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 |
|---|---|---|---|---|---|---|---|---|---|---|
filter_subseq a s : subseq (filter a s) s. | Proof. by apply/subseqP; exists (map a s); rewrite ?size_map ?filter_mask. Qed. | Lemma | filter_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"filter",
"filter_mask",
"map",
"size_map",
"subseq",
"subseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_memP s1 s2 : {subset s1 <= s2} -> {s3 | subseq s3 s2 & s1 =i s3}. | Proof.
move=> s12; exists (filter [in s1] s2); first by rewrite filter_subseq.
by move=> x; rewrite mem_filter andb_idr//; apply: s12.
Qed. | Lemma | subset_memP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"filter",
"filter_subseq",
"mem_filter",
"s1",
"s12",
"s2",
"s3",
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_filter s1 s2 a :
subseq s1 (filter a s2) = all a s1 && subseq s1 s2. | Proof.
elim: s2 s1 => [|x s2 IHs] [|y s1] //=; rewrite ?andbF ?sub0seq //.
by case a_x: (a x); rewrite /= !IHs /=; case: eqP => // ->; rewrite a_x.
Qed. | Lemma | subseq_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"filter",
"s1",
"s2",
"sub0seq",
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_uniqP s1 s2 :
uniq s2 -> reflect (s1 = filter [in s1] s2) (subseq s1 s2). | Proof.
move=> uniq_s2; apply: (iffP idP) => [ss12 | ->]; last exact: filter_subseq.
apply/eqP; rewrite -size_subseq_leqif ?subseq_filter ?(introT allP) //.
apply/eqP/esym/perm_size.
rewrite uniq_perm ?filter_uniq ?(subseq_uniq ss12) // => x.
by rewrite mem_filter; apply: andb_idr; apply: (mem_subseq ss12).
Qed. | Lemma | subseq_uniqP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"allP",
"apply",
"filter",
"filter_subseq",
"filter_uniq",
"last",
"mem_filter",
"mem_subseq",
"perm_size",
"s1",
"s2",
"size_subseq_leqif",
"subseq",
"subseq_filter",
"subseq_uniq",
"uniq",
"uniq_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_subseq_pivot x (s1 s2 s3 s4 : seq T) (s := s3 ++ x :: s4) :
uniq s -> subseq (s1 ++ x :: s2) s = (subseq s1 s3 && subseq s2 s4). | Proof.
move=> uniq_s; apply/idP/idP => [sub_s'_s|/andP[? ?]]; last first.
by rewrite cat_subseq //= eqxx.
have uniq_s' := subseq_uniq sub_s'_s uniq_s.
have/eqP {sub_s'_s uniq_s} := subseq_uniqP _ uniq_s sub_s'_s.
rewrite !filter_cat /= mem_cat inE eqxx orbT /=.
rewrite uniq_eqseq_pivotl // => /andP [/eqP -> /eqP ->].... | Lemma | uniq_subseq_pivot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"cat_subseq",
"eqxx",
"filter_cat",
"filter_subseq",
"inE",
"last",
"mem_cat",
"s1",
"s2",
"s3",
"s4",
"seq",
"subseq",
"subseq_uniq",
"subseq_uniqP",
"uniq",
"uniq_eqseq_pivotl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_to_subseq s1 s2 :
subseq s1 s2 -> {s3 | perm_eq s2 (s1 ++ s3)}. | Proof.
elim Ds2: s2 s1 => [|y s2' IHs] [|x s1] //=; try by exists s2; rewrite Ds2.
case: eqP => [-> | _] /IHs[s3 perm_s2] {IHs}.
by exists s3; rewrite perm_cons.
by exists (rcons s3 y); rewrite -cat_cons -perm_rcons -!cats1 catA perm_cat2r.
Qed. | Lemma | perm_to_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"catA",
"cat_cons",
"cats1",
"perm_cat2r",
"perm_cons",
"perm_eq",
"perm_rcons",
"rcons",
"s1",
"s2",
"s3",
"subseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subseq_rem x : {homo rem x : s1 s2 / @subseq T s1 s2}. | Proof.
move=> s1 s2; elim: s2 s1 => [|x2 s2 IHs2] [|x1 s1]; rewrite ?sub0seq //=.
have [->|_] := eqVneq x1 x2; first by case: eqP => //= _ /IHs2; rewrite eqxx.
move=> /IHs2/subseq_trans->//.
by have [->|_] := eqVneq x x2; [apply: rem_subseq|apply: subseq_cons].
