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
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|---|---|---|---|---|---|---|---|---|---|---|
split_find_nth x0 p s (i := find p s) :
has p s -> split_find_nth_spec p s (take i s) (drop i.+1 s) (nth x0 s i). | Proof.
move=> p_s; rewrite -[X in split_find_nth_spec _ X](cat_take_drop i s).
rewrite (drop_nth x0 _) -?has_find// -cat_rcons.
by constructor; [apply: nth_find | rewrite has_take -?leqNgt].
Qed. | Lemma | split_find_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"cat_rcons",
"cat_take_drop",
"drop",
"drop_nth",
"find",
"has",
"has_find",
"has_take",
"leqNgt",
"nth",
"nth_find",
"split_find_nth_spec",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
split_find_spec p : seq T -> seq T -> seq T -> Type | :=
FindSplit x s1 s2 of p x & ~~ has p s1 :
split_find_spec p (rcons s1 x ++ s2) s1 s2. | Variant | split_find_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 | |
split_find p s (i := find p s) :
has p s -> split_find_spec p s (take i s) (drop i.+1 s). | Proof.
by case: s => // x ? in i * => ?; case: split_find_nth => //; constructor.
Qed. | Lemma | split_find | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"find",
"has",
"split_find_nth",
"split_find_spec",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_rcons_cat_find x0 p s1 s2 x (s := rcons s1 x ++ s2) :
p x -> ~~ has p s1 -> nth x0 s (find p s) = x. | Proof.
move=> pz pNs1; rewrite /s cat_rcons find_cat (negPf pNs1).
by rewrite nth_cat/= pz addn0 subnn ltnn.
Qed. | Lemma | nth_rcons_cat_find | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"cat_rcons",
"find",
"find_cat",
"has",
"ltnn",
"nth",
"nth_cat",
"rcons",
"s1",
"s2",
"subnn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
incr_nth v i {struct i} | :=
if v is n :: v' then if i is i'.+1 then n :: incr_nth v' i' else n.+1 :: v'
else ncons i 0 [:: 1]. | Fixpoint | incr_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"ncons"
] | allows us to use nat seqs as bags of nats. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
nth_incr_nth v i j : nth 0 (incr_nth v i) j = (i == j) + nth 0 v j. | Proof.
elim: v i j => [|n v IHv] [|i] [|j] //=; rewrite ?eqSS ?addn0 //; try by case j.
elim: i j => [|i IHv] [|j] //=; rewrite ?eqSS //; by case j.
Qed. | Lemma | nth_incr_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn0",
"eqSS",
"incr_nth",
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_incr_nth v i :
size (incr_nth v i) = if i < size v then size v else i.+1. | Proof.
elim: v i => [|n v IHv] [|i] //=; first by rewrite size_ncons /= addn1.
by rewrite IHv; apply: fun_if.
Qed. | Lemma | size_incr_nth | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addn1",
"apply",
"incr_nth",
"size",
"size_ncons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
incr_nth_inj v : injective (incr_nth v). | Proof.
move=> i j /(congr1 (nth 0 ^~ i)); apply: contra_eq => neq_ij.
by rewrite !nth_incr_nth eqn_add2r eqxx /nat_of_bool ifN_eqC.
Qed. | Lemma | incr_nth_inj | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"contra_eq",
"eqn_add2r",
"eqxx",
"ifN_eqC",
"incr_nth",
"nat_of_bool",
"nth",
"nth_incr_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
incr_nthC v i j :
incr_nth (incr_nth v i) j = incr_nth (incr_nth v j) i. | Proof.
apply: (@eq_from_nth _ 0) => [|k _]; last by rewrite !nth_incr_nth addnCA.
by do !rewrite size_incr_nth leqNgt if_neg -/(maxn _ _); apply: maxnAC.
Qed. | Lemma | incr_nthC | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnCA",
"apply",
"eq_from_nth",
"incr_nth",
"last",
"leqNgt",
"maxn",
"maxnAC",
"nth_incr_nth",
"size_incr_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_eq s1 s2 | :=
all [pred x | count_mem x s1 == count_mem x s2] (s1 ++ s2). | Definition | perm_eq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"count_mem",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
permP s1 s2 : reflect (count^~ s1 =1 count^~ s2) (perm_eq s1 s2). | Proof.
apply: (iffP allP) => /= [eq_cnt1 a | eq_cnt x _]; last exact/eqP.
have [n le_an] := ubnP (count a (s1 ++ s2)); elim: n => // n IHn in a le_an *.
