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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|---|---|---|---|---|---|---|---|---|---|---|
"[ 'SubChoice_isSubZmodule' 'of' U 'by' <: ]" | :=
(SubChoice_isSubZmodule.Build _ _ U (zmodClosedP _))
(at level 0, format "[ 'SubChoice_isSubZmodule' 'of' U 'by' <: ]")
: form_scope. | Notation | [ 'SubChoice_isSubZmodule' 'of' U 'by' <: ] | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"Build",
"zmodClosedP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_add f g | := [ffun a => f a + g a]. | Definition | ffun_add | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_addrC : commutative (@ffun_add aT rT). | Proof. by move=> f1 f2; apply/ffunP => a; rewrite !ffunE addrC. Qed. | Fact | ffun_addrC | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"aT",
"addrC",
"apply",
"f1",
"f2",
"ffunE",
"ffunP",
"ffun_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_addrA : associative (@ffun_add aT rT). | Proof. by move=> f g h; apply/ffunP => a; rewrite !ffunE addrA. Qed. | Fact | ffun_addrA | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"aT",
"addrA",
"apply",
"ffunE",
"ffunP",
"ffun_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_zero | := [ffun a : aT => (0 : rT)]. | Definition | ffun_zero | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"aT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_add0r : left_id (@ffun_zero aT rT) (@ffun_add aT rT). | Proof. by move=> f; apply/ffunP => a; rewrite !ffunE add0r. Qed. | Fact | ffun_add0r | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"aT",
"add0r",
"apply",
"ffunE",
"ffunP",
"ffun_add",
"ffun_zero"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffunMnE f n x : (f *+ n) x = f x *+ n. | Proof.
elim: n => [|n IHn]; first by rewrite ffunE.
by rewrite !mulrS ffunE IHn.
Qed. | Lemma | ffunMnE | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"ffunE",
"mulrS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_opp f | := [ffun a => - f a]. | Definition | ffun_opp | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_addNr : left_inverse 0 ffun_opp +%R. | Proof. by move=> f; apply/ffunP => a; rewrite !ffunE addNr. Qed. | Fact | ffun_addNr | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"addNr",
"apply",
"ffunE",
"ffunP",
"ffun_opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_pair (a b : U * V) | := (a.1 + b.1, a.2 + b.2). | Definition | add_pair | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_addrC : commutative (@add_pair U V). | Proof. by move=> a b; congr pair; exact: addrC. Qed. | Fact | pair_addrC | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"add_pair",
"addrC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_addrA : associative (@add_pair U V). | Proof. by move=> [] al ar [] bl br [] cl cr; rewrite /add_pair !addrA. Qed. | Fact | pair_addrA | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"add_pair",
"addrA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_zero : U * V | := (0, 0). | Definition | pair_zero | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fst_is_zmod_morphism : nmod_morphism (@fst U V). | Proof. by []. Qed. | Fact | fst_is_zmod_morphism | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"nmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
snd_is_zmod_morphism : nmod_morphism (@snd U V). | Proof. by []. Qed. | Fact | snd_is_zmod_morphism | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"nmod_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_add0r : left_id (@pair_zero U V) (@add_pair U V). | Proof. by move=> [] al ar; rewrite /add_pair !add0r. Qed. | Fact | pair_add0r | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"add0r",
"add_pair",
"pair_zero"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_opp (a : U * V) | := (- a.1, - a.2). | Definition | pair_opp | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_addNr : left_inverse 0 pair_opp +%R. | Proof. by move=> [] al ar; rewrite /pair_opp; congr pair; apply/addNr. Qed. | Fact | pair_addNr | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"addNr",
"apply",
"pair_opp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natrE | := (natr0E, natrDE). | Definition | natrE | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"natr0E",
"natrDE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ord1 : all_equal_to (0 : 'I_1). | Proof. exact: ord1. Qed. | Lemma | ord1 | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [] | since 'I_n is now canonically a zmodType | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
lshift0 m n : lshift m (0 : 'I_n.+1) = (0 : 'I_(n + m).+1). | Proof. exact: val_inj. Qed. | Lemma | lshift0 | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"lshift",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rshift1 n : @rshift 1 n =1 lift (0 : 'I_n.+1). | Proof. by move=> i; apply: val_inj. Qed. | Lemma | rshift1 | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"apply",
"lift",
"rshift",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
split1 n i :
split (i : 'I_(1 + n)) = oapp (@inr _ _) (inl _ 0) (unlift 0 i). | Proof.
case: unliftP => [i'|] -> /=.
by rewrite -rshift1 (unsplitK (inr _ _)).
by rewrite -(lshift0 n 0) (unsplitK (inl _ _)).
