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 |
|---|---|---|---|---|---|---|---|---|---|---|
mask_uniq s : uniq s -> forall m, uniq (mask m s). | Proof.
elim: s => [|x s IHs] Uxs [|b m] //=.
case: b Uxs => //= /andP[s'x Us]; rewrite {}IHs // andbT.
by apply: contra s'x; apply: mem_mask.
Qed. | Lemma | mask_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"mask",
"mem_mask",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_mask_rot m s :
size m = size s -> mask (rot n0 m) (rot n0 s) =i mask m s. | Proof. by move=> Ems x; rewrite mask_rot // mem_rot. Qed. | Lemma | mem_mask_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mask",
"mask_rot",
"mem_rot",
"rot",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq s1 s2 | :=
if s2 is y :: s2' then
if s1 is x :: s1' then subseq (if x == y then s1' else s1) s2' else true
else s1 == [::]. | Fixpoint | subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub0seq s : subseq [::] s. | Proof. by case: s. Qed. | Lemma | sub0seq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq0 s : subseq s [::] = (s == [::]). | Proof. by []. Qed. | Lemma | subseq0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_refl s : subseq s s. | Proof. by elim: s => //= x s IHs; rewrite eqxx. Qed. | Lemma | subseq_refl | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqxx",
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseqP s1 s2 :
reflect (exists2 m, size m = size s2 & s1 = mask m s2) (subseq s1 s2). | Proof.
elim: s2 s1 => [|y s2 IHs2] [|x s1].
- by left; exists [::].
- by right=> -[m /eqP/nilP->].
- by left; exists (nseq (size s2).+1 false); rewrite ?size_nseq //= mask_false.
apply: {IHs2}(iffP (IHs2 _)) => [] [m sz_m def_s1].
by exists ((x == y) :: m); rewrite /= ?sz_m // -def_s1; case: eqP => // ->.
case: eqP =... | Lemma | subseqP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all_nthP",
"all_pred1P",
"apply",
"before_find",
"behead",
"cat_cons",
"cat_rcons",
"cat_take_drop",
"drop",
"drop_nth",
"geq_min",
"index",
"index_mem",
"last",
"lastI",
"ltnNge",
"mask",
"mask_cat",
"mask_false",
"negb_add",
"negb_eqb",
"nilP",
"nseq",
"nth_index",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_subseq m s : subseq (mask m s) s. | Proof. by apply/subseqP; have [m1] := resize_mask m s; exists m1. Qed. | Lemma | mask_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"mask",
"resize_mask",
"subseq",
"subseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_trans : transitive subseq. | Proof.
move=> _ _ s /subseqP[m2 _ ->] /subseqP[m1 _ ->].
elim: s => [|x s IHs] in m2 m1 *; first by rewrite !mask0.
case: m1 => [|[] m1]; first by rewrite mask0.
case: m2 => [|[] m2] //; first by rewrite /= eqxx IHs.
case/subseqP: (IHs m2 m1) => m sz_m def_s; apply/subseqP.
by exists (false :: m); rewrite //= sz_... | Lemma | subseq_trans | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eqxx",
"mask0",
"subseq",
"subseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat_subseq s1 s2 s3 s4 :
subseq s1 s3 -> subseq s2 s4 -> subseq (s1 ++ s2) (s3 ++ s4). | Proof.
case/subseqP=> m1 sz_m1 -> /subseqP [m2 sz_m2 ->]; apply/subseqP.
by exists (m1 ++ m2); rewrite ?size_cat ?mask_cat ?sz_m1 ?sz_m2.