have [/eqP|] := posnP (count a (s1 ++ s2)).
by rewrite count_cat addn_eq0; do 2!case: eqP => // ->.
rewrite -has_count => /hasP[x s12x a_x]; pose a' := predD1 a x.
h... | Lemma | permP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"add1n",
"addn0",
"addn_eq0",
"allP",
"apply",
"count",
"count_cat",
"count_mem",
"count_pred0",
"count_predUI",
"eq_count",
"eqnP",
"hasP",
"has_count",
"has_pred1",
"last",
"leq_add2r",
"leq_trans",
"ltnS",
"perm_eq",
"posnP",
"predD1",
"s1",
"s2",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_refl s : perm_eq s s. | Proof. exact/permP. Qed. | Lemma | perm_refl | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"permP",
"perm_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_sym : symmetric perm_eq. | Proof. by move=> s1 s2; apply/permP/permP=> eq_s12 a. Qed. | Lemma | perm_sym | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"permP",
"perm_eq",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_trans : transitive perm_eq. | Proof. by move=> s2 s1 s3 /permP-eq12 /permP/(ftrans eq12)/permP. Qed. | Lemma | perm_trans | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"permP",
"perm_eq",
"s1",
"s2",
"s3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_eql s1 s2 | := (perm_eq s1 =1 perm_eq s2). | Notation | perm_eql | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"perm_eq",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_eqr s1 s2 | := (perm_eq^~ s1 =1 perm_eq^~ s2). | Notation | perm_eqr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"perm_eq",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
permEl s1 s2 : perm_eql s1 s2 -> perm_eq s1 s2. | Proof. by move->. Qed. | Lemma | permEl | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"perm_eq",
"perm_eql",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
permPl s1 s2 : reflect (perm_eql s1 s2) (perm_eq s1 s2). | Proof.
apply: (iffP idP) => [eq12 s3 | -> //]; apply/idP/idP; last exact: perm_trans.
by rewrite -!(perm_sym s3) => /perm_trans; apply.
Qed. | Lemma | permPl | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"last",
"perm_eq",
"perm_eql",
"perm_sym",
"perm_trans",
"s1",
"s2",
"s3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
permPr s1 s2 : reflect (perm_eqr s1 s2) (perm_eq s1 s2). | Proof.
by apply/(iffP idP) => [/permPl eq12 s3| <- //]; rewrite !(perm_sym s3) eq12.
Qed. | Lemma | permPr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"permPl",
"perm_eq",
"perm_eqr",
"perm_sym",
"s1",
"s2",
"s3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_catC s1 s2 : perm_eql (s1 ++ s2) (s2 ++ s1). | Proof. by apply/permPl/permP=> a; rewrite !count_cat addnC. Qed. | Lemma | perm_catC | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnC",
"apply",
"count_cat",
"permP",
"permPl",
"perm_eql",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_cat2l s1 s2 s3 : perm_eq (s1 ++ s2) (s1 ++ s3) = perm_eq s2 s3. | Proof.
apply/permP/permP=> eq23 a; apply/eqP;
by move/(_ a)/eqP: eq23; rewrite !count_cat eqn_add2l.
