactor_dvdn // logn1.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | p_part_eq1 | |
p_part_gt1p n : (n`_p > 1) = (p \in \pi(n)).
Proof. by rewrite ltn_neqAle part_gt0 andbT eq_sym p_part_eq1 negbK. 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 | p_part_gt1 | |
primes_partpi n : primes n`_pi = filter [in pi] (primes n).
Proof.
have ltnT := ltn_trans; have [->|n_gt0] := posnP n; first by rewrite partn0.
apply: (irr_sorted_eq ltnT ltnn); rewrite ?(sorted_primes, sorted_filter) //.
move=> p; rewrite mem_filter /= !mem_primes n_gt0 part_gt0 /=.
apply/andP/and3P=> [[p_pr] | [pi_p p_pr dv_p_n]].
rewrite /partn; apply big_ind => [|n1 n2 IHn1 IHn2|q pi_q].
- by rewrite dvdn1; case: eqP p_pr => // ->.
- by rewrite Euclid_dvdM //; case/orP.
rewrite -{1}(expn1 p) pfactor_dvdn // lognX muln_gt0.
rewrite logn_gt0 mem_primes n_gt0 - andbA /=; case/and3P=> pr_q dv_q_n.
by rewrite logn_prime //; case: eqP => // ->.
have le_p_n: p < n.+1 by rewrite ltnS dvdn_leq.
rewrite [n`_pi]big_mkord (bigD1 (Ordinal le_p_n)) //= dvdn_mulr //.
by rewrite lognE p_pr n_gt0 dv_p_n expnS dvdn_mulr.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | primes_part | |
filter_pi_ofn m : n < m -> filter \pi(n) (index_iota 0 m) = primes n.
Proof.
move=> lt_n_m; have ltnT := ltn_trans; apply: (irr_sorted_eq ltnT ltnn).
- by rewrite sorted_filter // iota_ltn_sorted.
- exact: sorted_primes.
move=> p; rewrite mem_filter mem_index_iota /= mem_primes; case: and3P => //.
by case=> _ n_gt0 dv_p_n; apply: leq_ltn_trans lt_n_m; apply: dvdn_leq.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | filter_pi_of | |
partn_pin : n > 0 -> n`_\pi(n) = n.
Proof.
move=> n_gt0; rewrite [RHS]prod_prime_decomp // prime_decompE big_map.
by rewrite -[n`__]big_filter filter_pi_of.
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 | partn_pi | |
partnTn : n > 0 -> n`_predT = n.
Proof.
move=> n_gt0; rewrite -[RHS]partn_pi // [RHS]/partn big_mkcond /=.
by apply: eq_bigr => p _; rewrite -logn_gt0; case: (logn 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 | partnT | |
eqn_from_logm n : 0 < m -> 0 < n -> logn^~ m =1 logn^~ n -> m = n.
Proof.
by move=> ? ? /(@in1W _ predT)/eq_partn_from_log; rewrite !partnT// => ->.
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 | eqn_from_log | |
partnCpi n : n > 0 -> n`_pi * n`_pi^' = n.
Proof.
move=> n_gt0; rewrite -[RHS]partnT /partn //.
do 2!rewrite mulnC big_mkcond /=; rewrite -big_split; apply: eq_bigr => p _ /=.
by rewrite mulnC inE /=; case: (p \in pi); rewrite /= (muln1, mul1n).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | partnC | |
dvdn_partpi n : n`_pi %| n.
Proof. by case: n => // n; rewrite -{2}[n.+1](@partnC pi) // 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_part | |
logn_partp m : logn p m`_p = logn p m.
Proof.
case p_pr: (prime p); first by rewrite p_part pfactorK.
by rewrite lognE (lognE p m) p_pr.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | logn_part | |
partn_lcmpi m n : m > 0 -> n > 0 -> (lcmn m n)`_pi = lcmn m`_pi n`_pi.
Proof.
move=> m_gt0 n_gt0; have p_gt0: lcmn m n > 0 by rewrite lcmn_gt0 m_gt0.
apply/eqP; rewrite eqn_dvd dvdn_lcm !partn_dvd ?dvdn_lcml ?dvdn_lcmr //.
rewrite -(dvdn_pmul2r (part_gt0 pi^' (lcmn m n))) partnC // dvdn_lcm !andbT.
rewrite -[m in m %| _](partnC pi m_gt0) andbC -[n in n %| _](partnC pi n_gt0).
by rewrite !dvdn_mul ?partn_dvd ?dvdn_lcml ?dvdn_lcmr.
