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 |
|---|---|---|---|---|---|---|---|---|---|---|
aP : forall x, reflect (A x) (a x). | Hypothesis | aP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
hasPP : reflect (exists2 x, x \in s & A x) (has a s). | Proof. by apply: (iffP hasP) => -[x ? /aP]; exists x. Qed. | Lemma | hasPP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"aP",
"apply",
"has",
"hasP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
allPP : reflect {in s, forall x, A x} (all a s). | Proof. by apply: (iffP allP) => a_s x /a_s/aP. Qed. | Lemma | allPP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"aP",
"all",
"allP",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_filter s : {in s, a1 =1 a2} -> filter a1 s = filter a2 s. | Proof. by elim: s => //= x s IHs /forall_cons [-> /IHs ->]. Qed. | Lemma | eq_in_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"filter",
"forall_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_find s : {in s, a1 =1 a2} -> find a1 s = find a2 s. | Proof. by elim: s => //= x s IHs /forall_cons [-> /IHs ->]. Qed. | Lemma | eq_in_find | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"find",
"forall_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_count s : {in s, a1 =1 a2} -> count a1 s = count a2 s. | Proof. by move/eq_in_filter=> eq_a12; rewrite -!size_filter eq_a12. Qed. | Lemma | eq_in_count | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"count",
"eq_in_filter",
"size_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_all s : {in s, a1 =1 a2} -> all a1 s = all a2 s. | Proof. by move=> eq_a12; rewrite !all_count eq_in_count. Qed. | Lemma | eq_in_all | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"all",
"all_count",
"eq_in_count"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_has s : {in s, a1 =1 a2} -> has a1 s = has a2 s. | Proof. by move/eq_in_filter=> eq_a12; rewrite !has_filter eq_a12. Qed. | Lemma | eq_in_has | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"a1",
"a2",
"eq_in_filter",
"has",
"has_filter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_has_r s1 s2 : s1 =i s2 -> has^~ s1 =1 has^~ s2. | Proof.
by move=> Es a; apply/hasP/hasP=> -[x sx ax]; exists x; rewrite ?Es in sx *.
Qed. | Lemma | eq_has_r | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"has",
"hasP",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_all_r s1 s2 : s1 =i s2 -> all^~ s1 =1 all^~ s2. | Proof. by move=> Es a; apply/negb_inj; rewrite -!has_predC (eq_has_r Es). Qed. | Lemma | eq_all_r | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"apply",
"eq_has_r",
"has_predC",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_sym s1 s2 : has [in s1] s2 = has [in s2] s1. | Proof. by apply/hasP/hasP=> -[x]; exists x. Qed. | Lemma | has_sym | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"has",
"hasP",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_pred1 x s : has (pred1 x) s = (x \in s). | Proof. by rewrite -(eq_has (mem_seq1^~ x)) (has_sym [:: x]) /= orbF. Qed. | Lemma | has_pred1 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eq_has",
"has",
"has_sym",
"mem_seq1",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_rev s : rev s =i s. | Proof. by move=> a; rewrite -!has_pred1 has_rev. Qed. | Lemma | mem_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has_pred1",
"has_rev",
"rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constant s | := if s is x :: s' then all (pred1 x) s' else true. | Definition | constant | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"pred1"
] | Constant sequences, i.e., the image of nseq. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
all_pred1P x s : reflect (s = nseq (size s) x) (all (pred1 x) s). | Proof.
elim: s => [|y s IHs] /=; first by left.
case: eqP => [->{y} | ne_xy]; last by right=> [] [? _]; case ne_xy.
by apply: (iffP IHs) => [<- //| []].