Qed. | Lemma | perm_cat2l | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"count_cat",
"eqn_add2l",
"permP",
"perm_eq",
"s1",
"s2",
"s3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_catl s t1 t2 : perm_eq t1 t2 -> perm_eql (s ++ t1) (s ++ t2). | Proof. by move=> eq_t12; apply/permPl; rewrite perm_cat2l. Qed. | Lemma | perm_catl | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"permPl",
"perm_cat2l",
"perm_eq",
"perm_eql"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_cons x s1 s2 : perm_eq (x :: s1) (x :: s2) = perm_eq s1 s2. | Proof. exact: (perm_cat2l [::x]). Qed. | Lemma | perm_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"perm_cat2l",
"perm_eq",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_cat2r s1 s2 s3 : perm_eq (s2 ++ s1) (s3 ++ s1) = perm_eq s2 s3. | Proof. by do 2!rewrite perm_sym perm_catC; apply: perm_cat2l. Qed. | Lemma | perm_cat2r | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"perm_cat2l",
"perm_catC",
"perm_eq",
"perm_sym",
"s1",
"s2",
"s3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_catr s1 s2 t : perm_eq s1 s2 -> perm_eql (s1 ++ t) (s2 ++ t). | Proof. by move=> eq_s12; apply/permPl; rewrite perm_cat2r. Qed. | Lemma | perm_catr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"permPl",
"perm_cat2r",
"perm_eq",
"perm_eql",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_cat s1 s2 t1 t2 :
perm_eq s1 s2 -> perm_eq t1 t2 -> perm_eq (s1 ++ t1) (s2 ++ t2). | Proof. by move=> /perm_catr-> /perm_catl->. Qed. | Lemma | perm_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"perm_catl",
"perm_catr",
"perm_eq",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_catAC s1 s2 s3 : perm_eql ((s1 ++ s2) ++ s3) ((s1 ++ s3) ++ s2). | Proof. by apply/permPl; rewrite -!catA perm_cat2l perm_catC. Qed. | Lemma | perm_catAC | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"catA",
"permPl",
"perm_cat2l",
"perm_catC",
"perm_eql",
"s1",
"s2",
"s3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_catCA s1 s2 s3 : perm_eql (s1 ++ s2 ++ s3) (s2 ++ s1 ++ s3). | Proof. by apply/permPl; rewrite !catA perm_cat2r perm_catC. Qed. | Lemma | perm_catCA | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"catA",
"permPl",
"perm_cat2r",
"perm_catC",
"perm_eql",
"s1",
"s2",
"s3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_catACA s1 s2 s3 s4 :
perm_eql ((s1 ++ s2) ++ (s3 ++ s4)) ((s1 ++ s3) ++ (s2 ++ s4)). | Proof. by apply/permPl; rewrite perm_catAC !catA perm_catAC. Qed. | Lemma | perm_catACA | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"catA",
"permPl",
"perm_catAC",
"perm_eql",
"s1",
"s2",
"s3",
"s4"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_rcons x s : perm_eql (rcons s x) (x :: s). | Proof. by move=> /= s2; rewrite -cats1 perm_catC. Qed. | Lemma | perm_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"perm_catC",
"perm_eql",
"rcons",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_rot n s : perm_eql (rot n s) s. | Proof. by move=> /= s2; rewrite perm_catC cat_take_drop. Qed. | Lemma | perm_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"perm_catC",
"perm_eql",
"rot",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_rotr n s : perm_eql (rotr n s) s. | Proof. exact: perm_rot. Qed. | Lemma | perm_rotr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"perm_eql",
"perm_rot",
"rotr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_rev s : perm_eql (rev s) s. | Proof. by apply/permPl/permP=> i; rewrite count_rev. Qed. | Lemma | perm_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"count_rev",
"permP",
"permPl",
"perm_eql",
"rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_filter s1 s2 a :
perm_eq s1 s2 -> perm_eq (filter a s1) (filter a s2). | Proof. by move/permP=> s12_count; apply/permP=> Q; rewrite !count_filter. Qed. | Lemma | perm_filter | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"count_filter",
"filter",
"permP",
"perm_eq",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_filterC a s : perm_eql (filter a s ++ filter (predC a) s) s. | Proof.
apply/permPl; elim: s => //= x s IHs.
by case: (a x); last rewrite /= -cat1s perm_catCA; rewrite perm_cons.
Qed. | Lemma | perm_filterC | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"cat1s",
"filter",
"last",
"permPl",
"perm_catCA",
"perm_cons",
"perm_eql"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_size s1 s2 : perm_eq s1 s2 -> size s1 = size s2. | Proof. by move/permP=> eq12; rewrite -!count_predT eq12. Qed. | Lemma | perm_size | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"count_predT",
"permP",
"perm_eq",
"s1",
"s2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_mem s1 s2 : perm_eq s1 s2 -> s1 =i s2. | Proof. by move/permP=> eq12 x; rewrite -!has_pred1 !has_count eq12. Qed. | Lemma | perm_mem | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has_count",
"has_pred1",
"permP",
"perm_eq",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_nilP s : reflect (s = [::]) (perm_eq s [::]). | Proof. by apply: (iffP idP) => [/perm_size/eqP/nilP | ->]. Qed. | Lemma | perm_nilP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"nilP",
"perm_eq",
"perm_size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_consP x s t :
reflect (exists i u, rot i t = x :: u /\ perm_eq u s)
(perm_eq t (x :: s)). | Proof.
apply: (iffP idP) => [eq_txs | [i [u [Dt eq_us]]]].
have /rot_to[i u Dt]: x \in t by rewrite (perm_mem eq_txs) mem_head.
by exists i, u; rewrite -(perm_cons x) -Dt perm_rot.
by rewrite -(perm_rot i) Dt perm_cons.
