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 |
|---|---|---|---|---|---|---|---|---|---|---|
iter_finv_cycle n :
{in p, forall x, n <= order x -> iter n finv x = iter (order x - n) f x}. | Proof. exact: iter_finv_in. Qed. | Lemma | iter_finv_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"finv",
"iter",
"iter_finv_in",
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycle_orbit_cycle : {in p, forall x, fcycle f (orbit x)}. | Proof. exact: cycle_orbit_in. Qed. | Lemma | cycle_orbit_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"cycle_orbit_in",
"fcycle",
"orbit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fpath_finv_cycle q x : (x \in p) && (fpath finv x q) =
(last x q \in p) && fpath f (last x q) (rev (belast x q)). | Proof. exact: fpath_finv_in. Qed. | Lemma | fpath_finv_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"belast",
"finv",
"fpath",
"fpath_finv_in",
"last",
"rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fpath_finv_f_cycle q : {in p, forall x,
fpath finv x q -> fpath f (last x q) (rev (belast x q))}. | Proof. exact: fpath_finv_f_in. Qed. | Lemma | fpath_finv_f_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"belast",
"finv",
"fpath",
"fpath_finv_f_in",
"last",
"rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fpath_f_finv_cycle q x : last x q \in p ->
fpath f (last x q) (rev (belast x q)) -> fpath finv x q. | Proof. exact: fpath_f_finv_in. Qed. | Lemma | fpath_f_finv_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"belast",
"finv",
"fpath",
"fpath_f_finv_in",
"last",
"rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prevE x : x \in p -> prev p x = finv x. | Proof.
move=> x_p; have /eqP/(congr1 finv) := prev_cycle f_p x_p.
by rewrite finv_f_cycle// mem_prev.
Qed. | Lemma | prevE | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"finv",
"finv_f_cycle",
"mem_prev",
"prev",
"prev_cycle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcycle_rconsE : rcons (x :: p) x = traject f x (size p).+2. | Proof. by rewrite rcons_cons; have /fpathE-> := f_p; rewrite size_rcons. Qed. | Lemma | fcycle_rconsE | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fpathE",
"rcons",
"rcons_cons",
"size",
"size_rcons",
"traject"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcycle_consE : x :: p = traject f x (size p).+1. | Proof. by have := fcycle_rconsE; rewrite trajectSr => /rcons_inj[/= <-]. Qed. | Lemma | fcycle_consE | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fcycle_rconsE",
"rcons_inj",
"size",
"traject",
"trajectSr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcycle_consEflatten : exists k, x :: p = flatten (nseq k.+1 (orbit x)). | Proof.
move: f_p; rewrite fcycle_consE; elim/ltn_ind: (size p) => n IHn t_cycle.
have := order_le_cycle t_cycle (mem_head _ _); rewrite size_traject.
case: ltngtP => [||<-] //; last by exists 0; rewrite /= cats0.
rewrite ltnS => n_ge _; have := t_cycle.
rewrite -(subnKC n_ge) -addnS trajectD.
rewrite (iter_order_in (me... | Lemma | fcycle_consEflatten | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"addnS",
"cat_path",
"cat_rcons",
"cats0",
"fcycle_consE",
"flatten",
"inj_cycle",
"iter_order_in",
"last",
"last_rcons",
"leq_trans",
"ltnS",
"ltn_ind",
"ltn_subrL",
"ltngtP",
"mem_fcycle",
"mem_head",
"nseq",
"orbit",
"orderSpred",
"order_gt0",
"order_le_cycle",
"rcons_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
undup_cycle_cons : undup (x :: p) = orbit x. | Proof.
by have [n {1}->] := fcycle_consEflatten; rewrite undup_flatten_nseq ?undup_id.
Qed. | Lemma | undup_cycle_cons | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fcycle_consEflatten",
"orbit",
"undup",
"undup_flatten_nseq",
"undup_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcycleEflatten : exists k, p = flatten (nseq k (undup p)). | Proof.
case: p f_p => [//|x q] f_q; first by exists 0.
have [k {1}->] := @fcycle_consEflatten x q f_q.
by exists k.+1; rewrite undup_cycle_cons.