Qed. | Lemma | all_pred1P | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"apply",
"last",
"nseq",
"pred1",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_pred1_constant x s : all (pred1 x) s -> constant s. | Proof. by case: s => //= y s /andP[/eqP->]. Qed. | Lemma | all_pred1_constant | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"constant",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_pred1_nseq x n : all (pred1 x) (nseq n x). | Proof. by rewrite all_nseq /= eqxx orbT. Qed. | Lemma | all_pred1_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"all_nseq",
"eqxx",
"nseq",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_nseq n x y : (y \in nseq n x) = (0 < n) && (y == x). | Proof. by rewrite -has_pred1 has_nseq eq_sym. Qed. | Lemma | mem_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eq_sym",
"has_nseq",
"has_pred1",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nseqP n x y : reflect (y = x /\ n > 0) (y \in nseq n x). | Proof. by rewrite mem_nseq andbC; apply: (iffP andP) => -[/eqP]. Qed. | Lemma | nseqP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"mem_nseq",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constant_nseq n x : constant (nseq n x). | Proof. exact: all_pred1_constant (all_pred1_nseq x n). Qed. | Lemma | constant_nseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all_pred1_constant",
"all_pred1_nseq",
"constant",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
constantP s : reflect (exists x, s = nseq (size s) x) (constant s). | Proof.
apply: (iffP idP) => [| [x ->]]; last exact: constant_nseq.
case: s => [|x s] /=; first by exists x0.
by move/all_pred1P=> def_s; exists x; rewrite -def_s.
Qed. | Lemma | constantP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all_pred1P",
"apply",
"constant",
"constant_nseq",
"last",
"nseq",
"size"
] | Uses x0 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
uniq s | := if s is x :: s' then (x \notin s') && uniq s' else true. | Fixpoint | uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | Duplicate-freenes. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cons_uniq x s : uniq (x :: s) = (x \notin s) && uniq s. | Proof. by []. Qed. | Lemma | cons_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat_uniq s1 s2 :
uniq (s1 ++ s2) = [&& uniq s1, ~~ has [in s1] s2 & uniq s2]. | Proof.
elim: s1 => [|x s1 IHs]; first by rewrite /= has_pred0.
by rewrite has_sym /= mem_cat !negb_or has_sym IHs -!andbA; do !bool_congr.
Qed. | Lemma | cat_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has",
"has_pred0",
"has_sym",
"mem_cat",
"s1",
"s2",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_catC s1 s2 : uniq (s1 ++ s2) = uniq (s2 ++ s1). | Proof. by rewrite !cat_uniq has_sym andbCA andbA andbC. Qed. | Lemma | uniq_catC | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_uniq",
"has_sym",
"s1",
"s2",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_catCA s1 s2 s3 : uniq (s1 ++ s2 ++ s3) = uniq (s2 ++ s1 ++ s3). | Proof.
by rewrite !catA -!(uniq_catC s3) !(cat_uniq s3) uniq_catC !has_cat orbC.
Qed. | Lemma | uniq_catCA | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"catA",
"cat_uniq",
"has_cat",
"s1",
"s2",
"s3",
"uniq",
"uniq_catC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_cat_inLR x s1 s2 : uniq (s1 ++ s2) ->
x \in s1 ++ s2 -> (x \in s1) = (x \notin s2). | Proof.
rewrite mem_cat=> s_uniq /orP[] x_in; move: s_uniq.
by rewrite uniq_catC cat_uniq => /and3P[_ /hasPn /(_ _ x_in)->].
by rewrite cat_uniq => /and3P[_ /hasPn /(_ _ x_in) /= /negPf->]; rewrite x_in.
Qed. | Lemma | uniq_cat_inLR | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_uniq",
"hasPn",
"mem_cat",
"s1",
"s2",
"uniq",
"uniq_catC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_cat_inRL x s1 s2 : uniq (s1 ++ s2) ->
x \in s1 ++ s2 -> (x \in s2) = (x \notin s1). | Proof. by rewrite mem_catC uniq_catC; apply: uniq_cat_inLR. Qed. | Lemma | uniq_cat_inRL | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"mem_catC",
"s1",
"s2",
"uniq",
"uniq_catC",
"uniq_cat_inLR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcons_uniq s x : uniq (rcons s x) = (x \notin s) && uniq s. | Proof. by rewrite -cats1 uniq_catC. Qed. | Lemma | rcons_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"rcons",
"uniq",
"uniq_catC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_uniq s a : uniq s -> uniq (filter a s). | Proof.
elim: s => //= x s IHs /andP[s'x]; case: ifP => //= a_x /IHs->.
by rewrite mem_filter a_x s'x.