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 | partn_lcm | |
partn_gcdpi m n : m > 0 -> n > 0 -> (gcdn m n)`_pi = gcdn m`_pi n`_pi.
Proof.
move=> m_gt0 n_gt0; have p_gt0: gcdn m n > 0 by rewrite gcdn_gt0 m_gt0.
apply/eqP; rewrite eqn_dvd dvdn_gcd !partn_dvd ?dvdn_gcdl ?dvdn_gcdr //=.
rewrite -(dvdn_pmul2r (part_gt0 pi^' (gcdn m n))) partnC // dvdn_gcd.
rewrite -[m in _ %| m](partnC pi m_gt0) andbC -[n in _%| n](partnC pi n_gt0).
by rewrite !dvdn_mul ?partn_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 | partn_gcd | |
partn_biglcm(I : finType) (P : pred I) F pi :
(forall i, P i -> F i > 0) ->
(\big[lcmn/1%N]_(i | P i) F i)`_pi = \big[lcmn/1%N]_(i | P i) (F i)`_pi.
Proof.
move=> F_gt0; set m := \big[lcmn/1%N]_(i | P i) F i.
have m_gt0: 0 < m by elim/big_ind: m => // p q p_gt0; rewrite lcmn_gt0 p_gt0.
apply/eqP; rewrite eqn_dvd andbC; apply/andP; split.
by apply/dvdn_biglcmP=> i Pi; rewrite partn_dvd // (@biglcmn_sup _ i).
rewrite -(dvdn_pmul2r (part_gt0 pi^' m)) partnC //.
apply/dvdn_biglcmP=> i Pi; rewrite -(partnC pi (F_gt0 i Pi)) dvdn_mul //.
by rewrite (@biglcmn_sup _ i).
by rewrite partn_dvd // (@biglcmn_sup _ i).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | partn_biglcm | |
partn_biggcd(I : finType) (P : pred I) F pi :
#|SimplPred P| > 0 -> (forall i, P i -> F i > 0) ->
(\big[gcdn/0]_(i | P i) F i)`_pi = \big[gcdn/0]_(i | P i) (F i)`_pi.
Proof.
move=> ntP F_gt0; set d := \big[gcdn/0]_(i | P i) F i.
have d_gt0: 0 < d.
case/card_gt0P: ntP => i /= Pi; have:= F_gt0 i Pi.
rewrite !lt0n -!dvd0n; apply: contra => dv0d.
by rewrite (dvdn_trans dv0d) // (@biggcdn_inf _ i).
apply/eqP; rewrite eqn_dvd; apply/andP; split.
by apply/dvdn_biggcdP=> i Pi; rewrite partn_dvd ?F_gt0 // (@biggcdn_inf _ i).
rewrite -(dvdn_pmul2r (part_gt0 pi^' d)) partnC //.
apply/dvdn_biggcdP=> i Pi; rewrite -(partnC pi (F_gt0 i Pi)) dvdn_mul //.
by rewrite (@biggcdn_inf _ i).
by rewrite partn_dvd ?F_gt0 // (@biggcdn_inf _ i).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | partn_biggcd | |
logn_gcdp m n : 0 < m -> 0 < n ->
logn p (gcdn m n) = minn (logn p m) (logn p n).
Proof.
move=> m_gt0 n_gt0; case p_pr: (prime p); last by rewrite /logn p_pr.
by apply: (@expnI p); rewrite ?prime_gt1// expn_min -!p_part partn_gcd.
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_gcd | |
logn_lcmp m n : 0 < m -> 0 < n ->
logn p (lcmn m n) = maxn (logn p m) (logn p n).
Proof.
move=> m_gt0 n_gt0; rewrite /lcmn logn_div ?dvdn_mull ?dvdn_gcdr//.
by rewrite lognM// logn_gcd// -addn_min_max addnC 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_lcm | |
sub_in_pnatpi rho n :
{in \pi(n), {subset pi <= rho}} -> pi.-nat n -> rho.-nat n.