Qed. | Lemma | cat_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"mask_cat",
"s1",
"s2",
"s3",
"s4",
"size_cat",
"subseq",
"subseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prefix_subseq s1 s2 : subseq s1 (s1 ++ s2). | Proof. by rewrite -[s1 in subseq s1]cats0 cat_subseq ?sub0seq. Qed. | Lemma | prefix_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_subseq",
"cats0",
"s1",
"s2",
"sub0seq",
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
suffix_subseq s1 s2 : subseq s2 (s1 ++ s2). | Proof. exact: cat_subseq (sub0seq s1) _. Qed. | Lemma | suffix_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_subseq",
"s1",
"s2",
"sub0seq",
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_subseq s i : subseq (take i s) s. | Proof. by rewrite -[s in X in subseq _ X](cat_take_drop i) prefix_subseq. Qed. | Lemma | take_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"prefix_subseq",
"subseq",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_subseq s i : subseq (drop i s) s. | Proof. by rewrite -[s in X in subseq _ X](cat_take_drop i) suffix_subseq. Qed. | Lemma | drop_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"drop",
"subseq",
"suffix_subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_subseq s1 s2 : subseq s1 s2 -> {subset s1 <= s2}. | Proof. by case/subseqP=> m _ -> x; apply: mem_mask. Qed. | Lemma | mem_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"mem_mask",
"s1",
"s2",
"subseq",
"subseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub1seq x s : subseq [:: x] s = (x \in s). | Proof. by elim: s => //= y s /[1!inE]; case: ifP; rewrite ?sub0seq. Qed. | Lemma | sub1seq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"inE",
"sub0seq",
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_subseq s1 s2 : subseq s1 s2 -> size s1 <= size s2. | Proof. by case/subseqP=> m sz_m ->; rewrite size_mask -sz_m ?count_size. Qed. | Lemma | size_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count_size",
"s1",
"s2",
"size",
"size_mask",
"subseq",
"subseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_subseq_leqif s1 s2 :
subseq s1 s2 -> size s1 <= size s2 ?= iff (s1 == s2). | Proof.
move=> sub12; split; first exact: size_subseq.
apply/idP/eqP=> [|-> //]; case/subseqP: sub12 => m sz_m ->{s1}.
rewrite size_mask -sz_m // -all_count -(eq_all eqb_id).
by move/(@all_pred1P _ true)->; rewrite sz_m mask_true.
Qed. | Lemma | size_subseq_leqif | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all_count",
"all_pred1P",
"apply",
"eq_all",
"eqb_id",
"mask_true",
"s1",
"s2",
"size",
"size_mask",
"size_subseq",
"split",
"subseq",
"subseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_anti : antisymmetric subseq. | Proof.
move=> s1 s2 /andP[] /size_subseq_leqif /leqifP.
by case: eqP => [//|_] + /size_subseq; rewrite ltnNge => /negP.
Qed. | Lemma | subseq_anti | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"leqifP",
"ltnNge",
"s1",
"s2",
"size_subseq",
"size_subseq_leqif",
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_cons s x : subseq s (x :: s). | Proof. exact: suffix_subseq [:: x] s. Qed. | Lemma | subseq_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"subseq",
"suffix_subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cons_subseq s1 s2 x : subseq (x :: s1) s2 -> subseq s1 s2. | Proof. exact/subseq_trans/subseq_cons. Qed. | Lemma | cons_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"s1",
"s2",
"subseq",
"subseq_cons",
"subseq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_rcons s x : subseq s (rcons s x). | Proof. by rewrite -cats1 prefix_subseq. Qed. | Lemma | subseq_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"prefix_subseq",
"rcons",
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_uniq s1 s2 : subseq s1 s2 -> uniq s2 -> uniq s1. | Proof. by case/subseqP=> m _ -> Us2; apply: mask_uniq. Qed. | Lemma | subseq_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"mask_uniq",
"s1",
"s2",
"subseq",
"subseqP",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_uniq s n : uniq s -> uniq (take n s). | Proof. exact/subseq_uniq/take_subseq. Qed. | Lemma | take_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"subseq_uniq",
"take",
"take_subseq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_uniq s n : uniq s -> uniq (drop n s). | Proof. exact/subseq_uniq/drop_subseq. Qed. | Lemma | drop_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"drop_subseq",
"subseq_uniq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
undup_subseq s : subseq (undup s) s. | Proof.
elim: s => //= x s; case: (_ \in _); last by rewrite eqxx.
by case: (undup s) => //= y u; case: (_ == _) => //=; apply: cons_subseq.