Qed. | Lemma | subseq_rem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eqVneq",
"eqxx",
"rem",
"rem_subseq",
"s1",
"s2",
"sub0seq",
"subseq",
"subseq_cons",
"subseq_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_f s x : x \in s -> f x \in map f s. | Proof.
by elim: s => //= y s IHs /predU1P[->|/IHs]; [apply: predU1l | apply: predU1r].
Qed. | Lemma | map_f | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"map",
"predU1P",
"predU1l",
"predU1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mapP s y : reflect (exists2 x, x \in s & y = f x) (y \in map f s). | Proof.
elim: s => [|x s IHs]; [by right; case|rewrite /= inE].
exact: equivP (orPP eqP IHs) (iff_sym exists_cons).
Qed. | Lemma | mapP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"exists_cons",
"inE",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_mapP (s : seq T1) (s' : seq T2) :
{subset s' <= map f s} <-> exists2 t, all (mem s) t & s' = map f t. | Proof.
split => [|[r /allP/= rE ->] _ /mapP[x xr ->]]; last by rewrite map_f ?rE.
elim: s' => [|x s' IHs'] subss'; first by exists [::].
have /mapP[y ys ->] := subss' _ (mem_head _ _).
have [x' x's'|t st ->] := IHs'; first by rewrite subss'// inE x's' orbT.
by exists (y :: t); rewrite //= ys st.
Qed. | Lemma | subset_mapP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"allP",
"inE",
"last",
"map",
"mapP",
"map_f",
"mem_head",
"seq",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_uniq s : uniq (map f s) -> uniq s. | Proof.
elim: s => //= x s IHs /andP[not_sfx /IHs->]; rewrite andbT.
by apply: contra not_sfx => sx; apply/mapP; exists x.
Qed. | Lemma | map_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"map",
"mapP",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_inj_in_uniq s : {in s &, injective f} -> uniq (map f s) = uniq s. | Proof.
elim: s => //= x s IHs //= injf; congr (~~ _ && _).
apply/mapP/idP=> [[y sy /injf] | ]; last by exists x.
by rewrite mem_head mem_behead // => ->.
by apply: IHs => y z sy sz; apply: injf => //; apply: predU1r.
Qed. | Lemma | map_inj_in_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"injf",
"last",
"map",
"mapP",
"mem_behead",
"mem_head",
"predU1r",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_subseq s1 s2 : subseq s1 s2 -> subseq (map f s1) (map f s2). | Proof.
case/subseqP=> m sz_m ->; apply/subseqP.
by exists m; rewrite ?size_map ?map_mask.
Qed. | Lemma | map_subseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"map",
"map_mask",
"s1",
"s2",
"size_map",
"subseq",
"subseqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_map_in s x :
{in s &, injective f} -> x \in s -> index (f x) (map f s) = index x s. | Proof.
move=> f_inj x_in_s; rewrite /index find_map.
by apply: eq_in_find => y /= y_s; rewrite (inj_in_eq f_inj).
Qed. | Lemma | index_map_in | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eq_in_find",
"f_inj",
"find_map",
"index",
"inj_in_eq",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_map_inW s x : {in s, injective f} -> index (f x) (map f s) = index x s. | Proof.
move=> fI; have [/index_map_in-> // _ _ _ _ /fI-> //|xs] := boolP (x \in s).
rewrite !memNindex ?size_map//; apply/mapP => -[y ys].
by move=> /esym/fI -/(_ ys) yx; rewrite -yx ys in xs.
Qed. | Lemma | index_map_inW | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"index",
"index_map_in",
"map",
"mapP",
"memNindex",
"size_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_map_inj_in s : uniq (map f s) -> {in s &, injective f}. | Proof.
move=> f_uniq x y /(nthP x)[i ilt <-] /(nthP x)[j jlt <-].
by rewrite -!(nth_map _ (f x))// => /uniqP /[!(inE, size_map)] ->.