Qed. | Lemma | perm_consP | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"mem_head",
"perm_cons",
"perm_eq",
"perm_mem",
"perm_rot",
"rot",
"rot_to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_has s1 s2 a : perm_eq s1 s2 -> has a s1 = has a s2. | Proof. by move/perm_mem/eq_has_r. Qed. | Lemma | perm_has | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eq_has_r",
"has",
"perm_eq",
"perm_mem",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_all s1 s2 a : perm_eq s1 s2 -> all a s1 = all a s2. | Proof. by move/perm_mem/eq_all_r. Qed. | Lemma | perm_all | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"eq_all_r",
"perm_eq",
"perm_mem",
"s1",
"s2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_small_eq s1 s2 : size s2 <= 1 -> perm_eq s1 s2 -> s1 = s2. | Proof.
move=> s2_le1 eqs12; move/perm_size: eqs12 s2_le1 (perm_mem eqs12).
by case: s2 s1 => [|x []] // [|y []] // _ _ /(_ x) /[!(inE, eqxx)] /eqP->.
Qed. | Lemma | perm_small_eq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqxx",
"inE",
"perm_eq",
"perm_mem",
"perm_size",
"s1",
"s2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_leq_size s1 s2 : uniq s1 -> {subset s1 <= s2} -> size s1 <= size s2. | Proof.
elim: s1 s2 => //= x s1 IHs s2 /andP[not_s1x Us1] /forall_cons[s2x ss12].
have [i s3 def_s2] := rot_to s2x; rewrite -(size_rot i s2) def_s2.
apply: IHs => // y s1y; have:= ss12 y s1y.
by rewrite -(mem_rot i) def_s2 inE (negPf (memPn _ y s1y)).
Qed. | Lemma | uniq_leq_size | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"forall_cons",
"inE",
"memPn",
"mem_rot",
"rot_to",
"s1",
"s2",
"s3",
"size",
"size_rot",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_size_uniq s1 s2 :
uniq s1 -> {subset s1 <= s2} -> size s2 <= size s1 -> uniq s2. | Proof.
elim: s1 s2 => [[] | x s1 IHs s2] // Us1x; have /andP[not_s1x Us1] := Us1x.
case/forall_cons => /rot_to[i s3 def_s2] ss12 le_s21.
rewrite -(rot_uniq i) -(size_rot i) def_s2 /= in le_s21 *.
have ss13 y (s1y : y \in s1): y \in s3.
by have:= ss12 y s1y; rewrite -(mem_rot i) def_s2 inE (negPf (memPn _ y s1y)).
rew... | Lemma | leq_size_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"allP",
"apply",
"forall_cons",
"inE",
"leqNgt",
"memPn",
"mem_rot",
"rot_to",
"rot_uniq",
"s1",
"s2",
"s3",
"size",
"size_rot",
"uniq",
"uniq_leq_size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_size_uniq s1 s2 :
uniq s1 -> s1 =i s2 -> uniq s2 = (size s2 == size s1). | Proof.
move=> Us1 eqs12; apply/idP/idP=> [Us2 | /eqP eq_sz12].
by rewrite eqn_leq !uniq_leq_size // => y; rewrite eqs12.
by apply: (leq_size_uniq Us1) => [y|]; rewrite (eqs12, eq_sz12).
Qed. | Lemma | uniq_size_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eqn_leq",
"leq_size_uniq",
"s1",
"s2",
"size",
"uniq",
"uniq_leq_size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_min_size s1 s2 :
uniq s1 -> {subset s1 <= s2} -> size s2 <= size s1 ->
(size s1 = size s2) * (s1 =i s2). | Proof.
move=> Us1 ss12 le_s21; have Us2: uniq s2 := leq_size_uniq Us1 ss12 le_s21.
suffices: s1 =i s2 by split; first by apply/eqP; rewrite -uniq_size_uniq.
move=> x; apply/idP/idP=> [/ss12// | s2x]; apply: contraLR le_s21 => not_s1x.
rewrite -ltnNge (@uniq_leq_size (x :: s1)) /= ?not_s1x //.
by apply/allP; rewrite /= ... | Lemma | uniq_min_size | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"allP",
"apply",
"leq_size_uniq",
"ltnNge",
"s1",
"s2",
"size",
"split",
"uniq",
"uniq_leq_size",
"uniq_size_uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_uniq s1 s2 : size s1 = size s2 -> s1 =i s2 -> uniq s1 = uniq s2. | Proof.
move=> eq_sz12 eq_s12.
by apply/idP/idP=> Us; rewrite (uniq_size_uniq Us) ?eq_sz12 ?eqxx.
