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 |
|---|---|---|---|---|---|---|---|---|---|---|
pnat_1 pi n : pi.-nat n -> pi^'.-nat n -> n = 1. | Proof.
by move=> pi_n pi'_n; rewrite -(eqnP (pnat_coprime pi_n pi'_n)) gcdnn.
Qed. | Lemma | pnat_1 | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"eqnP",
"gcdnn",
"nat",
"pi",
"pnat_coprime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
part_pnat_id pi 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 | part_pnat_id | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"allP",
"apply",
"big_mkcond",
"eq_bigr",
"logn",
"logn_gt0",
"n_gt0",
"nat",
"partn",
"partnT",
"pi",
"posnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
part_p'nat pi n : pi^'.-nat n -> n`_pi = 1. | Proof.
case/andP=> n_gt0 pi'_n; apply: big1_seq => p /andP[pi_p _].
by have [-> //|] := posnP (logn p n); rewrite logn_gt0; case/(allP pi'_n)/negP.
Qed. | Lemma | part_p'nat | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"allP",
"apply",
"big1_seq",
"logn",
"logn_gt0",
"n_gt0",
"nat",
"pi",
"pi_p",
"posnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
partn_eq1 pi n : n > 0 -> (n`_pi == 1) = pi^'.-nat n. | Proof.
move=> n_gt0; apply/eqP/idP=> [pi_n_1|]; last exact: part_p'nat.
by rewrite -(partnC pi n_gt0) pi_n_1 mul1n part_pnat.
Qed. | Lemma | partn_eq1 | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"apply",
"last",
"mul1n",
"n_gt0",
"nat",
"part_p'nat",
"part_pnat",
"partnC",
"pi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pnatP pi n :
n > 0 -> reflect (forall p, prime p -> p %| n -> p \in pi) (pi.-nat n). | Proof.
move=> n_gt0; rewrite /pnat n_gt0.
apply: (iffP allP) => /= pi_n p => [pr_p p_n|].
by rewrite pi_n // mem_primes pr_p n_gt0.
by rewrite mem_primes n_gt0 /=; case/andP; move: p.
Qed. | Lemma | pnatP | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"allP",
"apply",
"mem_primes",
"n_gt0",
"nat",
"pi",
"pnat",
"pr_p",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_pnat pi p n : p.-nat n -> p \in pi -> pi.-nat n. | Proof.
move=> p_n pi_p; have [n_gt0 _] := andP p_n.
by apply/pnatP=> // q q_pr /(pnatP _ n_gt0 p_n _ q_pr)/eqnP->.
Qed. | Lemma | pi_pnat | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"apply",
"eqnP",
"n_gt0",
"nat",
"pi",
"pi_p",
"pnatP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p_natP p n : p.-nat n -> {k | n = p ^ k}. | Proof. by move=> p_n; exists (logn p n); rewrite -p_part part_pnat_id. Qed. | Lemma | p_natP | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"logn",
"nat",
"p_part",
"part_pnat_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi'_p'nat pi 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 | pi'_p'nat | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"apply",
"contraNneq",
"nat",
"pi",
"pi_p",
"sub_in_pnat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_p'nat p pi n : pi.-nat n -> p \in pi^' -> p^'.-nat n. | Proof. by move=> pi_n; apply: pi'_p'nat; rewrite pnatNK. Qed. | Lemma | pi_p'nat | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"apply",
"nat",
"pi",
"pi'_p'nat",
"pnatNK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
partn_part pi rho n : {subset pi <= rho} -> n`_rho`_pi = n`_pi. | Proof.
move=> pi_sub_rho; have [->|n_gt0] := posnP n; first by rewrite !partn0 partn1.
rewrite -[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 | partn_part | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"apply",
"muln1",
"n_gt0",
"nat",
"part_p'nat",
"part_pnat",
"partn0",
"partn1",
"partnC",
"partnM",
"pi",
"posnP",
"sub_in_pnat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
partnI pi rho n : n`_[predI pi & rho] = n`_pi`_rho. | Proof.
rewrite -(@partnC [predI pi & rho] _`_rho) //.
symmetry; rewrite 2?partn_part; try by move=> p /andP [].
rewrite mulnC part_p'nat ?mul1n // pnatNK pnatI part_pnat andbT.
exact: pnat_dvd (dvdn_part _ _) (part_pnat _ _).