Qed. | Lemma | split1 | boot | boot/nmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"choice",
"ssrnat",
"seq",
"bigop",
"fintype",
"finfun",
"monoid",
"div",
"AllExports"
] | [
"lshift0",
"rshift1",
"split",
"unlift",
"unliftP",
"unsplitK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path x (p : seq T) | :=
if p is y :: p' then e x y && path y p' else true. | Fixpoint | path | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat_path x p1 p2 : path x (p1 ++ p2) = path x p1 && path (last x p1) p2. | Proof. by elim: p1 x => [|y p1 Hrec] x //=; rewrite Hrec -!andbA. Qed. | Lemma | cat_path | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"last",
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcons_path x p y : path x (rcons p y) = path x p && e (last x p) y. | Proof. by rewrite -cats1 cat_path /= andbT. Qed. | Lemma | rcons_path | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"cat_path",
"cats1",
"last",
"path",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_path x p i : path x p -> path x (take i p). | Proof. elim: p x i => [//| x p] IHp x' [//| i] /= /andP[-> ?]; exact: IHp. Qed. | Lemma | take_path | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"path",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pathP x p x0 :
reflect (forall i, i < size p -> e (nth x0 (x :: p) i) (nth x0 p i))
(path x p). | Proof.
elim: p x => [|y p IHp] x /=; first by left.
apply: (iffP andP) => [[e_xy /IHp e_p [] //] | e_p].
by split; [apply: (e_p 0) | apply/(IHp y) => i; apply: e_p i.+1].
Qed. | Lemma | pathP | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"nth",
"path",
"size",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycle p | := if p is x :: p' then path x (rcons p' x) else true. | Definition | cycle | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"path",
"rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycle_path p : cycle p = path (last x0_cycle p) p. | Proof. by case: p => //= x p; rewrite rcons_path andbC. Qed. | Lemma | cycle_path | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"cycle",
"last",
"path",
"rcons_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycle_catC p q : cycle (p ++ q) = cycle (q ++ p). | Proof.
case: p q => [|x p] [|y q]; rewrite /= ?cats0 //=.
by rewrite !rcons_path !cat_path !last_cat /= -!andbA; do !bool_congr.
Qed. | Lemma | cycle_catC | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"cat_path",
"cats0",
"cycle",
"last_cat",
"rcons_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_cycle p : cycle (rot n0 p) = cycle p. | Proof. by rewrite cycle_catC cat_take_drop. Qed. | Lemma | rot_cycle | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"cat_take_drop",
"cycle",
"cycle_catC",
"rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotr_cycle p : cycle (rotr n0 p) = cycle p. | Proof. by rewrite -rot_cycle rotrK. Qed. | Lemma | rotr_cycle | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"cycle",
"rot_cycle",
"rotr",
"rotrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted s | := if s is x :: s' then path x s' else true. | Definition | sorted | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sortedP s x :
reflect (forall i, i.+1 < size s -> e (nth x s i) (nth x s i.+1)) (sorted s). | Proof. by case: s => *; [constructor|apply: (iffP (pathP _ _ _)); apply]. Qed. | Lemma | sortedP | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"nth",
"pathP",
"size",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_sorted x s : path x s -> sorted s. | Proof. by case: s => //= y s /andP[]. Qed. | Lemma | path_sorted | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"path",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_min_sorted x s : all (e x) s -> path x s = sorted s. | Proof. by case: s => //= y s /andP [->]. Qed. | Lemma | path_min_sorted | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"path",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pairwise_sorted s : pairwise e s -> sorted s. | Proof. by elim: s => //= x s IHs /andP[/path_min_sorted -> /IHs]. Qed. | Lemma | pairwise_sorted | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"pairwise",
"path_min_sorted",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_cat_cons s1 x s2 :
sorted (s1 ++ x :: s2) = sorted (rcons s1 x) && path x s2. | Proof.
by case: s1 => [ | e1 s1] //=; rewrite -cat_rcons cat_path last_rcons.