Qed. | Lemma | uniq_map_inj_in | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"inE",
"map",
"nthP",
"nth_map",
"size_map",
"uniq",
"uniqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_index_map s x0 x :
{in s &, injective f} -> x \in s -> nth x0 s (index (f x) (map f s)) = x. | Proof. by move=> f_inj x_in_s; rewrite index_map_in// nth_index. Qed. | Lemma | nth_index_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"f_inj",
"index",
"index_map_in",
"map",
"nth",
"nth_index"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_map s t : perm_eq s t -> perm_eq (map f s) (map f t). | Proof. by move/permP=> Est; apply/permP=> a; rewrite !count_map Est. Qed. | Lemma | perm_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"count_map",
"map",
"permP",
"perm_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_map s1 s2 : {subset s1 <= s2} -> {subset map f s1 <= map f s2}. | Proof. by move=> sub_s ? /mapP[x x_s ->]; rewrite map_f ?sub_s. Qed. | Lemma | sub_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"mapP",
"map_f",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mem_map s1 s2 : s1 =i s2 -> map f s1 =i map f s2. | Proof. by move=> Es x; apply/idP/idP; apply: sub_map => ?; rewrite Es. Qed. | Lemma | eq_mem_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"map",
"s1",
"s2",
"sub_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hf : injective f. | Hypothesis | Hf | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
mem_map s x : (f x \in map f s) = (x \in s). | Proof. by apply/mapP/idP=> [[y Hy /Hf->] //|]; exists x. Qed. | Lemma | mem_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"Hf",
"apply",
"map",
"mapP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_map s x : index (f x) (map f s) = index x s. | Proof. by apply: index_map_inW; apply: in1W. Qed. | Lemma | index_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"index",
"index_map_inW",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_inj_uniq s : uniq (map f s) = uniq s. | Proof. by apply: map_inj_in_uniq; apply: in2W. Qed. | Lemma | map_inj_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"map",
"map_inj_in_uniq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
undup_map_inj s : undup (map f s) = map f (undup s). | Proof. by elim: s => //= s0 s ->; rewrite mem_map //; case: (_ \in _). Qed. | Lemma | undup_map_inj | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map",
"mem_map",
"s0",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_map_inj s t : perm_eq (map f s) (map f t) -> perm_eq s t. | Proof.
move/permP=> Est; apply/allP=> x _ /=.
have Dx: pred1 x =1 preim f (pred1 (f x)) by move=> y /=; rewrite inj_eq.
by rewrite !(eq_count Dx) -!count_map Est.
Qed. | Lemma | perm_map_inj | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"Dx",
"allP",
"apply",
"count_map",
"eq_count",
"inj_eq",
"map",
"permP",
"perm_eq",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_of_seq (T1 : eqType) T2 (s : seq T1) (fs : seq T2) (y0 : T2) :
{f | uniq s -> size fs = size s -> map f s = fs}. | Proof.
exists (fun x => nth y0 fs (index x s)) => uAs eq_sz.
apply/esym/(@eq_from_nth _ y0); rewrite ?size_map eq_sz // => i ltis.
by have x0 : T1 by [case: (s) ltis]; rewrite (nth_map x0) // index_uniq.
Qed. | Lemma | map_of_seq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eq_from_nth",
"index",
"index_uniq",
"map",
"nth",
"nth_map",
"seq",
"size",
"size_map",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_id (s : seq T) : map id s = s. | Proof. by elim: s => //= x s ->. Qed. | Lemma | map_id | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"id",
"map",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_map (f g : S -> T) : f =1 g -> map f =1 map g. | Proof. by move=> Ef; elim=> //= x s ->; rewrite Ef. Qed. | Lemma | eq_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_comp (f : T -> U) (g : S -> T) s : map (f \o g) s = map f (map g s). | Proof. by elim: s => //= x s ->. Qed. | Lemma | map_comp | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mapK (f : S -> T) (g : T -> S) : cancel f g -> cancel (map f) (map g). | Proof. by move=> fK; elim=> //= x s ->; rewrite fK. Qed. | Lemma | mapK | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"fK",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mapK_in (A : {pred S}) (f : S -> T) (g : T -> S) :
{in A, cancel f g} -> {in [pred s | all [in A] s], cancel (map f) (map g)}. | Proof. by move=> fK; elim=> //= x s IHs /andP[/fK-> /IHs->]. Qed. | Lemma | mapK_in | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"fK",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_map (S : eqType) T (f g : S -> T) (s : seq S) :
{in s, f =1 g} <-> map f s = map g s. | Proof.
elim: s => //= x s IHs; split=> [/forall_cons[-> ?]|]; first by rewrite IHs.1.
by move=> -[? ?]; apply/forall_cons; split=> [//|]; apply: IHs.2.