Qed. | Lemma | undup_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"cons_subseq",
"eqxx",
"last",
"subseq",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_rev s1 s2 : subseq (rev s1) (rev s2) = subseq s1 s2. | Proof.
wlog suff W : s1 s2 / subseq s1 s2 -> subseq (rev s1) (rev s2).
by apply/idP/idP => /W //; rewrite !revK.
by case/subseqP => m size_m ->; rewrite rev_mask // mask_subseq.
Qed. | Lemma | subseq_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"mask_subseq",
"rev",
"revK",
"rev_mask",
"s1",
"s2",
"subseq",
"subseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_cat2l s s1 s2 : subseq (s ++ s1) (s ++ s2) = subseq s1 s2. | Proof. by elim: s => // x s IHs; rewrite !cat_cons /= eqxx. Qed. | Lemma | subseq_cat2l | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_cons",
"eqxx",
"s1",
"s2",
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_cat2r s s1 s2 : subseq (s1 ++ s) (s2 ++ s) = subseq s1 s2. | Proof. by rewrite -subseq_rev !rev_cat subseq_cat2l subseq_rev. Qed. | Lemma | subseq_cat2r | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rev_cat",
"s1",
"s2",
"subseq",
"subseq_cat2l",
"subseq_rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_rot p s n :
subseq p s -> exists2 k, k <= n & subseq (rot k p) (rot n s). | Proof.
move=> /subseqP[m size_m ->].
exists (count id (take n m)); last by rewrite -mask_rot // mask_subseq.
by rewrite (leq_trans (count_size _ _))// size_take_min geq_minl.
Qed. | Lemma | subseq_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"count_size",
"geq_minl",
"id",
"last",
"leq_trans",
"mask_rot",
"mask_subseq",
"rot",
"size_take_min",
"subseq",
"subseqP",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rem s | := if s is y :: t then (if y == x then t else y :: rem t) else s. | Fixpoint | rem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rem_cons y s : rem (y :: s) = if y == x then s else y :: rem s. | Proof. by []. Qed. | Lemma | rem_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
remE s : rem s = take (index x s) s ++ drop (index x s).+1 s. | Proof. by elim: s => //= y s ->; case: eqVneq; rewrite ?drop0. Qed. | Lemma | remE | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"drop0",
"eqVneq",
"index",
"rem",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rem_id s : x \notin s -> rem s = s. | Proof. by elim: s => //= y s IHs /norP[neq_yx /IHs->]; case: eqVneq neq_yx. Qed. | Lemma | rem_id | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqVneq",
"rem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_to_rem s : x \in s -> perm_eq s (x :: rem s). | Proof.
move=> xs; rewrite remE -[X in perm_eq X](cat_take_drop (index x s)).
by rewrite drop_index// -cat1s perm_catCA cat1s.
Qed. | Lemma | perm_to_rem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat1s",
"cat_take_drop",
"drop_index",
"index",
"perm_catCA",
"perm_eq",
"rem",
"remE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_rem s : x \in s -> size (rem s) = (size s).-1. | Proof. by move/perm_to_rem/perm_size->. Qed. | Lemma | size_rem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"perm_size",
"perm_to_rem",
"rem",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rem_subseq s : subseq (rem s) s. | Proof.
elim: s => //= y s IHs; rewrite eq_sym.
by case: ifP => _; [apply: subseq_cons | rewrite eqxx].
Qed. | Lemma | rem_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eq_sym",
"eqxx",
"rem",
"subseq",
"subseq_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rem_uniq s : uniq s -> uniq (rem s). | Proof. by apply: subseq_uniq; apply: rem_subseq. Qed. | Lemma | rem_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"rem",
"rem_subseq",
"subseq_uniq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_rem s : {subset rem s <= s}. | Proof. exact: mem_subseq (rem_subseq s). Qed. | Lemma | mem_rem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mem_subseq",
"rem",
"rem_subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rem_mem y s : y != x -> y \in s -> y \in rem s. | Proof.
move=> yx; elim: s => [//|z s IHs] /=.
rewrite inE => /orP[/eqP<-|ys]; first by rewrite (negbTE yx) inE eqxx.
by case: ifP => _ //; rewrite inE IHs ?orbT.