Qed. | Lemma | sorted_cat_cons | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"cat_path",
"cat_rcons",
"last_rcons",
"path",
"rcons",
"s1",
"s2",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_path x p :
path e (last x p) (rev (belast x p)) = path (fun z => e^~ z) x p. | Proof.
elim: p x => //= y p IHp x; rewrite rev_cons rcons_path -{}IHp andbC.
by rewrite -(last_cons x) -rev_rcons -lastI rev_cons last_rcons.
Qed. | Lemma | rev_path | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"belast",
"last",
"lastI",
"last_cons",
"last_rcons",
"path",
"rcons_path",
"rev",
"rev_cons",
"rev_rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_cycle p : cycle e (rev p) = cycle (fun z => e^~ z) p. | Proof.
case: p => //= x p; rewrite -rev_path last_rcons belast_rcons rev_cons.
by rewrite -[in LHS]cats1 cycle_catC.
Qed. | Lemma | rev_cycle | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"belast_rcons",
"cats1",
"cycle",
"cycle_catC",
"last_rcons",
"rev",
"rev_cons",
"rev_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rev_sorted p : sorted e (rev p) = sorted (fun z => e^~ z) p. | Proof. by case: p => //= x p; rewrite -rev_path lastI rev_rcons. Qed. | Lemma | rev_sorted | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"lastI",
"rev",
"rev_path",
"rev_rcons",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_relI x s :
path [rel x y | e x y && e' x y] x s = path e x s && path e' x s. | Proof. by elim: s x => //= y s IHs x; rewrite andbACA IHs. Qed. | Lemma | path_relI | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"e'",
"path",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycle_relI s :
cycle [rel x y | e x y && e' x y] s = cycle e s && cycle e' s. | Proof. by case: s => [|? ?]; last apply: path_relI. Qed. | Lemma | cycle_relI | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"cycle",
"e'",
"last",
"path_relI",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_relI s :
sorted [rel x y | e x y && e' x y] s = sorted e s && sorted e' s. | Proof. by case: s; last apply: path_relI. Qed. | Lemma | sorted_relI | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"e'",
"last",
"path_relI",
"rel",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(ee' : {in P &, subrel e e'}). | Hypothesis | ee' | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"e'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
sub_in_path x s : all P (x :: s) -> path e x s -> path e' x s. | Proof.
by elim: s x => //= y s ihs x /and3P [? ? ?] /andP [/ee' -> //]; apply/ihs/andP.
Qed. | Lemma | sub_in_path | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"e'",
"ee'",
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_in_cycle s : all P s -> cycle e s -> cycle e' s. | Proof.
case: s => //= x s /andP [Px Ps].
by apply: sub_in_path; rewrite /= all_rcons Px.
Qed. | Lemma | sub_in_cycle | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"Px",
"all",
"all_rcons",
"apply",
"cycle",
"e'",
"sub_in_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_in_sorted s : all P s -> sorted e s -> sorted e' s. | Proof. by case: s => //; apply: sub_in_path. Qed. | Lemma | sub_in_sorted | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"e'",
"sorted",
"sub_in_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(ee' : {in P &, e =2 e'}). | Hypothesis | ee' | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"e'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
e_e' : {in P &, subrel e e'}. | Proof. by move=> ? ? ? ?; rewrite ee'. Qed. | Let | e_e' | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"e'",
"ee'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
e'_e : {in P &, subrel e' e}. | Proof. by move=> ? ? ? ?; rewrite ee'. Qed. | Let | e'_e | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"e'",
"ee'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_path x s : all P (x :: s) -> path e x s = path e' x s. | Proof. by move=> Pxs; apply/idP/idP; apply: sub_in_path Pxs. Qed. | Lemma | eq_in_path | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"e'",
"path",