Qed. | Lemma | fcycleEflatten | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fcycle_consEflatten",
"flatten",
"nseq",
"undup",
"undup_cycle_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcycle_undup : fcycle f (undup p). | Proof.
case: p f_p => [//|x q] f_q; rewrite undup_cycle_cons//.
by rewrite (cycle_orbit_in (mem_fcycle f_q) (inj_cycle f_q)) ?mem_head.
Qed. | Lemma | fcycle_undup | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"cycle_orbit_in",
"fcycle",
"inj_cycle",
"mem_fcycle",
"mem_head",
"undup",
"undup_cycle_cons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p_undup_uniq | := undup_uniq p. | Let | p_undup_uniq | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"undup_uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_orbit_cycle : {in p &, forall x, orbit x =i p}. | Proof.
by move=> x y xp yp; rewrite (orbitE fcycle_undup)// ?mem_rot ?mem_undup.
Qed. | Lemma | in_orbit_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fcycle_undup",
"mem_rot",
"mem_undup",
"orbit",
"orbitE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_order_cycle : {in p &, forall x y, order y = order x}. | Proof. by move=> x y xp yp; rewrite !(order_cycle fcycle_undup) ?mem_undup. Qed. | Lemma | eq_order_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fcycle_undup",
"mem_undup",
"order",
"order_cycle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_order_cycle : {in p &, forall x y, iter (order x) f y = y}. | Proof.
by move=> x y xp yp; rewrite (eq_order_cycle yp) ?(iter_order_in homo_f f_inj).
Qed. | Lemma | iter_order_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"eq_order_cycle",
"f_inj",
"homo_f",
"iter",
"iter_order_in",
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect_eqVf x y : fconnect f x y = (x == y) || fconnect f (f x) y. | Proof.
apply/idP/idP => [/iter_findex <-|/predU1P [<-|] //]; last first.
exact/connect_trans/fconnect1.
by case: findex => [|i]; rewrite ?eqxx// iterSr fconnect_iter orbT.
Qed. | Lemma | fconnect_eqVf | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect_trans",
"eqxx",
"fconnect",
"fconnect1",
"fconnect_iter",
"findex",
"iterSr",
"iter_findex",
"last",
"predU1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orbitPcycle {x} : [<->
(* 0 *) fcycle f (orbit x);
(* 1 *) order (f x) = order x;
(* 2 *) x \in fconnect f (f x);
(* 3 *) exists k, iter k.+1 f x = x;
(* 4 *) iter (order x) f x = x;
(* 5 *) {in orbit x &, injective f}]. | Proof.
tfae=> [xorbit_cycle|||[k fkx]|fx y z|/injectivePcycle//].
- by apply: eq_order_cycle xorbit_cycle _ _ _ _; rewrite ?mem_orbit.
- move=> /subset_cardP/(_ _)->; rewrite ?inE//; apply/subsetP=> y.
by apply: connect_trans; apply: fconnect1.
- by exists (findex (f x) x); rewrite // iterSr iter_findex.
- apply: (@i... | Lemma | orbitPcycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"addnC",
"apply",
"connect_trans",
"eq_order_cycle",
"fconnect",
"fconnect1",
"fconnect_orbit",
"fcycle",
"findex",
"fpathP",
"inE",
"injectivePcycle",
"iter",
"iterD",
"iterSr",
"iter_findex",
"iter_order_cycle",
"mem_head",
"mem_orbit",
"orbit",
"order",
"orderSpred",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_id_cycle x : fcycle f (orbit x) -> order (f x) = order x. | Proof. by move/(orbitPcycle 0 1). Qed. | Lemma | order_id_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fcycle",
"orbit",
"orbitPcycle",
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_spec_cycle x : bool -> Type | :=
| OrderStepCycle of fcycle f (orbit x) & order x = order (f x) :
order_spec_cycle x true
| OrderStepNoCycle of ~~ fcycle f (orbit x) & order x = (order (f x)).+1 :
order_spec_cycle x false. | Inductive | order_spec_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fcycle",
"orbit",
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orderPcycle x : order_spec_cycle x (fcycle f (orbit x)). | Proof.