Qed. | Lemma | eq_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eqxx",
"s1",
"s2",
"size",
"uniq",
"uniq_size_uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_uniq s1 s2 : perm_eq s1 s2 -> uniq s1 = uniq s2. | Proof. by move=> eq_s12; apply/eq_uniq; [apply/perm_size | apply/perm_mem]. Qed. | Lemma | perm_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eq_uniq",
"perm_eq",
"perm_mem",
"perm_size",
"s1",
"s2",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
uniq_perm s1 s2 : uniq s1 -> uniq s2 -> s1 =i s2 -> perm_eq s1 s2. | Proof.
move=> Us1 Us2 eq12; apply/allP=> x _; apply/eqP.
by rewrite !count_uniq_mem ?eq12.
Qed. | Lemma | uniq_perm | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"allP",
"apply",
"count_uniq_mem",
"perm_eq",
"s1",
"s2",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_undup s1 s2 : s1 =i s2 -> perm_eq (undup s1) (undup s2). | Proof.
by move=> Es12; rewrite uniq_perm ?undup_uniq // => s; rewrite !mem_undup.
Qed. | Lemma | perm_undup | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mem_undup",
"perm_eq",
"s1",
"s2",
"undup",
"undup_uniq",
"uniq_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
count_mem_uniq s : (forall x, count_mem x s = (x \in s)) -> uniq s. | Proof.
move=> count1_s; have Uus := undup_uniq s.
suffices: perm_eq s (undup s) by move/perm_uniq->.
by apply/allP=> x _; apply/eqP; rewrite (count_uniq_mem x Uus) mem_undup.
Qed. | Lemma | count_mem_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"allP",
"apply",
"count_mem",
"count_uniq_mem",
"mem_undup",
"perm_eq",
"perm_uniq",
"undup",
"undup_uniq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_count_undup a s1 s2 :
{in a, s1 =i s2} -> count a (undup s1) = count a (undup s2). | Proof.
move=> s1_eq_s2; rewrite -!size_filter !filter_undup.
apply/perm_size/perm_undup => x.
by rewrite !mem_filter; case: (boolP (a x)) => //= /s1_eq_s2.
Qed. | Lemma | eq_count_undup | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"count",
"filter_undup",
"mem_filter",
"perm_size",
"perm_undup",
"s1",
"s2",
"size_filter",
"undup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
catCA_perm_ind P :
(forall s1 s2 s3, P (s1 ++ s2 ++ s3) -> P (s2 ++ s1 ++ s3)) ->
(forall s1 s2, perm_eq s1 s2 -> P s1 -> P s2). | Proof.
move=> PcatCA s1 s2 eq_s12; rewrite -[s1]cats0 -[s2]cats0.
elim: s2 nil => [|x s2 IHs] s3 in s1 eq_s12 *.
by case: s1 {eq_s12}(perm_size eq_s12).
have /rot_to[i s' def_s1]: x \in s1 by rewrite (perm_mem eq_s12) mem_head.
rewrite -(cat_take_drop i s1) -catA => /PcatCA.
rewrite catA -/(rot i s1) def_s1 /= -cat1s... | Lemma | catCA_perm_ind | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"cat1s",
"catA",
"cat_take_drop",
"cats0",
"mem_head",
"perm_cons",
"perm_eq",
"perm_mem",
"perm_rot",
"perm_size",
"rot",
"rot_to",
"s1",
"s2",
"s3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
catCA_perm_subst R F :
(forall s1 s2 s3, F (s1 ++ s2 ++ s3) = F (s2 ++ s1 ++ s3) :> R) ->
(forall s1 s2, perm_eq s1 s2 -> F s1 = F s2). | Proof.
move=> FcatCA s1 s2 /catCA_perm_ind => ind_s12.
by apply: (ind_s12 (eq _ \o F)) => //= *; rewrite FcatCA.