"sub_in_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_cycle s : all P s -> cycle e s = cycle e' s. | Proof. by move=> Ps; apply/idP/idP; apply: sub_in_cycle Ps. Qed. | Lemma | eq_in_cycle | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"cycle",
"e'",
"sub_in_cycle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_in_sorted s : all P s -> sorted e s = sorted e' s. | Proof. by move=> Ps; apply/idP/idP; apply: sub_in_sorted Ps. Qed. | Lemma | eq_in_sorted | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"e'",
"sorted",
"sub_in_sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_path : subrel e e' -> forall x p, path e x p -> path e' x p. | Proof. by move=> ? ? ?; apply/sub_in_path/all_predT; apply: in2W. Qed. | Lemma | sub_path | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all_predT",
"apply",
"e'",
"path",
"sub_in_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_cycle : subrel e e' -> subpred (cycle e) (cycle e'). | Proof. by move=> ee' [] // ? ?; apply: sub_path. Qed. | Lemma | sub_cycle | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"cycle",
"e'",
"ee'",
"sub_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_sorted : subrel e e' -> subpred (sorted e) (sorted e'). | Proof. by move=> ee' [] //=; apply: sub_path. Qed. | Lemma | sub_sorted | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"e'",
"ee'",
"sorted",
"sub_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_path : e =2 e' -> path e =2 path e'. | Proof. by move=> ? ? ?; apply/eq_in_path/all_predT; apply: in2W. Qed. | Lemma | eq_path | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all_predT",
"apply",
"e'",
"eq_in_path",
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_cycle : e =2 e' -> cycle e =1 cycle e'. | Proof. by move=> ee' [] // ? ?; apply: eq_path. Qed. | Lemma | eq_cycle | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"cycle",
"e'",
"ee'",
"eq_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_sorted : e =2 e' -> sorted e =1 sorted e'. | Proof. by move=> ee' [] // ? ?; apply: eq_path. Qed. | Lemma | eq_sorted | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"e'",
"ee'",
"eq_path",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_path_min_in x s :
{in P & &, transitive leT} -> all P (x :: s) -> path leT x s -> all (leT x) s. | Proof.
move=> leT_tr; elim: s => //= y s ihs /and3P [Px Py Ps] /andP [xy ys].
rewrite xy {}ihs ?Px //=; case: s Ps ys => //= z s /andP [Pz Ps] /andP [yz ->].
by rewrite (leT_tr _ _ _ Py Px Pz).
Qed. | Lemma | order_path_min_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"Px",
"all",
"leT_tr",
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leT_tr : {in P & &, transitive leT}. | Hypothesis | leT_tr | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
path_sorted_inE x s :
all P (x :: s) -> path leT x s = all (leT x) s && sorted leT s. | Proof.
move=> Pxs; apply/idP/idP => [xs|/andP[/path_min_sorted<-//]].
by rewrite (order_path_min_in leT_tr) //; apply: path_sorted xs.
Qed. | Lemma | path_sorted_inE | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"leT_tr",
"order_path_min_in",
"path",
"path_min_sorted",
"path_sorted",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_pairwise_in s : all P s -> sorted leT s = pairwise leT s. | Proof.
by elim: s => //= x s IHs /andP [Px Ps]; rewrite path_sorted_inE ?IHs //= Px.
Qed. | Lemma | sorted_pairwise_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"Px",
"all",
"pairwise",
"path_sorted_inE",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_pairwise_in x s :
all P (x :: s) -> path leT x s = pairwise leT (x :: s). | Proof. by move=> Pxs; rewrite -sorted_pairwise_in. Qed. | Lemma | path_pairwise_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"pairwise",
"path",
"sorted_pairwise_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cat_sorted2 s s' : sorted leT (s ++ s') -> sorted leT s * sorted leT s'. | Proof. by case: s => //= x s; rewrite cat_path => /andP[-> /path_sorted]. Qed. | Lemma | cat_sorted2 | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"cat_path",
"path_sorted",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_mask_in m s : all P s -> sorted leT s -> sorted leT (mask m s). | Proof.
by move=> Ps; rewrite !sorted_pairwise_in ?all_mask //; exact: pairwise_mask.