Qed. | Lemma | eq_in_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"forall_cons",
"map",
"seq",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_id_in (T : eqType) f (s : seq T) : {in s, f =1 id} -> map f s = s. | Proof. by move/eq_in_map->; apply: map_id. Qed. | Lemma | map_id_in | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eq_in_map",
"id",
"map",
"map_id",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmap s | :=
if s is x :: s' then let r := pmap s' in oapp (cons^~ r) r (f x) else [::]. | Fixpoint | pmap | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_pK : pcancel g f -> cancel (map g) pmap. | Proof. by move=> gK; elim=> //= x s ->; rewrite gK. Qed. | Lemma | map_pK | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"gK",
"map",
"pmap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_pmap s : size (pmap s) = count [eta f] s. | Proof. by elim: s => //= x s <-; case: (f _). Qed. | Lemma | size_pmap | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"pmap",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmapS_filter s : map some (pmap s) = map f (filter [eta f] s). | Proof. by elim: s => //= x s; case fx: (f x) => //= [u] <-; congr (_ :: _). Qed. | Lemma | pmapS_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter",
"map",
"pmap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fK : ocancel f g. | Hypothesis | fK | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
pmap_filter s : map g (pmap s) = filter [eta f] s. | Proof. by elim: s => //= x s <-; rewrite -{3}(fK x); case: (f _). Qed. | Lemma | pmap_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"fK",
"filter",
"map",
"pmap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmap_cat s t : pmap (s ++ t) = pmap s ++ pmap t. | Proof. by elim: s => //= x s ->; case/f: x. Qed. | Lemma | pmap_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"pmap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_pmap (p : pred rT) s :
all p (pmap s) = all [pred i | oapp p true (f i)] s. | Proof. by elim: s => //= x s <-; case: f. Qed. | Lemma | all_pmap | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"pmap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_pmap (aT : eqType) rT (f1 f2 : aT -> option rT) s :
{in s, f1 =1 f2} -> pmap f1 s = pmap f2 s. | Proof. by elim: s => //= a s IHs /forall_cons [-> /IHs ->]. Qed. | Lemma | eq_in_pmap | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"aT",
"f1",
"f2",
"forall_cons",
"pmap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_pmap aT rT (f1 f2 : aT -> option rT) :
f1 =1 f2 -> pmap f1 =1 pmap f2. | Proof. by move=> Ef; elim => //= a s ->; rewrite Ef. Qed. | Lemma | eq_pmap | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"aT",
"f1",
"f2",
"pmap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_pmap s u : (u \in pmap f s) = (Some u \in map f s). | Proof. by elim: s => //= x s IHs; rewrite in_cons -IHs; case: (f x). Qed. | Lemma | mem_pmap | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"in_cons",
"map",
"pmap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
can2_mem_pmap : pcancel g f -> forall s u, (u \in pmap f s) = (g u \in s). | Proof.
by move=> gK s u; rewrite -(mem_map (pcan_inj gK)) pmap_filter // mem_filter gK.
Qed. | Lemma | can2_mem_pmap | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"gK",
"mem_filter",
"mem_map",
"pmap",
"pmap_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmap_uniq s : uniq s -> uniq (pmap f s). | Proof. move/(filter_uniq f); rewrite -(pmap_filter fK); exact: map_uniq. Qed. | Lemma | pmap_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"fK",
"filter_uniq",
"map_uniq",
"pmap",
"pmap_filter",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_pmap s t : perm_eq s t -> perm_eq (pmap f s) (pmap f t). | Proof.
move=> eq_st; apply/(perm_map_inj Some_inj); rewrite !pmapS_filter.
exact/perm_map/perm_filter.