Qed. | Lemma | filter_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"filter",
"mem_filter",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_uniq s : uniq (rot n0 s) = uniq s. | Proof. by rewrite /rot uniq_catC cat_take_drop. Qed. | Lemma | rot_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"rot",
"uniq",
"uniq_catC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_uniq s : uniq (rev s) = uniq s. | Proof.
elim: s => // x s IHs.
by rewrite rev_cons -cats1 cat_uniq /= andbT andbC mem_rev orbF IHs.
Qed. | Lemma | rev_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_uniq",
"cats1",
"mem_rev",
"rev",
"rev_cons",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_memPn x s : reflect (count_mem x s = 0) (x \notin s). | Proof. by rewrite -has_pred1 has_count -eqn0Ngt; apply: eqP. Qed. | Lemma | count_memPn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"count_mem",
"eqn0Ngt",
"has_count",
"has_pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_uniq_mem s x : uniq s -> count_mem x s = (x \in s). | Proof.
elim: s => //= y s IHs /andP[/negbTE s'y /IHs-> {IHs}].
by rewrite in_cons; case: (eqVneq y x) => // <-; rewrite s'y.
Qed. | Lemma | count_uniq_mem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count_mem",
"eqVneq",
"in_cons",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_uniq_countP x s1 s2 : uniq s1 ->
reflect (x \in s1 -> x \in s2) (count_mem x s1 <= count_mem x s2). | Proof.
move/count_uniq_mem->; case: (boolP (_ \in _)) => //= _; last by constructor.
by rewrite -has_pred1 has_count; apply: (iffP idP) => //; apply.
Qed. | Lemma | leq_uniq_countP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"count_mem",
"count_uniq_mem",
"has_count",
"has_pred1",
"last",
"s1",
"s2",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_uniq_count s1 s2 : uniq s1 -> {subset s1 <= s2} ->
(forall x, count_mem x s1 <= count_mem x s2). | Proof. by move=> s1_uniq s1_s2 x; apply/leq_uniq_countP/s1_s2. Qed. | Lemma | leq_uniq_count | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"count_mem",
"leq_uniq_countP",
"s1",
"s2",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_pred1_uniq s x : uniq s -> x \in s -> filter (pred1 x) s = [:: x]. | Proof.
move=> uniq_s s_x; rewrite (all_pred1P _ _ (filter_all _ _)).
by rewrite size_filter count_uniq_mem ?s_x.
Qed. | Lemma | filter_pred1_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all_pred1P",
"count_uniq_mem",
"filter",
"filter_all",
"pred1",
"size_filter",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
undup s | :=
if s is x :: s' then if x \in s' then undup s' else x :: undup s' else [::]. | Fixpoint | undup | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | Removing duplicates | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
size_undup s : size (undup s) <= size s. | Proof. by elim: s => //= x s IHs; case: (x \in s) => //=; apply: ltnW. Qed. | Lemma | size_undup | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"ltnW",
"size",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_undup s : undup s =i s. | Proof.
move=> x; elim: s => //= y s IHs.
by case s_y: (y \in s); rewrite !inE IHs //; case: eqP => [->|].
Qed. | Lemma | mem_undup | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"inE",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
undup_uniq s : uniq (undup s). | Proof.
by elim: s => //= x s IHs; case s_x: (x \in s); rewrite //= mem_undup s_x.
Qed. | Lemma | undup_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mem_undup",
"undup",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
undup_id s : uniq s -> undup s = s. | Proof. by elim: s => //= x s IHs /andP[/negbTE-> /IHs->]. Qed. | Lemma | undup_id | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"undup",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltn_size_undup s : (size (undup s) < size s) = ~~ uniq s. | Proof.
by elim: s => //= x s IHs; case s_x: (x \in s); rewrite //= ltnS size_undup.