Qed. | Lemma | partnI | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"dvdn_part",
"mul1n",
"mulnC",
"part_p'nat",
"part_pnat",
"partnC",
"partn_part",
"pi",
"pnatI",
"pnatNK",
"pnat_dvd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd_2'nat n : odd n = 2^'.-nat n. | Proof. by case: n => // n; rewrite p'natE // dvdn2 negbK. Qed. | Lemma | odd_2'nat | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"dvdn2",
"nat",
"odd",
"p'natE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divisors_correct n : n > 0 ->
[/\ uniq (divisors n), sorted leq (divisors n)
& forall d, (d \in divisors n) = (d %| n)]. | Proof.
move/prod_prime_decomp=> def_n; rewrite {4}def_n {def_n}.
have: all prime (primes n) by apply/allP=> p; rewrite mem_primes; case/andP.
have:= primes_uniq n; rewrite /primes /divisors; move/prime_decomp: n.
elim=> [|[p e] pd] /=; first by split=> // d; rewrite big_nil dvdn1 mem_seq1.
rewrite big_cons /=; move: (f... | Lemma | divisors_correct | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"Euclid_dvdM",
"Gauss_dvdr",
"all",
"allP",
"apply",
"big_cons",
"big_ind",
"big_nil",
"big_seq",
"cat_uniq",
"coprimeXl",
"coprime_sym",
"def_n",
"divisors",
"dvdn1",
"dvdnP",
"dvdn_mulr",
"dvdn_pmul2l",
"dvdn_prime2",
"eqn_pmul2l",
"expnS",
"foldr",
"hasP",
"has_pred0... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_divisors n : sorted leq (divisors n). | Proof. by case: (posnP n) => [-> | /divisors_correct[]]. Qed. | Lemma | sorted_divisors | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"divisors",
"divisors_correct",
"leq",
"posnP",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divisors_uniq n : uniq (divisors n). | Proof. by case: (posnP n) => [-> | /divisors_correct[]]. Qed. | Lemma | divisors_uniq | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"divisors",
"divisors_correct",
"posnP",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_divisors_ltn n : sorted ltn (divisors n). | Proof. by rewrite ltn_sorted_uniq_leq divisors_uniq sorted_divisors. Qed. | Lemma | sorted_divisors_ltn | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"divisors",
"divisors_uniq",
"ltn",
"ltn_sorted_uniq_leq",
"sorted",
"sorted_divisors"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_divisors d m : 0 < m -> (d %| m) = (d \in divisors m). | Proof. by case/divisors_correct. Qed. | Lemma | dvdn_divisors | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"divisors",
"divisors_correct"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divisor1 n : 1 \in divisors n. | Proof. by case: n => // n; rewrite -dvdn_divisors // dvd1n. Qed. | Lemma | divisor1 | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"divisors",
"dvd1n",
"dvdn_divisors"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
divisors_id n : 0 < n -> n \in divisors n. | Proof. by move/dvdn_divisors <-. Qed. | Lemma | divisors_id | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"divisors",
"dvdn_divisors"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdn_sum d I r (K : pred I) F :
(forall i, K i -> d %| F i) -> d %| \sum_(i <- r | K i) F i. | Proof. by move=> dF; elim/big_ind: _ => //; apply: dvdn_add. Qed. | Lemma | dvdn_sum | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"apply",
"big_ind",
"dvdn_add"
] | Big sum / product lemmas | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
dvdn_partP n m : 0 < n ->
reflect (forall p, p \in \pi(n) -> n`_p %| m) (n %| m). | Proof.