Qed. | Lemma | catCA_perm_subst | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"catCA_perm_ind",
"perm_eq",
"s1",
"s2",
"s3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_rotr s : size (rotr n0 s) = size s. | Proof. by rewrite size_rot. Qed. | Lemma | size_rotr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rotr",
"size",
"size_rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_rotr (s : seq T') : rotr n0 s =i s. | Proof. by move=> x; rewrite mem_rot. Qed. | Lemma | mem_rotr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"T'",
"mem_rot",
"rotr",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotr_size_cat s1 s2 : rotr (size s2) (s1 ++ s2) = s2 ++ s1. | Proof. by rewrite /rotr size_cat addnK rot_size_cat. Qed. | Lemma | rotr_size_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnK",
"rot_size_cat",
"rotr",
"s1",
"s2",
"size",
"size_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotr1_rcons x s : rotr 1 (rcons s x) = x :: s. | Proof. by rewrite -rot1_cons rotK. Qed. | Lemma | rotr1_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rcons",
"rot1_cons",
"rotK",
"rotr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_rotr a s : has a (rotr n0 s) = has a s. | Proof. by rewrite has_rot. Qed. | Lemma | has_rotr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has",
"has_rot",
"rotr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotr_uniq (s : seq T') : uniq (rotr n0 s) = uniq s. | Proof. by rewrite rot_uniq. Qed. | Lemma | rotr_uniq | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"T'",
"rot_uniq",
"rotr",
"seq",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotrK : cancel (@rotr T n0) (rot n0). | Proof.
move=> s; have [lt_n0s | ge_n0s] := ltnP n0 (size s).
by rewrite -{1}(subKn (ltnW lt_n0s)) -{1}[size s]size_rotr; apply: rotK.
by rewrite -[in RHS](rot_oversize ge_n0s) /rotr (eqnP ge_n0s) rot0.
Qed. | Lemma | rotrK | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"eqnP",
"ltnP",
"ltnW",
"rot",
"rot0",
"rotK",
"rot_oversize",
"rotr",
"size",
"size_rotr",
"subKn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotr_inj : injective (@rotr T n0). | Proof. exact (can_inj rotrK). Qed. | Lemma | rotr_inj | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rotr",
"rotrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_rev s : take n0 (rev s) = rev (drop (size s - n0) s). | Proof.
set m := _ - n0; rewrite -[s in LHS](cat_take_drop m) rev_cat take_cat.
rewrite size_rev size_drop -minnE minnC leq_min ltnn /m.
by have [_|/eqnP->] := ltnP; rewrite ?subnn take0 cats0.
Qed. | Lemma | take_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"cats0",
"drop",
"eqnP",
"leq_min",
"ltnP",
"ltnn",
"minnC",
"minnE",
"rev",
"rev_cat",
"size",
"size_drop",
"size_rev",
"subnn",
"take",
"take0",
"take_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_take s : rev (take n0 s) = drop (size s - n0) (rev s). | Proof. by rewrite -[s in take _ s]revK take_rev revK size_rev. Qed. | Lemma | rev_take | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"rev",
"revK",
"size",
"size_rev",
"take",
"take_rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_rev s : drop n0 (rev s) = rev (take (size s - n0) s). | Proof.
set m := _ - n0; rewrite -[s in LHS](cat_take_drop m) rev_cat drop_cat.
rewrite size_rev size_drop -minnE minnC leq_min ltnn /m.
by have [_|/eqnP->] := ltnP; rewrite ?take0 // subnn drop0.
Qed. | Lemma | drop_rev | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"drop",
"drop0",
"drop_cat",
"eqnP",
"leq_min",
"ltnP",
"ltnn",
"minnC",
"minnE",
"rev",
"rev_cat",
"size",
"size_drop",
"size_rev",
"subnn",
"take",
"take0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_drop s : rev (drop n0 s) = take (size s - n0) (rev s). | Proof. by rewrite -[s in drop _ s]revK drop_rev revK size_rev. Qed. | Lemma | rev_drop | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"drop_rev",
"rev",
"revK",
"size",
"size_rev",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_rotr s : rev (rotr n0 s) = rot n0 (rev s). | Proof. by rewrite rev_cat -take_rev -drop_rev. Qed. | Lemma | rev_rotr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop_rev",
"rev",
"rev_cat",
"rot",
"rotr",
"take_rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_rot s : rev (rot n0 s) = rotr n0 (rev s). | Proof. by apply: canLR revK _; rewrite rev_rotr revK. Qed. | Lemma | rev_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"rev",
"revK",
"rev_rotr",
"rot",
"rotr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotD m n s : m + n <= size s -> rot (m + n) s = rot m (rot n s). | Proof.
move=> sz_s; rewrite [LHS]/rot -[take _ s](cat_take_drop n).
rewrite 5!(catA, =^~ rot_size_cat) !cat_take_drop.
by rewrite size_drop !size_takel ?leq_addl ?addnK.