Qed. | Lemma | ltn_size_undup | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"ltnS",
"size",
"size_undup",
"undup",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
filter_undup p s : filter p (undup s) = undup (filter p s). | Proof.
elim: s => //= x s IHs; rewrite (fun_if undup) [_ = _]fun_if /= mem_filter /=.
by rewrite (fun_if (filter p)) /= IHs; case: ifP => -> //=; apply: if_same.
Qed. | Lemma | filter_undup | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"filter",
"mem_filter",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
undup_nil s : undup s = [::] -> s = [::]. | Proof. by case: s => //= x s; rewrite -mem_undup; case: ifP; case: undup. Qed. | Lemma | undup_nil | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mem_undup",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
undup_cat s t :
undup (s ++ t) = [seq x <- undup s | x \notin t] ++ undup t. | Proof. by elim: s => //= x s ->; rewrite mem_cat; do 2 case: in_mem => //=. Qed. | Lemma | undup_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mem_cat",
"seq",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
undup_rcons s x : undup (rcons s x) = rcons [seq y <- undup s | y != x] x. | Proof.
by rewrite -!cats1 undup_cat; congr cat; apply: eq_filter => y; rewrite inE.
Qed. | Lemma | undup_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"cat",
"cats1",
"eq_filter",
"inE",
"rcons",
"seq",
"undup",
"undup_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_undup s p : count p (undup s) <= count p s. | Proof. by rewrite -!size_filter filter_undup size_undup. Qed. | Lemma | count_undup | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count",
"filter_undup",
"size_filter",
"size_undup",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_undup p s : has p (undup s) = has p s. | Proof. by apply: eq_has_r => x; rewrite mem_undup. Qed. | Lemma | has_undup | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eq_has_r",
"has",
"mem_undup",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_undup p s : all p (undup s) = all p s. | Proof. by apply: eq_all_r => x; rewrite mem_undup. Qed. | Lemma | all_undup | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"apply",
"eq_all_r",
"mem_undup",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index x | := find (pred1 x). | Definition | index | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find",
"pred1"
] | Lookup | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
index_size x s : index x s <= size s. | Proof. by rewrite /index find_size. Qed. | Lemma | index_size | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find_size",
"index",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_mem x s : (index x s < size s) = (x \in s). | Proof. by rewrite -has_pred1 has_find. Qed. | Lemma | index_mem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has_find",
"has_pred1",
"index",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
memNindex x s : x \notin s -> index x s = size s. | Proof. by rewrite -has_pred1 => /hasNfind. Qed. | Lemma | memNindex | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"hasNfind",
"has_pred1",
"index",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_index x s : x \in s -> nth s (index x s) = x. | Proof. by rewrite -has_pred1 => /(nth_find x0)/eqP. Qed. | Lemma | nth_index | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has_pred1",
"index",
"nth",
"nth_find"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_inj s : {in s &, injective (index ^~ s)}. | Proof.
by move=> x y x_s y_s eidx; rewrite -(nth_index x_s) eidx nth_index.
Qed. | Lemma | index_inj | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"index",
"nth_index"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_cat x s1 s2 :
index x (s1 ++ s2) = if x \in s1 then index x s1 else size s1 + index x s2. | Proof. by rewrite /index find_cat has_pred1. Qed. | Lemma | index_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"find_cat",
"has_pred1",
"index",
"s1",
"s2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_ltn x s i : x \in take i s -> index x s < i. | Proof. by rewrite -has_pred1; apply: find_ltn. Qed. | Lemma | index_ltn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"find_ltn",
"has_pred1",
"index",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_take x s i : x \in s -> (x \in take i s) = (index x s < i). | Proof. by rewrite -?has_pred1; apply: has_take. Qed. | Lemma | in_take | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"has_pred1",
"has_take",
"index",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_take_leq x s i : i <= size s -> (x \in take i s) = (index x s < i). | Proof. by rewrite -?has_pred1; apply: has_take_leq. Qed. | Lemma | in_take_leq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"has_pred1",
"has_take_leq",
"index",
"size",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_nth i s : i < size s -> index (nth s i) s <= i. | Proof.
move=> lti; rewrite -ltnS index_ltn// -(@nth_take i.+1)// mem_nth // size_take.
by case: ifP.