Qed. | Lemma | sorted_mask_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"all_mask",
"mask",
"pairwise_mask",
"sorted",
"sorted_pairwise_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_filter_in a s : all P s -> sorted leT s -> sorted leT (filter a s). | Proof. rewrite filter_mask; exact: sorted_mask_in. Qed. | Lemma | sorted_filter_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"filter",
"filter_mask",
"sorted",
"sorted_mask_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_mask_in x m s :
all P (x :: s) -> path leT x s -> path leT x (mask m s). | Proof. exact/(sorted_mask_in (true :: m)). Qed. | Lemma | path_mask_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"mask",
"path",
"sorted_mask_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_filter_in x a s :
all P (x :: s) -> path leT x s -> path leT x (filter a s). | Proof. by move=> Pxs; rewrite filter_mask; exact: path_mask_in. Qed. | Lemma | path_filter_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"filter",
"filter_mask",
"path",
"path_mask_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_ltn_nth_in x0 s : all P s -> sorted leT s ->
{in [pred n | n < size s] &, {homo nth x0 s : i j / i < j >-> leT i j}}. | Proof. by move=> Ps; rewrite sorted_pairwise_in //; apply/pairwiseP. Qed. | Lemma | sorted_ltn_nth_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"nth",
"pairwiseP",
"size",
"sorted",
"sorted_pairwise_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leT_refl : {in P, reflexive leT}. | Hypothesis | leT_refl | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
sorted_leq_nth_in x0 s : all P s -> sorted leT s ->
{in [pred n | n < size s] &, {homo nth x0 s : i j / i <= j >-> leT i j}}. | Proof.
move=> Ps s_sorted x y xs ys; rewrite leq_eqVlt=> /predU1P[->|].
exact/leT_refl/all_nthP.
exact: sorted_ltn_nth_in.
Qed. | Lemma | sorted_leq_nth_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"all_nthP",
"leT_refl",
"leq_eqVlt",
"nth",
"predU1P",
"size",
"sorted",
"sorted_ltn_nth_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_path_min x s : transitive leT -> path leT x s -> all (leT x) s. | Proof.
by move=> leT_tr; apply/order_path_min_in/all_predT => //; apply: in3W.
Qed. | Lemma | order_path_min | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"all_predT",
"apply",
"leT_tr",
"order_path_min_in",
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leT_tr : transitive leT. | Hypothesis | leT_tr | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
path_le x x' s : leT x x' -> path leT x' s -> path leT x s. | Proof.
by case: s => [//| x'' s xlex' /= /andP[x'lex'' ->]]; rewrite (leT_tr xlex').
Qed. | Lemma | path_le | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"leT_tr",
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leT_tr' : {in predT & &, transitive leT}. | Proof. exact: in3W. Qed. | Let | leT_tr' | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_sortedE x s : path leT x s = all (leT x) s && sorted leT s. | Proof. exact/path_sorted_inE/all_predT. Qed. | Lemma | path_sortedE | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"all_predT",
"path",
"path_sorted_inE",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_pairwise s : sorted leT s = pairwise leT s. | Proof. exact/sorted_pairwise_in/all_predT. Qed. | Lemma | sorted_pairwise | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all_predT",
"pairwise",
"sorted",
"sorted_pairwise_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_pairwise x s : path leT x s = pairwise leT (x :: s). | Proof. exact/path_pairwise_in/all_predT. Qed. | Lemma | path_pairwise | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all_predT",
"pairwise",
"path",
"path_pairwise_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_mask m s : sorted leT s -> sorted leT (mask m s). | Proof. exact/sorted_mask_in/all_predT. Qed. | Lemma | sorted_mask | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all_predT",
"mask",
"sorted",
"sorted_mask_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_filter a s : sorted leT s -> sorted leT (filter a s). | Proof. exact/sorted_filter_in/all_predT. Qed. | Lemma | sorted_filter | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all_predT",
"filter",
"sorted",
"sorted_filter_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_mask x m s : path leT x s -> path leT x (mask m s). | Proof. exact/path_mask_in/all_predT. Qed. | Lemma | path_mask | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all_predT",
"mask",
"path",
"path_mask_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_filter x a s : path leT x s -> path leT x (filter a s). | Proof. exact/path_filter_in/all_predT. Qed. | Lemma | path_filter | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all_predT",
"filter",
"path",