Qed. | Lemma | rotD | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnK",
"catA",
"cat_take_drop",
"leq_addl",
"rot",
"rot_size_cat",
"size",
"size_drop",
"size_takel",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotS n s : n < size s -> rot n.+1 s = rot 1 (rot n s). | Proof. exact: (@rotD 1). Qed. | Lemma | rotS | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rot",
"rotD",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_add_mod m n s : n <= size s -> m <= size s ->
rot m (rot n s) = rot (if m + n <= size s then m + n else m + n - size s) s. | Proof.
move=> Hn Hm; case: leqP => [/rotD // | /ltnW Hmn]; symmetry.
by rewrite -{2}(rotK n s) /rotr -rotD size_rot addnBA ?subnK ?addnK.
Qed. | Lemma | rot_add_mod | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnBA",
"addnK",
"leqP",
"ltnW",
"rot",
"rotD",
"rotK",
"rotr",
"size",
"size_rot",
"subnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_minn n s : rot n s = rot (minn n (size s)) s. | Proof.
by case: (leqP n (size s)) => // /leqW ?; rewrite rot_size rot_oversize.
Qed. | Lemma | rot_minn | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"leqP",
"leqW",
"minn",
"rot",
"rot_oversize",
"rot_size",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_add s n m (k := size s) (p := minn m k + minn n k) | :=
locked (if p <= k then p else p - k). | Definition | rot_add | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"minn",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_rot_add n m s : rot_add s n m <= size s. | Proof.
by unlock rot_add; case: ifP; rewrite // leq_subLR leq_add // geq_minr.
Qed. | Lemma | leq_rot_add | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"geq_minr",
"leq_add",
"leq_subLR",
"rot_add",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_addC n m s : rot_add s n m = rot_add s m n. | Proof. by unlock rot_add; rewrite ![minn n _ + _]addnC. Qed. | Lemma | rot_addC | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"addnC",
"minn",
"rot_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_rot_add n m s : rot m (rot n s) = rot (rot_add s n m) s. | Proof.
unlock rot_add.
by rewrite (rot_minn n) (rot_minn m) rot_add_mod ?size_rot ?geq_minr.
Qed. | Lemma | rot_rot_add | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"geq_minr",
"rot",
"rot_add",
"rot_add_mod",
"rot_minn",
"size_rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_rot m n s : rot m (rot n s) = rot n (rot m s). | Proof. by rewrite rot_rot_add rot_addC -rot_rot_add. Qed. | Lemma | rot_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rot",
"rot_addC",
"rot_rot_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_rotr m n s : rot m (rotr n s) = rotr n (rot m s). | Proof. by rewrite [RHS]/rotr size_rot rot_rot. Qed. | Lemma | rot_rotr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rot",
"rot_rot",
"rotr",
"size_rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotr_rotr m n s : rotr m (rotr n s) = rotr n (rotr m s). | Proof. by rewrite /rotr !size_rot rot_rot. Qed. | Lemma | rotr_rotr | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"rot_rot",
"rotr",
"size_rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask m s {struct m} | :=
match m, s with
| b :: m', x :: s' => if b then x :: mask m' s' else mask m' s'
| _, _ => [::]
end. | Fixpoint | mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_false s n : mask (nseq n false) s = [::]. | Proof. by elim: s n => [|x s IHs] [|n] /=. Qed. | Lemma | mask_false | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mask",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_true s n : size s <= n -> mask (nseq n true) s = s. | Proof. by elim: s n => [|x s IHs] [|n] //= Hn; congr (_ :: _); apply: IHs. Qed. | Lemma | mask_true | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"mask",
"nseq",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask0 m : mask m [::] = [::]. | Proof. by case: m. Qed. | Lemma | mask0 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask0s s : mask [::] s = [::]. | Proof. by []. Qed. | Lemma | mask0s | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask1 b x : mask [:: b] [:: x] = nseq b x. | Proof. by case: b. Qed. | Lemma | mask1 | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mask",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_cons b m x s : mask (b :: m) (x :: s) = nseq b x ++ mask m s. | Proof. by case: b. Qed. | Lemma | mask_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mask",
"nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_mask m s : size m = size s -> size (mask m s) = count id m. | Proof. by move: m s; apply: seq_ind2 => // -[] x m s /= _ ->. Qed. | Lemma | size_mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"count",
"id",
"mask",
"seq_ind2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_cat m1 m2 s1 s2 :
size m1 = size s1 -> mask (m1 ++ m2) (s1 ++ s2) = mask m1 s1 ++ mask m2 s2. | Proof. by move: m1 s1; apply: seq_ind2 => // -[] m1 x1 s1 /= _ ->. Qed. | Lemma | mask_cat | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"mask",
"s1",
"s2",
"seq_ind2",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_rcons b m x s : size m = size s ->
mask (rcons m b) (rcons s x) = mask m s ++ nseq b x. | Proof. by move=> ms; rewrite -!cats1 mask_cat//; case: b. Qed. | Lemma | mask_rcons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cats1",
"mask",
"mask_cat",
"nseq",
"rcons",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
all_mask a m s : all a s -> all a (mask m s). | Proof. by elim: s m => [|x s IHs] [|[] m]//= /andP[ax /IHs->]; rewrite ?ax. Qed. | Lemma | all_mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all",
"mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_mask_cons a b m x s :
has a (mask (b :: m) (x :: s)) = b && a x || has a (mask m s). | Proof. by case: b. Qed. | Lemma | has_mask_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has",
"mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
has_mask a m s : has a (mask m s) -> has a s. | Proof. by apply/contraTT; rewrite -!all_predC; apply: all_mask. Qed. | Lemma | has_mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"all_mask",
"all_predC",
"apply",
"has",
"mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_mask m s : size m = size s -> rev (mask m s) = mask (rev m) (rev s). | Proof.
move: m s; apply: seq_ind2 => //= b x m s eq_size_sm IH.
by case: b; rewrite !rev_cons mask_rcons ?IH ?size_rev// (cats1, cats0).
Qed. | Lemma | rev_mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"apply",
"cats0",
"cats1",
"mask",
"mask_rcons",
"rev",
"rev_cons",
"seq_ind2",
"size",
"size_rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mask_rot m s : size m = size s ->
mask (rot n0 m) (rot n0 s) = rot (count id (take n0 m)) (mask m s). | Proof.
move=> Ems; rewrite mask_cat ?size_drop ?Ems // -rot_size_cat.
by rewrite size_mask -?mask_cat ?size_take ?Ems // !cat_take_drop.
Qed. | Lemma | mask_rot | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"cat_take_drop",
"count",
"id",
"mask",
"mask_cat",
"rot",
"rot_size_cat",
"size",
"size_drop",
"size_mask",
"size_take",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
resize_mask m s : {m1 | size m1 = size s & mask m s = mask m1 s}. | Proof.
exists (take (size s) m ++ nseq (size s - size m) false).
by elim: s m => [|x s IHs] [|b m] //=; rewrite (size_nseq, IHs).
by elim: s m => [|x s IHs] [|b m] //=; rewrite (mask_false, IHs).
Qed. | Lemma | resize_mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mask",
"mask_false",
"nseq",
"size",
"size_nseq",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
takeEmask i s : take i s = mask (nseq i true) s. | Proof. by elim: i s => [s|i IHi []// ? ?]; rewrite ?take0 //= IHi. Qed. | Lemma | takeEmask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mask",
"nseq",
"take",
"take0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dropEmask i s :
drop i s = mask (nseq i false ++ nseq (size s - i) true) s. | Proof. by elim: i s => [s|? ? []//]; rewrite drop0/= mask_true// subn0. Qed. | Lemma | dropEmask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"drop",
"drop0",
"mask",
"mask_true",
"nseq",
"size",
"subn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_mask_cons x b m y s :
(x \in mask (b :: m) (y :: s)) = b && (x == y) || (x \in mask m s). | Proof. by case: b. Qed. | Lemma | mem_mask_cons | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_mask x m s : x \in mask m s -> x \in s. | Proof. by rewrite -!has_pred1 => /has_mask. Qed. | Lemma | mem_mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"has_mask",
"has_pred1",
"mask"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_mask x m s :
uniq s -> (x \in mask m s) = (x \in s) && nth false m (index x s). | Proof.
elim: s m => [|y s IHs] [|[] m]//= /andP[yNs ?]; rewrite ?in_cons ?IHs //=;
by have [->|neq_xy] //= := eqVneq; rewrite ?andbF // (negPf yNs).
Qed. | Lemma | in_mask | boot | boot/seq.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat"
] | [
"eqVneq",
"in_cons",
"index",
"mask",
"nth",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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