Qed. | Lemma | index_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"index",
"index_ltn",
"ltnS",
"mem_nth",
"nth",
"nth_take",
"size",
"size_take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nthK s: uniq s -> {in gtn (size s), cancel (nth s) (index^~ s)}. | Proof.
elim: s => //= x s IHs /andP[s'x Us] i; rewrite inE ltnS eq_sym -if_neg.
by case: i => /= [_|i lt_i_s]; rewrite ?eqxx ?IHs ?(memPn s'x) ?mem_nth.
Qed. | Lemma | nthK | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eq_sym",
"eqxx",
"gtn",
"inE",
"index",
"ltnS",
"memPn",
"mem_nth",
"nth",
"size",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_uniq i s : i < size s -> uniq s -> index (nth s i) s = i. | Proof. by move/nthK. Qed. | Lemma | index_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"index",
"nth",
"nthK",
"size",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_head x s : index x (x :: s) = 0. | Proof. by rewrite /= eqxx. Qed. | Lemma | index_head | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqxx",
"index"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_last x s : uniq (x :: s) -> index (last x s) (x :: s) = size s. | Proof.
rewrite lastI rcons_uniq -cats1 index_cat size_belast.
by case: ifP => //=; rewrite eqxx addn0.
Qed. | Lemma | index_last | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"cats1",
"eqxx",
"index",
"index_cat",
"last",
"lastI",
"rcons_uniq",
"size",
"size_belast",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_uniq s i j :
i < size s -> j < size s -> uniq s -> (nth s i == nth s j) = (i == j). | Proof. by move=> lti ltj /nthK/can_in_eq->. Qed. | Lemma | nth_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"can_in_eq",
"nth",
"nthK",
"size",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniqPn s :
reflect (exists i j, [/\ i < j, j < size s & nth s i = nth s j]) (~~ uniq s). | Proof.
apply: (iffP idP) => [|[i [j [ltij ltjs]]]]; last first.
by apply: contra_eqN => Us; rewrite nth_uniq ?ltn_eqF // (ltn_trans ltij).
elim: s => // x s IHs /nandP[/negbNE | /IHs[i [j]]]; last by exists i.+1, j.+1.
by exists 0, (index x s).+1; rewrite !ltnS index_mem /= nth_index.
Qed. | Lemma | uniqPn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"contra_eqN",
"index",
"index_mem",
"last",
"ltnS",
"ltn_eqF",
"ltn_trans",
"nth",
"nth_index",
"nth_uniq",
"size",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniqP s : reflect {in gtn (size s) &, injective (nth s)} (uniq s). | Proof.
apply: (iffP idP) => [/nthK/can_in_inj// | nth_inj].
apply/uniqPn => -[i [j [ltij ltjs /nth_inj/eqP/idPn]]].
by rewrite !inE (ltn_trans ltij ltjs) ltn_eqF //=; case.