"path_filter_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_ltn_nth x0 s : sorted leT s ->
{in [pred n | n < size s] &, {homo nth x0 s : i j / i < j >-> leT i j}}. | Proof. exact/sorted_ltn_nth_in/all_predT. Qed. | Lemma | sorted_ltn_nth | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all_predT",
"nth",
"size",
"sorted",
"sorted_ltn_nth_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leT_refl : reflexive leT. | Hypothesis | leT_refl | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
sorted_leq_nth x0 s : sorted leT s ->
{in [pred n | n < size s] &, {homo nth x0 s : i j / i <= j >-> leT i j}}. | Proof. exact/sorted_leq_nth_in/all_predT. Qed. | Lemma | sorted_leq_nth | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all_predT",
"nth",
"size",
"sorted",
"sorted_leq_nth_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
take_sorted n s : sorted leT s -> sorted leT (take n s). | Proof. by rewrite -[s in sorted _ s](cat_take_drop n) => /cat_sorted2[]. Qed. | Lemma | take_sorted | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"cat_sorted2",
"cat_take_drop",
"sorted",
"take"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
drop_sorted n s : sorted leT s -> sorted leT (drop n s). | Proof. by rewrite -[s in sorted _ s](cat_take_drop n) => /cat_sorted2[]. Qed. | Lemma | drop_sorted | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"cat_sorted2",
"cat_take_drop",
"drop",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_map x s : path e' (f x) (map f s) = path (relpre f e') x s. | Proof. by elim: s x => //= y s <-. Qed. | Lemma | path_map | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"e'",
"map",
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycle_map s : cycle e' (map f s) = cycle (relpre f e') s. | Proof. by case: s => //= ? ?; rewrite -map_rcons path_map. Qed. | Lemma | cycle_map | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"cycle",
"e'",
"map",
"map_rcons",
"path_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sorted_map s : sorted e' (map f s) = sorted (relpre f e') s. | Proof. by case: s; last apply: path_map. Qed. | Lemma | sorted_map | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"e'",
"last",
"map",
"path_map",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homo_path_in x s : {in P &, {homo f : x y / e x y >-> e' x y}} ->
all P (x :: s) -> path e x s -> path e' (f x) (map f s). | Proof. by move=> f_mono; rewrite path_map; apply: sub_in_path. Qed. | Lemma | homo_path_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"e'",
"f_mono",
"map",
"path",
"path_map",
"sub_in_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homo_cycle_in s : {in P &, {homo f : x y / e x y >-> e' x y}} ->
all P s -> cycle e s -> cycle e' (map f s). | Proof. by move=> f_mono; rewrite cycle_map; apply: sub_in_cycle. Qed. | Lemma | homo_cycle_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"cycle",
"cycle_map",
"e'",
"f_mono",
"map",
"sub_in_cycle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homo_sorted_in s : {in P &, {homo f : x y / e x y >-> e' x y}} ->
all P s -> sorted e s -> sorted e' (map f s). | Proof. by move=> f_mono; rewrite sorted_map; apply: sub_in_sorted. Qed. | Lemma | homo_sorted_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"e'",
"f_mono",
"map",
"sorted",
"sorted_map",
"sub_in_sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mono_path_in x s : {in P &, {mono f : x y / e x y >-> e' x y}} ->
all P (x :: s) -> path e' (f x) (map f s) = path e x s. | Proof. by move=> f_mono; rewrite path_map; apply: eq_in_path. Qed. | Lemma | mono_path_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"e'",
"eq_in_path",
"f_mono",
"map",
"path",
"path_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mono_cycle_in s : {in P &, {mono f : x y / e x y >-> e' x y}} ->
all P s -> cycle e' (map f s) = cycle e s. | Proof. by move=> f_mono; rewrite cycle_map; apply: eq_in_cycle. Qed. | Lemma | mono_cycle_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"cycle",
"cycle_map",
"e'",
"eq_in_cycle",
"f_mono",
"map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mono_sorted_in s : {in P &, {mono f : x y / e x y >-> e' x y}} ->
all P s -> sorted e' (map f s) = sorted e s. | Proof. by case: s => // x s; apply: mono_path_in. Qed. | Lemma | mono_sorted_in | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"all",
"apply",
"e'",
"map",
"mono_path_in",
"sorted"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homo_path x s : {homo f : x y / e x y >-> e' x y} ->
path e x s -> path e' (f x) (map f s). | Proof. by move=> f_homo; rewrite path_map; apply: sub_path. Qed. | Lemma | homo_path | boot | boot/path.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq"
] | [
"apply",
"e'",
"map",
"path",
"path_map",
"sub_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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