Proof.
rewrite /pnat => subpi /andP[-> pi_n].
by apply/allP=> p pr_p; apply: subpi => //; apply: (allP pi_n).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | sub_in_pnat | |
eq_in_pnatpi rho n : {in \pi(n), pi =i rho} -> pi.-nat n = rho.-nat n.
Proof. by move=> eqpi; apply/idP/idP; apply: sub_in_pnat => p /eqpi->. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | eq_in_pnat | |
eq_pnatpi rho n : pi =i rho -> pi.-nat n = rho.-nat n.
Proof. by move=> eqpi; apply: eq_in_pnat => 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 | eq_pnat | |
pnatNKpi n : pi^'^'.-nat n = pi.-nat n.
Proof. exact: eq_pnat (negnK pi). 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 | pnatNK | |
pnatIpi rho n : [predI pi & rho].-nat n = pi.-nat n && rho.-nat n.
Proof. by rewrite /pnat andbCA all_predI !andbA andbb. 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 | pnatI | |
pnatMpi m n : pi.-nat (m * n) = pi.-nat m && pi.-nat n.
Proof.
rewrite /pnat muln_gt0 andbCA -andbA andbCA.
case: posnP => // n_gt0; case: posnP => //= m_gt0.
apply/allP/andP=> [pi_mn | [pi_m pi_n] p].
by split; apply/allP=> p m_p; apply: pi_mn; rewrite primesM // m_p ?orbT.
by rewrite primesM // => /orP[]; [apply: (allP pi_m) | apply: (allP pi_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 | pnatM | |
pnatXpi m n : pi.-nat (m ^ n) = pi.-nat m || (n == 0).
Proof. by case: n => [|n]; rewrite orbC // /pnat expn_gt0 orbC primesX. 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 | pnatX | |
part_pnatpi n : pi.-nat n`_pi.
Proof.
rewrite /pnat primes_part part_gt0.
by apply/allP=> p; rewrite mem_filter => /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 | part_pnat | |
pnatEpi p : prime p -> pi.-nat p = (p \in pi).
Proof. by move=> pr_p; rewrite /pnat prime_gt0 ?primes_prime //= andbT. 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 | pnatE | |
pnat_idp : prime p -> p.-nat p.
Proof. by move=> pr_p; rewrite pnatE ?inE /=. 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 | pnat_id | |
coprime_pi'm n : m > 0 -> n > 0 -> coprime m n = \pi(m)^'.-nat n.
Proof.
by move=> m_gt0 n_gt0; rewrite /pnat n_gt0 all_predC coprime_has_primes.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | coprime_pi' | |
pnat_pin : n > 0 -> \pi(n).-nat n.
Proof. by rewrite /pnat => ->; apply/allP. 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 | pnat_pi | |
pi_of_dvdm n : m %| n -> n > 0 -> {subset \pi(m) <= \pi(n)}.
Proof.
move=> m_dv_n n_gt0 p; rewrite !mem_primes n_gt0 => /and3P[-> _ p_dv_m].
exact: dvdn_trans p_dv_m m_dv_n.
Qed. | Lemma | 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_dvd | |
pi_ofMm n : m > 0 -> n > 0 -> \pi(m * n) =i [predU \pi(m) & \pi(n)].
Proof. by move=> m_gt0 n_gt0 p; apply: primesM. 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_ofM | |
pi_of_partpi n : n > 0 -> \pi(n`_pi) =i [predI \pi(n) & pi].
Proof. by move=> n_gt0 p; rewrite /pi_of primes_part mem_filter andbC. Qed. | Lemma | 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_part | |
pi_of_expp n : n > 0 -> \pi(p ^ n) = \pi(p).
Proof. by move=> n_gt0; rewrite /pi_of primesX. 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_of_exp | |
pi_of_primep : prime p -> \pi(p) =i (p : nat_pred).
Proof. by move=> pr_p q; rewrite /pi_of primes_prime // mem_seq1. 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_of_prime | |
p'natEpip n : n > 0 -> p^'.-nat n = (p \notin \pi(n)).
Proof. by case: n => // n _; rewrite /pnat all_predC has_pred1. 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 | p'natEpi | |
p'natEp n : prime p -> p^'.-nat n = ~~ (p %| n).