move=> n_gt0; apply: (iffP idP) => n_dvd_m => [p _|].
by apply: dvdn_trans n_dvd_m; apply: dvdn_part.
have [-> // | m_gt0] := posnP m.
rewrite -(partnT n_gt0) -(partnT m_gt0).
rewrite !(@widen_partn (m + n)) ?leq_addl ?leq_addr // /in_mem /=.
elim/big_ind2: _ => // [* | q _]; first exact: dvdn_mul.
have [-> //... | Lemma | dvdn_partP | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"apply",
"big_ind2",
"dvdn_mul",
"dvdn_part",
"dvdn_trans",
"leq_addl",
"leq_addr",
"logn",
"logn_gt0",
"mem_primes",
"n_gt0",
"p_part",
"partnT",
"pfactorK",
"pfactor_dvdn",
"pi",
"posnP",
"prime",
"widen_partn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modn_partP n a b : 0 < n ->
reflect (forall p : nat, p \in \pi(n) -> a = b %[mod n`_p]) (a == b %[mod n]). | Proof.
move=> n_gt0; wlog le_b_a: a b / b <= a.
move=> IH; case: (leqP b a) => [|/ltnW] /IH {IH}// IH.
by rewrite eq_sym; apply: (iffP IH) => eqab p /eqab.
rewrite eqn_mod_dvd //; apply: (iffP (dvdn_partP _ n_gt0)) => eqab p /eqab;
by rewrite -eqn_mod_dvd // => /eqP.
Qed. | Lemma | modn_partP | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"apply",
"dvdn_partP",
"eq_sym",
"eqn_mod_dvd",
"leqP",
"ltnW",
"n_gt0",
"nat",
"pi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
totientE n :
n > 0 -> totient n = \prod_(p <- primes n) (p.-1 * p ^ (logn p n).-1). | Proof.
move=> n_gt0; rewrite /totient n_gt0 prime_decompE unlock.
by elim: (primes n) => //= [p pr ->]; rewrite !natTrecE.
Qed. | Lemma | totientE | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"logn",
"n_gt0",
"natTrecE",
"prime_decompE",
"primes",
"totient"
] | The Euler totient function | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
totient_gt0 n : (0 < totient n) = (0 < n). | Proof.
case: n => // n; rewrite totientE // big_seq_cond prodn_cond_gt0 // => p.
by rewrite mem_primes muln_gt0 expn_gt0; case: p => [|[|]].
Qed. | Lemma | totient_gt0 | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"big_seq_cond",
"expn_gt0",
"mem_primes",
"muln_gt0",
"prodn_cond_gt0",
"totient",
"totientE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
totient_pfactor p e :
prime p -> e > 0 -> totient (p ^ e) = p.-1 * p ^ e.-1. | Proof.
move=> p_pr e_gt0; rewrite totientE ?expn_gt0 ?prime_gt0 //.
by rewrite primesX // primes_prime // unlock /= muln1 pfactorK.
Qed. | Lemma | totient_pfactor | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"expn_gt0",
"muln1",
"p_pr",
"pfactorK",
"prime",
"prime_gt0",
"primesX",
"primes_prime",
"totient",
"totientE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
totient_prime p : prime p -> totient p = p.-1. | Proof. by move=> p_prime; rewrite -{1}[p]expn1 totient_pfactor // muln1. Qed. | Lemma | totient_prime | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"expn1",
"muln1",
"prime",
"totient",
"totient_pfactor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
totient_coprime m n :
coprime m n -> totient (m * n) = totient m * totient n. | Proof.
move=> co_mn; have [-> //| m_gt0] := posnP m.
have [->|n_gt0] := posnP n; first by rewrite !muln0.
rewrite !totientE ?muln_gt0 ?m_gt0 //.
have /(perm_big _)->: perm_eq (primes (m * n)) (primes m ++ primes n).
apply: uniq_perm => [||p]; first exact: primes_uniq.