Qed. | Lemma | perm_pmap | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"perm_eq",
"perm_filter",
"perm_map",
"perm_map_inj",
"pmap",
"pmapS_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_pmap_sub s : size (pmap (insub : T -> option sT) s) = count p s. | Proof. by rewrite size_pmap (eq_count (isSome_insub _)). Qed. | Lemma | size_pmap_sub | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"eq_count",
"insub",
"isSome_insub",
"pmap",
"sT",
"size",
"size_pmap"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insT : T -> option sT | := insub. | Let | insT | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"insub",
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_pmap_sub s u : (u \in pmap insT s) = (val u \in s). | Proof. exact/(can2_mem_pmap (insubK _))/valK. Qed. | Lemma | mem_pmap_sub | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"can2_mem_pmap",
"insT",
"insubK",
"pmap",
"val",
"valK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pmap_sub_uniq s : uniq s -> uniq (pmap insT s). | Proof. exact: (pmap_uniq (insubK _)). Qed. | Lemma | pmap_sub_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"insT",
"insubK",
"pmap",
"pmap_uniq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iota m n | := if n is n'.+1 then m :: iota m.+1 n' else [::]. | Fixpoint | iota | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"n'"
] | Index sequence | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
size_iota m n : size (iota m n) = n. | Proof. by elim: n m => //= n IHn m; rewrite IHn. Qed. | Lemma | size_iota | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"iota",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iotaD m n1 n2 : iota m (n1 + n2) = iota m n1 ++ iota (m + n1) n2. | Proof. by elim: n1 m => [|n1 IHn1] m; rewrite ?addn0 // -addSnnS /= -IHn1. Qed. | Lemma | iotaD | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addSnnS",
"addn0",
"iota"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iotaDl m1 m2 n : iota (m1 + m2) n = map (addn m1) (iota m2 n). | Proof. by elim: n m2 => //= n IHn m2; rewrite -addnS IHn. Qed. | Lemma | iotaDl | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn",
"addnS",
"iota",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_iota p m n i : i < n -> nth p (iota m n) i = m + i. | Proof.
by move/subnKC <-; rewrite addSnnS iotaD nth_cat size_iota ltnn subnn.
Qed. | Lemma | nth_iota | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addSnnS",
"iota",
"iotaD",
"ltnn",
"nth",
"nth_cat",
"size_iota",
"subnKC",
"subnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_iota m n i : (i \in iota m n) = (m <= i < m + n). | Proof.
elim: n m => [|n IHn] /= m; first by rewrite addn0 ltnNge andbN.
by rewrite in_cons IHn addnS ltnS; case: ltngtP => // ->; rewrite leq_addr.
Qed. | Lemma | mem_iota | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"addnS",
"in_cons",
"iota",
"leq_addr",
"ltnNge",
"ltnS",
"ltngtP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iota_uniq m n : uniq (iota m n). | Proof. by elim: n m => //= n IHn m; rewrite mem_iota ltnn /=. Qed. | Lemma | iota_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"iota",
"ltnn",
"mem_iota",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_iota k m n : take k (iota m n) = iota m (minn k n). | Proof.
have [lt_k_n|le_n_k] := ltnP.
by elim: k n lt_k_n m => [|k IHk] [|n] //= H m; rewrite IHk.
by apply: take_oversize; rewrite size_iota.
Qed. | Lemma | take_iota | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"iota",
"ltnP",
"minn",
"size_iota",
"take",
"take_oversize"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_iota k m n : drop k (iota m n) = iota (m + k) (n - k). | Proof.
by elim: k m n => [|k IHk] m [|n] //=; rewrite ?addn0 // IHk addnS subSS.
Qed. | Lemma | drop_iota | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"addnS",
"drop",
"iota",
"subSS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_iota_ltn m n j : j <= n ->
[seq i <- iota m n | i < m + j] = iota m j. | Proof.
elim: n m j => [m j|n IHn m [|j] jlen]; first by rewrite leqn0 => /eqP ->.
rewrite (@eq_in_filter _ _ pred0) ?filter_pred0// => i.
by rewrite addn0 ltnNge mem_iota => /andP[->].
by rewrite /= addnS leq_addr -addSn IHn.
Qed. | Lemma | filter_iota_ltn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addSn",
"addn0",
"addnS",
"eq_in_filter",
"filter_pred0",
"iota",
"leq_addr",
"leqn0",
"ltnNge",
"mem_iota",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_iota_leq n m j : j < n ->
[seq i <- iota m n | i <= m + j] = iota m j.+1. | Proof.
elim: n m j => [//|n IHn] m [|j] jlen /=; rewrite leq_addr.
rewrite (@eq_in_filter _ _ pred0) ?filter_pred0// => i.
by rewrite addn0 leqNgt mem_iota => /andP[->].
by rewrite addnS -addSn IHn -1?ltnS.