Proof.
case: n => [|n] p_pr; first by case: p p_pr.
by rewrite p'natEpi // mem_primes p_pr.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | p'natE | |
pnatPpipi n p : pi.-nat n -> p \in \pi(n) -> p \in pi.
Proof. by case/andP=> _ /allP; apply. 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 | pnatPpi | |
pnat_dvdm n pi : m %| n -> pi.-nat n -> pi.-nat m.
Proof. by case/dvdnP=> q ->; rewrite pnatM; case/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 | pnat_dvd | |
pnat_divm n pi : m %| n -> pi.-nat n -> pi.-nat (n %/ m).
Proof.
case/dvdnP=> q ->; rewrite pnatM andbC => /andP[].
by case: m => // m _; rewrite mulnK.
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 | pnat_div | |
pnat_coprimepi m n : pi.-nat m -> pi^'.-nat n -> coprime m n.
Proof.
case/andP=> m_gt0 pi_m /andP[n_gt0 pi'_n]; rewrite coprime_has_primes //.
by apply/hasPn=> p /(allP pi'_n); apply/contra/allP.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pnat_coprime | |
p'nat_coprimepi m n : pi^'.-nat m -> pi.-nat n -> coprime m n.
Proof. by move=> pi'm pi_n; rewrite (pnat_coprime pi'm) ?pnatNK. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | p'nat_coprime | |
sub_pnat_coprimepi rho m n :
{subset rho <= pi^'} -> pi.-nat m -> rho.-nat n -> coprime m n.
Proof.
by move=> pi'rho pi_m /(sub_in_pnat (in1W pi'rho)); apply: pnat_coprime.
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 | sub_pnat_coprime | |
coprime_partCpi m n : coprime m`_pi n`_pi^'.
Proof. by apply: (@pnat_coprime pi); apply: part_pnat. 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_partC | |
pnat_1pi n : pi.-nat n -> pi^'.-nat n -> n = 1.
Proof.
by move=> pi_n pi'_n; rewrite -(eqnP (pnat_coprime pi_n pi'_n)) gcdnn.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pnat_1 | |
part_pnat_idpi n : pi.-nat n -> n`_pi = n.
Proof.
case/andP=> n_gt0 pi_n; rewrite -[RHS]partnT // /partn big_mkcond /=.
apply: eq_bigr=> p _; have [->|] := posnP (logn p n); first by rewrite if_same.
by rewrite logn_gt0 => /(allP pi_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 | part_pnat_id | |
part_p'natpi n : pi^'.-nat n -> n`_pi = 1.
Proof.
case/andP=> n_gt0 pi'_n; apply: big1_seq => p /andP[pi_p _].
by have [-> //|] := posnP (logn p n); rewrite logn_gt0; case/(allP pi'_n)/negP.
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 | part_p'nat | |
partn_eq1pi n : n > 0 -> (n`_pi == 1) = pi^'.-nat n.
Proof.
move=> n_gt0; apply/eqP/idP=> [pi_n_1|]; last exact: part_p'nat.
by rewrite -(partnC pi n_gt0) pi_n_1 mul1n part_pnat.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | partn_eq1 | |
pnatPpi n :
n > 0 -> reflect (forall p, prime p -> p %| n -> p \in pi) (pi.-nat n).
Proof.
move=> n_gt0; rewrite /pnat n_gt0.
apply: (iffP allP) => /= pi_n p => [pr_p p_n|].
by rewrite pi_n // mem_primes pr_p n_gt0.
by rewrite mem_primes n_gt0 /=; case/andP; move: p.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pnatP | |
pi_pnatpi p n : p.-nat n -> p \in pi -> pi.-nat n.
Proof.
move=> p_n pi_p; have [n_gt0 _] := andP p_n.
by apply/pnatP=> // q q_pr /(pnatP _ n_gt0 p_n _ q_pr)/eqnP->.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | pi_pnat | |
p_natPp n : p.-nat n -> {k | n = p ^ k}.
Proof. by move=> p_n; exists (logn p n); rewrite -p_part part_pnat_id. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | p_natP | |
pi'_p'natpi p n : pi^'.-nat n -> p \in pi -> p^'.-nat n.
Proof.
by move=> pi'n pi_p; apply: sub_in_pnat pi'n => q _; apply: contraNneq => ->.
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'_p'nat | |
pi_p'natp pi n : pi.-nat n -> p \in pi^' -> p^'.-nat n.