by rewrite cat_uniq !primes_uniq -coprime_has... | Lemma | totient_coprime | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"apply",
"big_cat",
"big_seq",
"cat_uniq",
"coprime",
"coprimeMl",
"coprime_has_primes",
"coprime_sym",
"divnK",
"eq_bigr",
"logn_Gauss",
"mem_cat",
"mem_primes",
"muln0",
"mulnC",
"muln_gt0",
"n_gt0",
"perm_big",
"perm_eq",
"posnP",
"primes",
"primesM",
"primes_uniq",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
totient_count_coprime n : totient n = \sum_(0 <= d < n) coprime n d. | Proof.
elim/ltn_ind: n => // n IHn.
case: (leqP n 1) => [|lt1n]; first by rewrite unlock; case: (n) => [|[]].
pose p := pdiv n; have p_pr: prime p by apply: pdiv_prime.
have p1 := prime_gt1 p_pr; have p0 := ltnW p1.
pose np := n`_p; pose np' := n`_p^'.
have co_npp': coprime np np' by rewrite coprime_partC.
have [n0 np0... | Lemma | totient_count_coprime | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"addnI",
"apply",
"big_mkcond",
"big_mkord",
"cardC",
"card_in_image",
"card_ord",
"chinese",
"chinese_mod",
"chinese_modl",
"chinese_modr",
"codom",
"codomP",
"coprime",
"coprimeMl",
"coprime_modr",
"coprime_partC",
"coprime_pexpl",
"def_n",
"dvdnP",
"dvdn_mulr",
"dvdn_par... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
totient_gt1 n : (totient n > 1) = (n > 2). | Proof.
case: n => [|[|[|[|n']]]]//=; set n := n'.+4; rewrite [RHS]isT.
wlog [q] : / exists k, k.+3 \in primes n; last first.
rewrite mem_primes => /and3P[qp ngt0 qn].
have [[|k]// cqk ->] := pfactor_coprime qp ngt0.
rewrite totient_coprime 1?coprime_sym ?coprimeXl//.
rewrite totient_pfactor// -?pfactor_dvdn// m... | Lemma | totient_gt1 | boot | boot/prime.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"fintype",
"div",
"bigop",
"NatTrec"
] | [
"all_prime_primes",
"apply",
"big_map",
"big_nil",
"big_seq1",
"coprimeXl",
"coprime_sym",
"eqxx",
"expn_gt0",
"in_cons",
"last",
"leq_pexp2l",
"leq_pmulr",
"leq_trans",
"logn",
"mem_head",
"mem_primes",
"mul1n",
"mulnCA",
"muln_gt0",
"n'",
"pfactor_coprime",
"pfactor_dvd... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seq | := list. | Notation | seq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | Inductive seq (T : Type) : Type := Nil | Cons of T & seq T. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Cons T | := (@cons T) (only parsing). | Notation | Cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Nil T | := (@nil T) (only parsing). | Notation | Nil | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ :: ]" | := nil (format "[ :: ]") : seq_scope. | Notation | [ :: ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ :: x1 ]" | := (x1 :: [::]) (format "[ :: x1 ]") : seq_scope. | Notation | [ :: x1 ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ :: x & s ]" | := (x :: s) (only parsing) : seq_scope. | Notation | [ :: x & s ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ :: x1 , x2 , .. , xn & s ]" | := (x1 :: x2 :: .. (xn :: s) ..)
(format
"'[hv' [ :: '[' x1 , '/' x2 , '/' .. , '/' xn ']' '/ ' & s ] ']'"
) : seq_scope. | Notation | [ :: x1 , x2 , .. , xn & s ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ :: x1 ; x2 ; .. ; xn ]" | := (x1 :: x2 :: .. [:: xn] ..)