Qed. | Lemma | filter_iota_leq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addSn",
"addn0",
"addnS",
"eq_in_filter",
"filter_pred0",
"iota",
"leqNgt",
"leq_addr",
"ltnS",
"mem_iota",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mindex m | := nth (size m) (mask m (iota 0 (size m))). | Definition | mindex | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"iota",
"mask",
"nth",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_mask {T} m x0 (s : seq T) : size m >= size s ->
forall i, nth x0 (mask m s) i = nth x0 s (mindex m i). | Proof.
rewrite /mindex => sm i.
by elim: m s sm i => [|[] m IHm]//= [|x s]//= sm [|i];
rewrite ?nth_nil ?(iotaDl 1)//= -map_mask -map_nth/= IHm.
Qed. | Lemma | nth_mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"iotaDl",
"map_mask",
"map_nth",
"mask",
"mindex",
"nth",
"nth_nil",
"seq",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mkseq f n : seq T | := map f (iota 0 n). | Definition | mkseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"iota",
"map",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_mkseq f n : size (mkseq f n) = n. | Proof. by rewrite size_map size_iota. Qed. | Lemma | size_mkseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mkseq",
"size",
"size_iota",
"size_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mkseqS f n :
mkseq f n.+1 = rcons (mkseq f n) (f n). | Proof. by rewrite /mkseq -addn1 iotaD add0n map_cat cats1. Qed. | Lemma | mkseqS | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"add0n",
"addn1",
"cats1",
"iotaD",
"map_cat",
"mkseq",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_mkseq f g : f =1 g -> mkseq f =1 mkseq g. | Proof. by move=> Efg n; apply: eq_map Efg _. Qed. | Lemma | eq_mkseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eq_map",
"mkseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_mkseq f n i : i < n -> nth x0 (mkseq f n) i = f i. | Proof. by move=> Hi; rewrite (nth_map 0) ?nth_iota ?size_iota. Qed. | Lemma | nth_mkseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mkseq",
"nth",
"nth_iota",
"nth_map",
"size_iota"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mkseq_nth s : mkseq (nth x0 s) (size s) = s. | Proof.
by apply: (@eq_from_nth _ x0); rewrite size_mkseq // => i Hi; rewrite nth_mkseq.
Qed. | Lemma | mkseq_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eq_from_nth",
"mkseq",
"nth",
"nth_mkseq",
"size",
"size_mkseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mkseq_spec s : seq T -> Type | :=
| MapIota n f : s = mkseq f n -> mkseq_spec s (mkseq f n). | Variant | mkseq_spec | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mkseq",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mkseqP s : mkseq_spec s s. | Proof. by rewrite -[s]mkseq_nth; constructor. Qed. | Lemma | mkseqP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mkseq_nth",
"mkseq_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_nth_iota0 s i :
i <= size s -> [seq nth x0 s j | j <- iota 0 i] = take i s. | Proof.
by move=> ile; rewrite -[s in RHS]mkseq_nth -map_take take_iota (minn_idPl _).
Qed. | Lemma | map_nth_iota0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"iota",
"map_take",
"minn_idPl",
"mkseq_nth",
"nth",
"seq",
"size",
"take",
"take_iota"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_nth_iota s i j : j <= size s - i ->
[seq nth x0 s k | k <- iota i j] = take j (drop i s). | Proof.
elim: i => [|i IH] in s j *; first by rewrite subn0 drop0 => /map_nth_iota0->.
case: s => [|x s /IH<-]; first by rewrite leqn0 => /eqP->.
by rewrite -add1n iotaDl -map_comp.