Qed. | Lemma | uniqP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"gtn",
"inE",
"ltn_eqF",
"ltn_trans",
"nth",
"nthK",
"size",
"uniq",
"uniqPn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_rot s : rot n0 s =i s. | Proof. by move=> x; rewrite -[s in RHS](cat_take_drop n0) !mem_cat /= orbC. Qed. | Lemma | mem_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"mem_cat",
"rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqseq_rot s1 s2 : (rot n0 s1 == rot n0 s2) = (s1 == s2). | Proof. exact/inj_eq/rot_inj. Qed. | Lemma | eqseq_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"inj_eq",
"rot",
"rot_inj",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_index s (n := index x0 s) : x0 \in s -> drop n s = x0 :: drop n.+1 s. | Proof. by move=> xs; rewrite (drop_nth x0) ?index_mem ?nth_index. Qed. | Lemma | drop_index | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"drop_nth",
"index",
"index_mem",
"nth_index"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
index_pivot x s1 s2 (s := s1 ++ x :: s2) : x \notin s1 ->
index x s = size s1. | Proof. by rewrite index_cat/= eqxx addn0; case: ifPn. Qed. | Lemma | index_pivot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"eqxx",
"index",
"index_cat",
"s1",
"s2",
"size"
] | lemmas about the pivot pattern [_ ++ _ :: _] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
take_pivot x s2 s1 (s := s1 ++ x :: s2) : x \notin s1 ->
take (index x s) s = s1. | Proof. by move=> /index_pivot->; rewrite take_size_cat. Qed. | Lemma | take_pivot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"index",
"index_pivot",
"s1",
"s2",
"take",
"take_size_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_pivot x s1 s2 : rev (s1 ++ x :: s2) = rev s2 ++ x :: rev s1. | Proof. by rewrite rev_cat rev_cons cat_rcons. Qed. | Lemma | rev_pivot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_rcons",
"rev",
"rev_cat",
"rev_cons",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqseq_pivot2l x s1 s2 s3 s4 : x \notin s1 -> x \notin s3 ->
(s1 ++ x :: s2 == s3 ++ x :: s4) = (s1 == s3) && (s2 == s4). | Proof.
move=> xNs1 xNs3; apply/idP/idP => [E|/andP[/eqP-> /eqP->]//].
suff S : size s1 = size s3 by rewrite eqseq_cat// eqseq_cons eqxx in E.
by rewrite -(index_pivot s2 xNs1) (eqP E) index_pivot.
Qed. | Lemma | eqseq_pivot2l | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eqseq_cat",
"eqseq_cons",
"eqxx",
"index_pivot",
"s1",
"s2",
"s3",
"s4",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqseq_pivot2r x s1 s2 s3 s4 : x \notin s2 -> x \notin s4 ->
(s1 ++ x :: s2 == s3 ++ x :: s4) = (s1 == s3) && (s2 == s4). | Proof.
move=> xNs2 xNs4; rewrite -(can_eq revK) !rev_pivot.
by rewrite eqseq_pivot2l ?mem_rev // !(can_eq revK) andbC.
Qed. | Lemma | eqseq_pivot2r | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"can_eq",
"eqseq_pivot2l",
"mem_rev",
"revK",
"rev_pivot",
"s1",
"s2",
"s3",
"s4"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqseq_pivotl x s1 s2 s3 s4 : x \notin s1 -> x \notin s2 ->
(s1 ++ x :: s2 == s3 ++ x :: s4) = (s1 == s3) && (s2 == s4). | Proof.
move=> xNs1 xNs2; apply/idP/idP => [E|/andP[/eqP-> /eqP->]//].
rewrite -(@eqseq_pivot2l x)//; have /eqP/(congr1 (count_mem x)) := E.
rewrite !count_cat/= eqxx !addnS (count_memPn _ _ xNs1) (count_memPn _ _ xNs2).
by move=> -[/esym/eqP]; rewrite addn_eq0 => /andP[/eqP/count_memPn].
Qed. | Lemma | eqseq_pivotl | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnS",
"addn_eq0",
"apply",
"count_cat",
"count_mem",
"count_memPn",
"eqseq_pivot2l",
"eqxx",
"s1",
"s2",
"s3",
"s4"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqseq_pivotr x s1 s2 s3 s4 : x \notin s3 -> x \notin s4 ->
(s1 ++ x :: s2 == s3 ++ x :: s4) = (s1 == s3) && (s2 == s4). | Proof. by move=> *; rewrite eq_sym eqseq_pivotl//; case: eqVneq => /=. Qed. | Lemma | eqseq_pivotr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqVneq",
"eq_sym",
"eqseq_pivotl",
"s1",
"s2",
"s3",
"s4"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_eqseq_pivotl x s1 s2 s3 s4 : uniq (s1 ++ x :: s2) ->
(s1 ++ x :: s2 == s3 ++ x :: s4) = (s1 == s3) && (s2 == s4). | Proof.
by rewrite uniq_catC/= mem_cat => /andP[/norP[? ?] _]; rewrite eqseq_pivotl.