(format "[ :: '[' x1 ; '/' x2 ; '/' .. ; '/' xn ']' ]"
) : seq_scope. | Notation | [ :: x1 ; x2 ; .. ; xn ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size s | := if s is _ :: s' then (size s').+1 else 0. | Fixpoint | size | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size0nil s : size s = 0 -> s = [::]. | Proof. by case: s. Qed. | Lemma | size0nil | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nilp s | := size s == 0. | Definition | nilp | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nilP s : reflect (s = [::]) (nilp s). | Proof. by case: s => [|x s]; constructor. Qed. | Lemma | nilP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nilp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ohead s | := if s is x :: _ then Some x else None. | Definition | ohead | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
head s | := if s is x :: _ then x else x0. | Definition | head | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
behead s | := if s is _ :: s' then s' else [::]. | Definition | behead | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_behead s : size (behead s) = (size s).-1. | Proof. by case: s. Qed. | Lemma | size_behead | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"behead",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ncons n x | := iter n (cons x). | Definition | ncons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"iter"
] | Factories | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
nseq n x | := ncons n x [::]. | Definition | nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"ncons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_ncons n x s : size (ncons n x s) = n + size s. | Proof. by elim: n => //= n ->. Qed. | Lemma | size_ncons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"ncons",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_nseq n x : size (nseq n x) = n. | Proof. by rewrite size_ncons addn0. Qed. | Lemma | size_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"nseq",
"size",
"size_ncons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seqn_type n | := if n is n'.+1 then T -> seqn_type n' else seq T. | Fixpoint | seqn_type | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"n'",
"seq"
] | n-ary, dependently typed constructor. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
seqn_rec f n : seqn_type n | :=
if n is n'.+1 return seqn_type n then
fun x => seqn_rec (fun s => f (x :: s)) n'
else f [::]. | Fixpoint | seqn_rec | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"n'",
"seqn_type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
seqn | := seqn_rec id. | Definition | seqn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"id",
"seqn_rec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat s1 s2 | := if s1 is x :: s1' then x :: s1' ++ s2 else s2
where "s1 ++ s2" := (cat s1 s2) : seq_scope. | Fixpoint | cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"s1",
"s2"
] | Sequence catenation "cat". | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cat0s s : [::] ++ s = s. | Proof. by []. Qed. | Lemma | cat0s | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat1s x s : [:: x] ++ s = x :: s. | Proof. by []. Qed. | Lemma | cat1s | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat_cons x s1 s2 : (x :: s1) ++ s2 = x :: s1 ++ s2. | Proof. by []. Qed. | Lemma | cat_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat_nseq n x s : nseq n x ++ s = ncons n x s. | Proof. by elim: n => //= n ->. Qed. | Lemma | cat_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"ncons",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nseqD n1 n2 x : nseq (n1 + n2) x = nseq n1 x ++ nseq n2 x. | Proof. by rewrite cat_nseq /nseq /ncons iterD. Qed. | Lemma | nseqD | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_nseq",
"iterD",
"ncons",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cats0 s : s ++ [::] = s. | Proof. by elim: s => //= x s ->. Qed. | Lemma | cats0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
catA s1 s2 s3 : s1 ++ s2 ++ s3 = (s1 ++ s2) ++ s3. | Proof. by elim: s1 => //= x s1 ->. Qed. | Lemma | catA | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"s1",
"s2",