Qed. | Lemma | map_nth_iota | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"add1n",
"drop",
"drop0",
"iota",
"iotaDl",
"leqn0",
"map_comp",
"map_nth_iota0",
"nth",
"seq",
"size",
"subn0",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_mkseq f n i : take i (mkseq f n) = mkseq f (minn i n). | Proof. by rewrite /mkseq -map_take take_iota. Qed. | Lemma | take_mkseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"map_take",
"minn",
"mkseq",
"take",
"take_iota"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_mkseq f n i :
drop i (mkseq f n) = mkseq (fun k => f (i + k)) (n - i). | Proof. by rewrite /mkseq -map_drop drop_iota addnC iotaDl -map_comp. Qed. | Lemma | drop_mkseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnC",
"drop",
"drop_iota",
"iotaDl",
"map_comp",
"map_drop",
"mkseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mkseq_uniqP (f : nat -> T) n :
reflect {in gtn n &, injective f} (uniq (mkseq f n)). | Proof.
apply: (equivP (uniqP (f 0))); rewrite size_mkseq.
by split=> injf i j lti ltj; have:= injf i j lti ltj; rewrite !nth_mkseq.
Qed. | Lemma | mkseq_uniqP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"gtn",
"injf",
"mkseq",
"nat",
"nth_mkseq",
"size_mkseq",
"split",
"uniq",
"uniqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mkseq_uniq (f : nat -> T) n : injective f -> uniq (mkseq f n). | Proof. by move/map_inj_uniq->; apply: iota_uniq. Qed. | Lemma | mkseq_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"iota_uniq",
"map_inj_uniq",
"mkseq",
"nat",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_iotaP {s t : seq T} x0 (It := iota 0 (size t)) :
reflect (exists2 Is, perm_eq Is It & s = map (nth x0 t) Is) (perm_eq s t). | Proof.
apply: (iffP idP) => [Est | [Is eqIst ->]]; last first.
by rewrite -{2}[t](mkseq_nth x0) perm_map.
elim: t => [|x t IHt] in s It Est *.
by rewrite (perm_small_eq _ Est) //; exists [::].
have /rot_to[k s1 Ds]: x \in s by rewrite (perm_mem Est) mem_head.
have [|Is1 eqIst1 Ds1] := IHt s1; first by rewrite -(per... | Lemma | perm_iotaP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eq_map",
"iota",
"iotaDl",
"last",
"map",
"map_comp",
"map_rotr",
"mem_head",
"mkseq_nth",
"nth",
"perm_cons",
"perm_eq",
"perm_map",
"perm_mem",
"perm_rot",
"perm_small_eq",
"rotK",
"rot_to",
"rotr",
"s1",
"seq",
"size",
"succn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldr s | := if s is x :: s' then f x (foldr s') else z0. | Fixpoint | foldr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"z0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldr_cat s1 s2 : foldr f z0 (s1 ++ s2) = foldr f (foldr f z0 s2) s1. | Proof. by elim: s1 => //= x s1 ->. Qed. | Lemma | foldr_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"foldr",
"s1",
"s2",
"z0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldr_rcons s x : foldr f z0 (rcons s x) = foldr f (f x z0) s. | Proof. by rewrite -cats1 foldr_cat. Qed. | Lemma | foldr_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"foldr",
"foldr_cat",
"rcons",
"z0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldr_map s : foldr f z0 (map h s) = foldr (fun x z => f (h x) z) z0 s. | Proof. by elim: s => //= x s ->. Qed. | Lemma | foldr_map | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"foldr",
"map",
"z0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumn | := foldr addn 0. | Definition | sumn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn",
"foldr"
] | Quick characterization of the null sequence. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
sumn_ncons x n s : sumn (ncons n x s) = x * n + sumn s. | Proof. by rewrite mulnC; elim: n => //= n ->; rewrite addnA. Qed. | Lemma | sumn_ncons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnA",
"mulnC",
"ncons",
"sumn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumn_nseq x n : sumn (nseq n x) = x * n. | Proof. by rewrite sumn_ncons addn0. Qed. | Lemma | sumn_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"nseq",
"sumn",
"sumn_ncons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumn_cat s1 s2 : sumn (s1 ++ s2) = sumn s1 + sumn s2. | Proof. by elim: s1 => //= x s1 ->; rewrite addnA. Qed. | Lemma | sumn_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnA",
"s1",
"s2",
"sumn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumn_count T (a : pred T) s : sumn [seq a i : nat | i <- s] = count a s. | Proof. by elim: s => //= s0 s /= ->. Qed. | Lemma | sumn_count | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"nat",
"s0",
"seq",
"sumn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumn_rcons s n : sumn (rcons s n) = sumn s + n. | Proof. by rewrite -cats1 sumn_cat /= addn0. Qed. | Lemma | sumn_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"cats1",
"rcons",
"sumn",
"sumn_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_sumn s1 s2 : perm_eq s1 s2 -> sumn s1 = sumn s2. | Proof.
by apply/catCA_perm_subst: s1 s2 => s1 s2 s3; rewrite !sumn_cat addnCA.