Qed. | Lemma | rem_mem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqxx",
"inE",
"rem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rem_filter s : uniq s -> rem s = filter (predC1 x) s. | Proof.
elim: s => //= y s IHs /andP[not_s_y /IHs->].
by case: eqP => //= <-; apply/esym/all_filterP; rewrite all_predC has_pred1.
Qed. | Lemma | rem_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all_filterP",
"all_predC",
"apply",
"filter",
"has_pred1",
"predC1",
"rem",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_rem_uniq s : uniq s -> rem s =i [predD1 s & x]. | Proof. by move/rem_filter=> -> y; rewrite mem_filter. Qed. | Lemma | mem_rem_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mem_filter",
"predD1",
"rem",
"rem_filter",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_rem_uniqF s : uniq s -> (x \in rem s) = false. | Proof. by move/mem_rem_uniq->; rewrite inE eqxx. Qed. | Lemma | mem_rem_uniqF | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqxx",
"inE",
"mem_rem_uniq",
"rem",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_rem P s : count P (rem s) = count P s - (x \in s) && P x. | Proof.
have [/perm_to_rem/permP->|xNs]/= := boolP (x \in s); first by rewrite addKn.
by rewrite subn0 rem_id.
Qed. | Lemma | count_rem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addKn",
"count",
"permP",
"perm_to_rem",
"rem",
"rem_id",
"subn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_mem_rem y s : count_mem y (rem s) = count_mem y s - (x == y). | Proof.
rewrite count_rem; have []//= := boolP (x \in s).
by case: eqP => // <- /count_memPn->.
Qed. | Lemma | count_mem_rem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count_mem",
"count_memPn",
"count_rem",
"rem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map s | := if s is x :: s' then f x :: map s' else [::]. | Fixpoint | map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_cons x s : map (x :: s) = f x :: map s. | Proof. by []. Qed. | Lemma | map_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_nseq x : map (nseq n0 x) = nseq n0 (f x). | Proof. by elim: n0 => // *; congr (_ :: _). Qed. | Lemma | map_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_cat s1 s2 : map (s1 ++ s2) = map s1 ++ map s2. | Proof. by elim: s1 => [|x s1 IHs] //=; rewrite IHs. Qed. | Lemma | map_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_map s : size (map s) = size s. | Proof. by elim: s => //= x s ->. Qed. | Lemma | size_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
behead_map s : behead (map s) = map (behead s). | Proof. by case: s. Qed. | Lemma | behead_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"behead",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_map n s : n < size s -> nth x2 (map s) n = f (nth x1 s n). | Proof. by elim: s n => [|x s IHs] []. Qed. | Lemma | nth_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"nth",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_nth x n s : f (nth x s n) = nth (f x) (map s) n. | Proof. by elim: n s => [|n IHn] [|y s] /=. Qed. | Lemma | map_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_rcons s x : map (rcons s x) = rcons (map s) (f x). | Proof. by rewrite -!cats1 map_cat. Qed. | Lemma | map_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"map",
"map_cat",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
last_map s x : last (f x) (map s) = f (last x s). | Proof. by elim: s x => /=. Qed. | Lemma | last_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"last",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
belast_map s x : belast (f x) (map s) = map (belast x s). | Proof. by elim: s x => //= y s IHs x; rewrite IHs. Qed. | Lemma | belast_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"belast",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_map a s : filter a (map s) = map (filter (preim f a) s). | Proof. by elim: s => //= x s IHs; rewrite (fun_if map) /= IHs. Qed. | Lemma | filter_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
find_map a s : find a (map s) = find (preim f a) s. | Proof. by elim: s => //= x s ->. Qed. | Lemma | find_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_map a s : has a (map s) = has (preim f a) s. | Proof. by elim: s => //= x s ->. Qed. | Lemma | has_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_map a s : all a (map s) = all (preim f a) s. | Proof. by elim: s => //= x s ->. Qed. | Lemma | all_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_mapT (a : pred T2) s : (forall x, a (f x)) -> all a (map s). | Proof. by rewrite all_map => /allT->. Qed. | Lemma | all_mapT | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"allT",