"s3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_cat s1 s2 : size (s1 ++ s2) = size s1 + size s2. | Proof. by elim: s1 => //= x s1 ->. Qed. | Lemma | size_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"s1",
"s2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat_nilp s1 s2 : nilp (s1 ++ s2) = nilp s1 && nilp s2. | Proof. by case: s1. Qed. | Lemma | cat_nilp | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nilp",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcons s z | := if s is x :: s' then x :: rcons s' z else [:: z]. | Fixpoint | rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | last, belast, rcons, and last induction. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rcons_cons x s z : rcons (x :: s) z = x :: rcons s z. | Proof. by []. Qed. | Lemma | rcons_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cats1 s z : s ++ [:: z] = rcons s z. | Proof. by elim: s => //= x s ->. Qed. | Lemma | cats1 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
last x s | := if s is x' :: s' then last x' s' else x. | Fixpoint | last | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
belast x s | := if s is x' :: s' then x :: (belast x' s') else [::]. | Fixpoint | belast | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lastI x s : x :: s = rcons (belast x s) (last x s). | Proof. by elim: s x => [|y s IHs] x //=; rewrite IHs. Qed. | Lemma | lastI | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"belast",
"last",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
last_cons x y s : last x (y :: s) = last y s. | Proof. by []. Qed. | Lemma | last_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_rcons s x : size (rcons s x) = (size s).+1. | Proof. by rewrite -cats1 size_cat addnC. Qed. | Lemma | size_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnC",
"cats1",
"rcons",
"size",
"size_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_belast x s : size (belast x s) = size s. | Proof. by elim: s x => [|y s IHs] x //=; rewrite IHs. Qed. | Lemma | size_belast | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"belast",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
last_cat x s1 s2 : last x (s1 ++ s2) = last (last x s1) s2. | Proof. by elim: s1 x => [|y s1 IHs] x //=; rewrite IHs. Qed. | Lemma | last_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"last",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
last_rcons x s z : last x (rcons s z) = z. | Proof. by rewrite -cats1 last_cat. Qed. | Lemma | last_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"last",
"last_cat",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
belast_cat x s1 s2 :
belast x (s1 ++ s2) = belast x s1 ++ belast (last x s1) s2. | Proof. by elim: s1 x => [|y s1 IHs] x //=; rewrite IHs. Qed. | Lemma | belast_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"belast",
"last",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
belast_rcons x s z : belast x (rcons s z) = x :: s. | Proof. by rewrite lastI -!cats1 belast_cat. Qed. | Lemma | belast_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"belast",
"belast_cat",
"cats1",
"lastI",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat_rcons x s1 s2 : rcons s1 x ++ s2 = s1 ++ x :: s2. | Proof. by rewrite -cats1 -catA. Qed. | Lemma | cat_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"catA",
"cats1",
"rcons",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcons_cat x s1 s2 : rcons (s1 ++ s2) x = s1 ++ rcons s2 x. | Proof. by rewrite -!cats1 catA. Qed. | Lemma | rcons_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"catA",
"cats1",
"rcons",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
last_spec : seq T -> Type | :=
| LastNil : last_spec [::]
| LastRcons s x : last_spec (rcons s x). | Variant | last_spec | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rcons",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lastP s : last_spec s. | Proof. case: s => [|x s]; [left | rewrite lastI; right]. Qed. | Lemma | lastP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"lastI",
"last_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
last_ind P :
P [::] -> (forall s x, P s -> P (rcons s x)) -> forall s, P s. | Proof.
move=> Hnil Hlast s; rewrite -(cat0s s).
elim: s [::] Hnil => [|x s2 IHs] s1 Hs1; first by rewrite cats0.
by rewrite -cat_rcons; apply/IHs/Hlast.