Qed. | Lemma | uniq_eqseq_pivotl | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqseq_pivotl",
"mem_cat",
"s1",
"s2",
"s3",
"s4",
"uniq",
"uniq_catC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_eqseq_pivotr x s1 s2 s3 s4 : uniq (s3 ++ x :: s4) ->
(s1 ++ x :: s2 == s3 ++ x :: s4) = (s1 == s3) && (s2 == s4). | Proof. by move=> ?; rewrite eq_sym uniq_eqseq_pivotl//; case: eqVneq => /=. Qed. | Lemma | uniq_eqseq_pivotr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqVneq",
"eq_sym",
"s1",
"s2",
"s3",
"s4",
"uniq",
"uniq_eqseq_pivotl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'has_ view" | := (hasPP _ (fun _ => view))
(at level 4, right associativity, format "''has_' view"). | Notation | 'has_ view | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"hasPP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'all_ view" | := (allPP _ (fun _ => view))
(at level 4, right associativity, format "''all_' view"). | Notation | 'all_ view | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"allPP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_index s x (i := index x s) : x \in s ->
rot i s = x :: (drop i.+1 s ++ take i s). | Proof. by move=> x_s; rewrite /rot drop_index. Qed. | Lemma | rot_index | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"drop_index",
"index",
"rot",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_to_spec s x | := RotToSpec i s' of rot i s = x :: s'. | Variant | rot_to_spec | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_to s x : x \in s -> rot_to_spec s x. | Proof. by move=> /rot_index /RotToSpec. Qed. | Lemma | rot_to | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rot_index",
"rot_to_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nthP (T : eqType) (s : seq T) x x0 :
reflect (exists2 i, i < size s & nth x0 s i = x) (x \in s). | Proof.
apply: (iffP idP) => [|[n Hn <-]]; last exact: mem_nth.
by exists (index x s); [rewrite index_mem | apply nth_index].
Qed. | Lemma | nthP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"index",
"index_mem",
"last",
"mem_nth",
"nth",
"nth_index",
"seq",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_nthP a s x0 :
reflect (exists2 i, i < size s & a (nth x0 s i)) (has a s). | Proof.
elim: s => [|x s IHs] /=; first by right; case.
case nax: (a x); first by left; exists 0.
by apply: (iffP IHs) => [[i]|[[|i]]]; [exists i.+1 | rewrite nax | exists i].
Qed. | Lemma | has_nthP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"has",
"nth",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_nthP a s x0 :
reflect (forall i, i < size s -> a (nth x0 s i)) (all a s). | Proof.
rewrite -(eq_all (fun x => negbK (a x))) all_predC.
case: (has_nthP _ _ x0) => [na_s | a_s]; [right=> a_s | left=> i lti].
by case: na_s => i lti; rewrite a_s.
by apply/idPn=> na_si; case: a_s; exists i.
Qed. | Lemma | all_nthP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"all_predC",
"apply",
"eq_all",
"has_nthP",
"nth",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_nthE s x0 n x :
set_nth x0 s n x = if n < size s
then take n s ++ x :: drop n.+1 s
else s ++ ncons (n - size s) x0 [:: x]. | Proof.
elim: s n => [|a s IH] n /=; first by rewrite subn0 set_nth_nil.
case: n => [|n]; first by rewrite drop0.
by rewrite ltnS /=; case: ltnP (IH n) => _ ->.