"all_map",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_map a s : count a (map s) = count (preim f a) s. | Proof. by elim: s => //= x s ->. Qed. | Lemma | count_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_take s : map (take n0 s) = take n0 (map s). | Proof. by elim: n0 s => [|n IHn] [|x s] //=; rewrite IHn. Qed. | Lemma | map_take | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_drop s : map (drop n0 s) = drop n0 (map s). | Proof. by elim: n0 s => [|n IHn] [|x s] //=; rewrite IHn. Qed. | Lemma | map_drop | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_rot s : map (rot n0 s) = rot n0 (map s). | Proof. by rewrite /rot map_cat map_take map_drop. Qed. | Lemma | map_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"map_cat",
"map_drop",
"map_take",
"rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_rotr s : map (rotr n0 s) = rotr n0 (map s). | Proof. by apply: canRL (rotK n0) _; rewrite -map_rot rotrK. Qed. | Lemma | map_rotr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"map",
"map_rot",
"rotK",
"rotr",
"rotrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_rev s : map (rev s) = rev (map s). | Proof. by elim: s => //= x s IHs; rewrite !rev_cons -!cats1 map_cat IHs. Qed. | Lemma | map_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"map",
"map_cat",
"rev",
"rev_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_mask m s : map (mask m s) = mask m (map s). | Proof. by elim: m s => [|[|] m IHm] [|x p] //=; rewrite IHm. Qed. | Lemma | map_mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_map : injective f -> injective map. | Proof. by move=> injf; elim=> [|x s IHs] [|y t] //= [/injf-> /IHs->]. Qed. | Lemma | inj_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"injf",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_in_map (A : {pred T1}) :
{in A &, injective f} -> {in [pred s | all [in A] s] &, injective map}. | Proof.
move=> injf; elim=> [|x s IHs] [|y t] //= /andP[Ax As] /andP[Ay At].
by case=> /injf-> // /IHs->.
Qed. | Lemma | inj_in_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"injf",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onth s n {struct n} : option T | :=
if s isn't x :: s then None else
if n isn't n.+1 then Some x else onth s n. | Fixpoint | onth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odflt_onth x0 s n : odflt x0 (onth s n) = nth x0 s n. | Proof. by elim: n s => [|? ?] []. Qed. | Lemma | odflt_onth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nth",
"onth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onthE s : onth s =1 nth None (map Some s). | Proof. by move=> n; elim: n s => [|? ?] []. Qed. | Lemma | onthE | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"nth",
"onth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onth_nth x0 x t n : onth t n = Some x -> nth x0 t n = x. | Proof. by move=> tn; rewrite -odflt_onth tn. Qed. | Lemma | onth_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nth",
"odflt_onth",
"onth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onth0n n : onth [::] n = None. | Proof. by case: n. Qed. | Lemma | onth0n | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"onth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onth1P x y n : onth [:: x] n = Some y <-> n = 0 /\ x = y. | Proof. by case: n => [|[]]; split=> // -[] // _ ->. Qed. | Lemma | onth1P | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"onth",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onthTE s n : onth s n = (n < size s) :> bool. | Proof. by elim: n s => [|? ?] []. Qed. | Lemma | onthTE | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"onth",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onthNE s n: ~~ onth s n = (size s <= n). | Proof. by rewrite onthTE -leqNgt. Qed. | Lemma | onthNE | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"leqNgt",
"onth",
"onthTE",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onth_default n s : size s <= n -> onth s n = None. | Proof. by rewrite -onthNE; case: onth. Qed. | Lemma | onth_default | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"onth",
"onthNE",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onth_cat s1 s2 n :
onth (s1 ++ s2) n = if n < size s1 then onth s1 n else onth s2 (n - size s1). | Proof. by elim: n s1 => [|? ?] []. Qed. | Lemma | onth_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"onth",
"s1",
"s2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onth_nseq x n m : onth (nseq n x) m = if m < n then Some x else None. | Proof. by rewrite onthE/= -nth_nseq map_nseq. Qed. | Lemma | onth_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map_nseq",
"nseq",
"nth_nseq",
"onth",
"onthE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_onthP {s1 s2} :
[<-> s1 = s2;
forall i : nat, i < maxn (size s1) (size s2) -> onth s1 i = onth s2 i;
forall i : nat, onth s1 i = onth s2 i]. | Proof.