Qed. | Lemma | last_ind | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"cat0s",
"cat_rcons",
"cats0",
"rcons",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth s n {struct n} | :=
if s is x :: s' then if n is n'.+1 then @nth s' n' else x else x0. | Fixpoint | nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"n'"
] | Sequence indexing. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
set_nth s n y {struct n} | :=
if s is x :: s' then if n is n'.+1 then x :: @set_nth s' n' y else y :: s'
else ncons n x0 [:: y]. | Fixpoint | set_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"n'",
"ncons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth0 s : nth s 0 = head s. | Proof. by []. Qed. | Lemma | nth0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"head",
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_default s n : size s <= n -> nth s n = x0. | Proof. by elim: s n => [|x s IHs] []. Qed. | Lemma | nth_default | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nth",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
if_nth s b n : b || (size s <= n) ->
(if b then nth s n else x0) = nth s n. | Proof. by case: leqP; case: ifP => //= *; rewrite nth_default. Qed. | Lemma | if_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"leqP",
"nth",
"nth_default",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_nil n : nth [::] n = x0. | Proof. by case: n. Qed. | Lemma | nth_nil | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_seq1 n x : nth [:: x] n = if n == 0 then x else x0. | Proof. by case: n => [|[]]. Qed. | Lemma | nth_seq1 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
last_nth x s : last x s = nth (x :: s) (size s). | Proof. by elim: s x => [|y s IHs] x /=. Qed. | Lemma | last_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"last",
"nth",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_last s : nth s (size s).-1 = last x0 s. | Proof. by case: s => //= x s; rewrite last_nth. Qed. | Lemma | nth_last | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"last",
"last_nth",
"nth",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_behead s n : nth (behead s) n = nth s n.+1. | Proof. by case: s n => [|x s] [|n]. Qed. | Lemma | nth_behead | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"behead",
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_cat s1 s2 n :
nth (s1 ++ s2) n = if n < size s1 then nth s1 n else nth s2 (n - size s1). | Proof. by elim: s1 n => [|x s1 IHs] []. Qed. | Lemma | nth_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nth",
"s1",
"s2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_cons x s n : nth (x :: s) n = if n > 0 then nth s n.-1 else x. | Proof. by case: n. Qed. | Lemma | nth_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_rcons s x n :
nth (rcons s x) n =
if n < size s then nth s n else if n == size s then x else x0. | Proof. by elim: s n => [|y s IHs] [] //=; apply: nth_nil. Qed. | Lemma | nth_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"nth",
"nth_nil",
"rcons",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_rcons_default s i : nth (rcons s x0) i = nth s i. | Proof.
by rewrite nth_rcons; case: ltngtP => //[/ltnW ?|->]; rewrite nth_default.
Qed. | Lemma | nth_rcons_default | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"ltnW",
"ltngtP",
"nth",
"nth_default",
"nth_rcons",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_ncons m x s n :
nth (ncons m x s) n = if n < m then x else nth s (n - m). | Proof. by elim: m n => [|m IHm] []. Qed. | Lemma | nth_ncons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"ncons",
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_nseq m x n : nth (nseq m x) n = (if n < m then x else x0). | Proof. by elim: m n => [|m IHm] []. Qed. | Lemma | nth_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nseq",
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_from_nth s1 s2 :
size s1 = size s2 -> (forall i, i < size s1 -> nth s1 i = nth s2 i) ->
s1 = s2. | Proof.
elim: s1 s2 => [|x1 s1 IHs1] [|x2 s2] //= [eq_sz] eq_s12.
by rewrite [x1](eq_s12 0) // (IHs1 s2) // => i; apply: (eq_s12 i.+1).
Qed. | Lemma | eq_from_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"nth",
"s1",
"s2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_set_nth s n y : size (set_nth s n y) = maxn n.+1 (size s). | Proof.
rewrite maxnC; elim: s n => [|x s IHs] [|n] //=.
- by rewrite size_ncons addn1.
- by rewrite IHs maxnSS.
Qed. | Lemma | size_set_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn1",
"maxn",
"maxnC",
"maxnSS",
"set_nth",
"size",
"size_ncons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_nth_nil n y : set_nth [::] n y = ncons n x0 [:: y]. | Proof. by case: n. Qed. | Lemma | set_nth_nil | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"ncons",
"set_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_set_nth s n y : nth (set_nth s n y) =1 [eta nth s with n |-> y]. | Proof.
elim: s n => [|x s IHs] [|n] [|m] //=; rewrite ?nth_nil ?IHs // nth_ncons eqSS.
case: ltngtP => // [lt_nm | ->]; last by rewrite subnn.
by rewrite nth_default // subn_gt0.
Qed. | Lemma | nth_set_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqSS",
"last",
"ltngtP",
"nth",
"nth_default",
"nth_ncons",
"nth_nil",
"set_nth",
"subn_gt0",
"subnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.