Proof. by move=> pi_n; apply: pi'_p'nat; rewrite pnatNK. 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_p'nat | |
partn_partpi rho n : {subset pi <= rho} -> n`_rho`_pi = n`_pi.
Proof.
move=> pi_sub_rho; have [->|n_gt0] := posnP n; first by rewrite !partn0 partn1.
rewrite -[in RHS](partnC rho n_gt0) partnM //.
suffices: pi^'.-nat n`_rho^' by move/part_p'nat->; rewrite muln1.
by apply: sub_in_pnat (part_pnat _ _) => q _; apply/contra/pi_sub_rho.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | partn_part | |
partnIpi rho n : n`_[predI pi & rho] = n`_pi`_rho.
Proof.
rewrite -(@partnC [predI pi & rho] _`_rho) //.
symmetry; rewrite 2?partn_part; try by move=> p /andP [].
rewrite mulnC part_p'nat ?mul1n // pnatNK pnatI part_pnat andbT.
exact: pnat_dvd (dvdn_part _ _) (part_pnat _ _).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | partnI | |
odd_2'natn : odd n = 2^'.-nat n.
Proof. by case: n => // n; rewrite p'natE // dvdn2 negbK. 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_2'nat | |
divisors_correctn : n > 0 ->
[/\ uniq (divisors n), sorted leq (divisors n)
& forall d, (d \in divisors n) = (d %| n)].
Proof.
move/prod_prime_decomp=> def_n; rewrite {4}def_n {def_n}.
have: all prime (primes n) by apply/allP=> p; rewrite mem_primes; case/andP.
have:= primes_uniq n; rewrite /primes /divisors; move/prime_decomp: n.
elim=> [|[p e] pd] /=; first by split=> // d; rewrite big_nil dvdn1 mem_seq1.
rewrite big_cons /=; move: (foldr _ _ pd) => divs.
move=> IHpd /andP[npd_p Upd] /andP[pr_p pr_pd].
have lt0p: 0 < p by apply: prime_gt0.
have {IHpd Upd}[Udivs Odivs mem_divs] := IHpd Upd pr_pd.
have ndivs_p m: p * m \notin divs.
suffices: p \notin divs; rewrite !mem_divs.
by apply: contra => /dvdnP[n ->]; rewrite mulnCA dvdn_mulr.
have ndv_p_1: ~~(p %| 1) by rewrite dvdn1 neq_ltn orbC prime_gt1.
rewrite big_seq; elim/big_ind: _ => [//|u v npu npv|[q f] /= pd_qf].
by rewrite Euclid_dvdM //; apply/norP.
elim: (f) => // f'; rewrite expnS Euclid_dvdM // orbC negb_or => -> {f'}/=.
have pd_q: q \in unzip1 pd by apply/mapP; exists (q, f).
by apply: contra npd_p; rewrite dvdn_prime2 // ?(allP pr_pd) // => /eqP->.
elim: e => [|e] /=; first by split=> // d; rewrite mul1n.
have Tmulp_inj: injective (NatTrec.mul p).
by move=> u v /eqP; rewrite !natTrecE eqn_pmul2l // => /eqP.
move: (iter e _ _) => divs' [Udivs' Odivs' mem_divs']; split=> [||d].
- rewrite merge_uniq cat_uniq map_inj_uniq // Udivs Udivs' andbT /=.
apply/hasP=> [[d dv_d /mapP[d' _ def_d]]].
by case/idPn:
... | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype div bigop"
] | boot/prime.v | divisors_correct | |
sorted_divisorsn : sorted leq (divisors n).
Proof. by case: (posnP n) => [-> | /divisors_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 | sorted_divisors | |
divisors_uniqn : uniq (divisors n).
Proof. by case: (posnP n) => [-> | /divisors_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 | divisors_uniq | |
sorted_divisors_ltnn : sorted ltn (divisors n).
Proof. by rewrite ltn_sorted_uniq_leq divisors_uniq sorted_divisors. 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_divisors_ltn | |
dvdn_divisorsd m : 0 < m -> (d %| m) = (d \in divisors m).
Proof. by case/divisors_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 | dvdn_divisors | |
divisor1n : 1 \in divisors n.
Proof. by case: n => // n; rewrite -dvdn_divisors // 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 | divisor1 |
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