tfae=> [->//|eqs12 i|eqs12].
have := eqs12 i; case: ltnP => [_ ->//|].
by rewrite geq_max => /andP[is1 is2] _; rewrite !onth_default.
have /eqP eq_size_12 : size s1 == size s2.
by rewrite eqn_leq -!onthNE eqs12 onthNE -eqs12 onthNE !leqnn.
apply/(inj_map Some_inj)/(@eq_from_nth _ None); rewrite !size_map//... | Lemma | eq_onthP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eq_from_nth",
"eqn_leq",
"geq_max",
"inj_map",
"leqnn",
"ltnP",
"maxn",
"nat",
"onth",
"onthE",
"onthNE",
"onth_default",
"s1",
"s2",
"size",
"size_map",
"tfae"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_from_onth [s1 s2 : seq T] :
(forall i : nat, onth s1 i = onth s2 i) -> s1 = s2. | Proof. by move/(eq_onthP 0 2). Qed. | Lemma | eq_from_onth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eq_onthP",
"nat",
"onth",
"s1",
"s2",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_from_onth_le [s1 s2 : seq T] :
(forall i : nat, i < maxn (size s1) (size s2) -> onth s1 i = onth s2 i) ->
s1 = s2. | Proof. by move/(eq_onthP 0 1). Qed. | Lemma | eq_from_onth_le | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eq_onthP",
"maxn",
"nat",
"onth",
"s1",
"s2",
"seq",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onth_map {T S} n (s : seq T) (f : T -> S) :
onth (map f s) n = omap f (onth s n). | Proof. by elim: s n => [|x s IHs] []. Qed. | Lemma | onth_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"onth",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_onth_map {T S} n (s : seq T) (f : T -> S) x :
injective f -> onth (map f s) n = Some (f x) -> onth s n = Some x. | Proof. by rewrite onth_map => /inj_omap + fs; apply. Qed. | Lemma | inj_onth_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"inj_omap",
"map",
"onth",
"onth_map",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onthP s x : reflect (exists i, onth s i = Some x) (x \in s). | Proof.
elim: s => [|y s IHs]; first by constructor=> -[] [].
rewrite in_cons; case: eqVneq => [->|/= Nxy]; first by constructor; exists 0.
apply: (iffP idP) => [/IHs[i <-]|[[|i]//=]]; first by exists i.+1.
by move=> [eq_xy]; rewrite eq_xy eqxx in Nxy.
by move=> six; apply/IHs; exists i.
Qed. | Lemma | onthP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eqVneq",
"eqxx",
"in_cons",
"onth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onthPn s x : reflect (forall i, onth s i != Some x) (x \notin s). | Proof.
apply: (iffP idP); first by move=> /onthP + i; apply: contra_not_neq; exists i.
by move=> nsix; apply/onthP => -[n /eqP/negPn]; rewrite nsix.
Qed. | Lemma | onthPn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"contra_not_neq",
"onth",
"onthP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
onth_inj s n m : uniq s -> minn m n < size s ->
onth s n = onth s m -> n = m. | Proof.
elim: s m n => [|x s IHs]//= [|m] [|n]//=; rewrite ?minnSS !ltnS.
- by move=> /andP[+ _] _ /eqP => /onthPn/(_ _)/negPf->.
- by move=> /andP[+ _] _ /esym /eqP => /onthPn/(_ _)/negPf->.
by move=> /andP[xNs /IHs]/[apply]/[apply]->.