Qed. | Lemma | perm_sumn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnCA",
"apply",
"catCA_perm_subst",
"perm_eq",
"s1",
"s2",
"s3",
"sumn",
"sumn_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumn_rot s n : sumn (rot n s) = sumn s. | Proof. by apply/perm_sumn; rewrite perm_rot. Qed. | Lemma | sumn_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"perm_rot",
"perm_sumn",
"rot",
"sumn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumn_rev s : sumn (rev s) = sumn s. | Proof. by apply/perm_sumn; rewrite perm_rev. Qed. | Lemma | sumn_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"perm_rev",
"perm_sumn",
"rev",
"sumn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natnseq0P s : reflect (s = nseq (size s) 0) (sumn s == 0). | Proof.
apply: (iffP idP) => [|->]; last by rewrite sumn_nseq.
by elim: s => //= x s IHs; rewrite addn_eq0 => /andP[/eqP-> /IHs <-].
Qed. | Lemma | natnseq0P | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn_eq0",
"apply",
"last",
"nseq",
"size",
"sumn",
"sumn_nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumn_set_nth s x0 n x :
sumn (set_nth x0 s n x) =
sumn s + x - (nth x0 s n) * (n < size s) + x0 * (n - size s). | Proof.
rewrite set_nthE; case: ltnP => [nlts|nges]; last first.
by rewrite sumn_cat sumn_ncons /= addn0 muln0 subn0 addnAC addnA.
have -> : n - size s = 0 by apply/eqP; rewrite subn_eq0 ltnW.
rewrite -[in sumn s](cat_take_drop n s) [drop n s](drop_nth x0)//.
by rewrite !sumn_cat /= muln1 muln0 addn0 addnAC !addnA [in... | Lemma | sumn_set_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"addnA",
"addnAC",
"addnK",
"apply",
"cat_take_drop",
"drop",
"drop_nth",
"last",
"ltnP",
"ltnW",
"muln0",
"muln1",
"nth",
"set_nth",
"set_nthE",
"size",
"subn0",
"subn_eq0",
"sumn",
"sumn_cat",
"sumn_ncons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumn_set_nth_ltn s x0 n x : n < size s ->
sumn (set_nth x0 s n x) = sumn s + x - nth x0 s n. | Proof.
move=> nlts; rewrite sumn_set_nth nlts muln1.
have -> : n - size s = 0 by apply/eqP; rewrite subn_eq0 ltnW.
by rewrite muln0 addn0.
Qed. | Lemma | sumn_set_nth_ltn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"apply",
"ltnW",
"muln0",
"muln1",
"nth",
"set_nth",
"size",
"subn_eq0",
"sumn",
"sumn_set_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sumn_set_nth0 s n x : sumn (set_nth 0 s n x) = sumn s + x - nth 0 s n. | Proof.
rewrite sumn_set_nth mul0n addn0.
by case: ltnP => [_|nges]; rewrite ?muln1// nth_default.
Qed. | Lemma | sumn_set_nth0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"ltnP",
"mul0n",
"muln1",
"nth",
"nth_default",
"set_nth",
"sumn",
"sumn_set_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldl z s | := if s is x :: s' then foldl (f z x) s' else z. | Fixpoint | foldl | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldl_rev z s : foldl z (rev s) = foldr (fun x z => f z x) z s. | Proof.
by elim/last_ind: s z => // s x IHs z; rewrite rev_rcons -cats1 foldr_cat -IHs.
Qed. | Lemma | foldl_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"foldl",
"foldr",
"foldr_cat",
"last_ind",
"rev",
"rev_rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
foldl_cat z s1 s2 : foldl z (s1 ++ s2) = foldl (foldl z s1) s2. | Proof.
by rewrite -(revK (s1 ++ s2)) foldl_rev rev_cat foldr_cat -!foldl_rev !revK.
Qed. | Lemma | foldl_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"foldl",
"foldl_rev",
"foldr_cat",
"revK",
"rev_cat",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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