Qed. | Lemma | set_nthE | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"drop0",
"ltnP",
"ltnS",
"ncons",
"set_nth",
"set_nth_nil",
"size",
"subn0",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_set_nth a s x0 n x :
count a (set_nth x0 s n x) =
count a s + a x - a (nth x0 s n) * (n < size s) + (a x0) * (n - size s). | Proof.
rewrite set_nthE; case: ltnP => [nlts|nges]; last first.
rewrite -cat_nseq !count_cat count_nseq /=.
by rewrite muln0 addn0 subn0 addnAC addnA.
have -> : n - size s = 0 by apply/eqP; rewrite subn_eq0 ltnW.
rewrite -[in count a s](cat_take_drop n s) [drop n s](drop_nth x0)//.
by rewrite !count_cat/= muln1 mul... | Lemma | count_set_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"addnA",
"addnAC",
"addnK",
"apply",
"cat_nseq",
"cat_take_drop",
"count",
"count_cat",
"count_nseq",
"drop",
"drop_nth",
"last",
"ltnP",
"ltnW",
"muln0",
"muln1",
"nth",
"set_nth",
"set_nthE",
"size",
"subn0",
"subn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_set_nth_ltn a s x0 n x : n < size s ->
count a (set_nth x0 s n x) = count a s + a x - a (nth x0 s n). | Proof.
move=> nlts; rewrite count_set_nth nlts muln1.
have -> : n - size s = 0 by apply/eqP; rewrite subn_eq0 ltnW.
by rewrite muln0 addn0.
Qed. | Lemma | count_set_nth_ltn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"apply",
"count",
"count_set_nth",
"ltnW",
"muln0",
"muln1",
"nth",
"set_nth",
"size",
"subn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_set_nthF a s x0 n x : ~~ a x0 ->
count a (set_nth x0 s n x) = count a s + a x - a (nth x0 s n). | Proof.
move=> /negbTE ax0; rewrite count_set_nth ax0 mul0n addn0.
case: ltnP => [_|nges]; first by rewrite muln1.
by rewrite nth_default// ax0 subn0.
Qed. | Lemma | count_set_nthF | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"count",
"count_set_nth",
"ltnP",
"mul0n",
"muln1",
"nth",
"nth_default",
"set_nth",
"subn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_nth_default n : n < size s -> nth x0 s n = nth y0 s n. | Proof. by elim: s n => [|y s' IHs] [|n] //= /IHs. Qed. | Lemma | set_nth_default | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"nth",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_last_default : size s > 0 -> last x0 s = last y0 s. | Proof. by elim: s => [|y s' IHs] //= /IHs. Qed. | Lemma | set_last_default | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"last",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
headI T s (x : T) : rcons s x = head x s :: behead (rcons s x). | Proof. by case: s. Qed. | Lemma | headI | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"behead",
"head",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
last_take x n s : last x (take n s) =
if n > 0 then if n <= size s then nth x s n.-1 else last x s else x. | Proof.
elim: n s => [|n ihn] [|y s] //= in x *; rewrite {}ihn ltnS; case: n => //=.
by move=> n; case: ltnP => // n_lt; apply/set_nth_default.
Qed. | Lemma | last_take | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"last",
"ltnP",
"ltnS",
"nth",
"set_nth_default",
"size",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
last_drop x n s : last x (drop n s) = if n < size s then last x s else x. | Proof.
case: ltnP => sn; last by rewrite drop_oversize.
rewrite -[s in RHS](cat_take_drop n) last_cat.
by rewrite (@set_last_default _ _ x) ?size_drop ?subn_gt0.
Qed. | Lemma | last_drop | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"drop",
"drop_oversize",
"last",
"last_cat",
"ltnP",
"set_last_default",
"size",
"size_drop",
"subn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bitseq | := seq bool. | Definition | bitseq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bitseq_predType | := Eval hnf in [predType of bitseq]. | Canonical | bitseq_predType | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"bitseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
split_find_nth_spec p : seq T -> seq T -> seq T -> T -> Type | :=
FindNth x s1 s2 of p x & ~~ has p s1 :
split_find_nth_spec p (rcons s1 x ++ s2) s1 s2 x. | Variant | split_find_nth_spec | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has",
"rcons",
"s1",
"s2",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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