Qed. | Lemma | onth_inj | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"ltnS",
"minn",
"minnSS",
"onth",
"onthPn",
"size",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'seq' E | i <- s ]" | := (map (fun i => E) s)
(i binder, format "[ '[hv' 'seq' E '/ ' | i <- s ] ']'") : seq_scope. | Notation | [ 'seq' E | i <- s ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'seq' E | i <- s & C ]" | := [seq E | i <- [seq i <- s | C]]
(i binder,
format "[ '[hv' 'seq' E '/ ' | i <- s '/ ' & C ] ']'") : seq_scope. | Notation | [ 'seq' E | i <- s & C ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'seq' E : R | i <- s ]" | := (@map _ R (fun i => E) s)
(i binder, only parsing) : seq_scope. | Notation | [ 'seq' E : R | i <- s ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'seq' E : R | i <- s & C ]" | := [seq E : R | i <- [seq i <- s | C]]
(i binder, only parsing) : seq_scope. | Notation | [ 'seq' E : R | i <- s & C ] | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_mask T a (s : seq T) : filter a s = mask (map a s) s. | Proof. by elim: s => //= x s <-; case: (a x). Qed. | Lemma | filter_mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter",
"map",
"mask",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_sigP T a (s : seq T) : all a s -> {s' : seq (sig a) | s = map sval s'}. | Proof.
elim: s => /= [_|x s ihs /andP [ax /ihs [s' ->]]]; first by exists [::].
by exists (exist a x ax :: s').
Qed. | Lemma | all_sigP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"map",
"seq",
"sig"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_count_mask T (P : {pred T}) m s : count P (mask m s) <= count P s. | Proof.
by elim: s m => [|x s IHs] [|[] m]//=;
rewrite ?leq_add2l (leq_trans (IHs _)) ?leq_addl.
Qed. | Lemma | leq_count_mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"leq_add2l",
"leq_addl",
"leq_trans",
"mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_filter s m : uniq s -> mask m s = [seq i <- s | i \in mask m s]. | Proof.
elim: m s => [|[] m IH] [|x s /= /andP[/negP xS uS]]; rewrite ?filter_pred0 //.
rewrite inE eqxx /=; congr cons; rewrite [LHS]IH//.
by apply/eq_in_filter => ? /[1!inE]; case: eqP => [->|].
by case: ifP => [/mem_mask //|_]; apply: IH.
Qed. | Lemma | mask_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eq_in_filter",
"eqxx",
"filter_pred0",
"inE",
"mask",
"mem_mask",
"seq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_count_subseq P s1 s2 : subseq s1 s2 -> count P s1 <= count P s2. | Proof. by move=> /subseqP[m _ ->]; rewrite leq_count_mask. Qed. | Lemma | leq_count_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"leq_count_mask",
"s1",
"s2",
"subseq",
"subseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_maskP s1 s2 :
(forall x, count_mem x s1 <= count_mem x s2) <->
exists2 m : bitseq, size m = size s2 & perm_eq s1 (mask m s2). | Proof.
split=> [s1_le|[m _ /permP s1ms2 x]]; last by rewrite s1ms2 leq_count_mask.
suff [m mP]: exists m, perm_eq s1 (mask m s2).
by have [m' sm' eqm] := resize_mask m s2; exists m'; rewrite -?eqm.
elim: s2 => [|x s2 IHs]//= in s1 s1_le *.
by exists [::]; apply/allP => x _/=; rewrite eqn_leq s1_le.
have [y|m s1s2] ... | Lemma | count_maskP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"allP",
"apply",
"bitseq",
"count_mem",
"count_mem_rem",
"eqn_leq",
"last",
"leq_count_mask",
"leq_subLR",
"mask",
"permP",
"permPl",
"perm_cons",
"perm_eq",
"perm_to_rem",
"rem",
"rem_id",
"resize_mask",
"s1",
"s2",
"size",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_subseqP s1 s2 :
(forall x, count_mem x s1 <= count_mem x s2) <->
exists2 s, subseq s s2 & perm_eq s1 s. | Proof.
split=> [/count_maskP[m _]|]; first by exists (mask m s2); rewrite ?mask_subseq.
by move=> -[_/subseqP[m sm ->] ?]; apply/count_maskP; exists m.
Qed. | Lemma | count_subseqP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"count_maskP",
"count_mem",
"mask",
"mask_subseq",
"perm_eq",
"s1",
"s2",
"split",
"subseq",
"subseqP"
] | 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.