have [xcycle|Ncycle] := boolP (fcycle f (orbit x)); constructor => //.
by rewrite order_id_cycle.
rewrite /order (eq_card (_ : _ =1 [predU1 x & fconnect f (f x)])).
by move=> y; rewrite !inE fconnect_eqVf eq_sym.
by rewrite cardU1 inE (contraNN (all_iffLR orbitPcycle 2 0)).
Qed. | Lemma | orderPcycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"all_iffLR",
"cardU1",
"eq_card",
"eq_sym",
"fconnect",
"fconnect_eqVf",
"fcycle",
"inE",
"orbit",
"orbitPcycle",
"order",
"order_id_cycle",
"order_spec_cycle",
"predU1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect_f x : fconnect f (f x) x = fcycle f (orbit x). | Proof. by apply/idP/idP => /(orbitPcycle 0 2). Qed. | Lemma | fconnect_f | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"fconnect",
"fcycle",
"orbit",
"orbitPcycle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect_findex x y :
fconnect f x y -> y != x -> findex x y = (findex (f x) y).+1. | Proof.
rewrite /findex fconnect_orbit /orbit -orderSpred /= inE => /orP [-> //|].
rewrite eq_sym; move=> yin /negPf->; have [_ eq_o|_ ->//] := orderPcycle x.
by rewrite -(orderSpred (f x)) trajectSr -cats1 index_cat -eq_o yin.
Qed. | Lemma | fconnect_findex | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"cats1",
"eq_sym",
"fconnect",
"fconnect_orbit",
"findex",
"inE",
"index_cat",
"orbit",
"orderPcycle",
"orderSpred",
"trajectSr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect_id (x : T) : fconnect id x =1 xpred1 x. | Proof. by move=> y; rewrite (@fconnect_cycle _ _ [:: x]) //= ?inE ?eqxx. Qed. | Lemma | fconnect_id | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"eqxx",
"fconnect",
"fconnect_cycle",
"id",
"inE",
"xpred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_id (x : T) : order id x = 1. | Proof. by rewrite /order (eq_card (fconnect_id x)) card1. Qed. | Lemma | order_id | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"card1",
"eq_card",
"fconnect_id",
"id",
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orbit_id (x : T) : orbit id x = [:: x]. | Proof. by rewrite /orbit order_id. Qed. | Lemma | orbit_id | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"id",
"orbit",
"order_id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
froots_id (x : T) : froots id x. | Proof. by rewrite /roots -fconnect_id connect_root. Qed. | Lemma | froots_id | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect_root",
"fconnect_id",
"froots",
"id",
"roots"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
froot_id (x : T) : froot id x = x. | Proof. by apply/eqP; apply: froots_id. Qed. | Lemma | froot_id | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"froot",
"froots_id",
"id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcard_id (a : {pred T}) : fcard id a = #|a|. | Proof. by apply: eq_card => x; rewrite inE froots_id. Qed. | Lemma | fcard_id | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"eq_card",
"fcard",
"froots_id",
"id",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finv_eq_can : cancel f f' -> finv f =1 f'. | Proof.
move=> fK; have inj_f := can_inj fK.
by apply: bij_can_eq fK; [apply: injF_bij | apply: finv_f].
Qed. | Lemma | finv_eq_can | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"fK",
"finv",
"finv_f",
"injF_bij",
"inj_f"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_f : f =1 f'. | Hypothesis | eq_f | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
eq_rf | := eq_frel eq_f. | Let | eq_rf | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"eq_f",
"eq_frel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_fconnect : fconnect f =2 fconnect f'. | Proof. exact: eq_connect eq_rf. Qed. | Lemma | eq_fconnect | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"eq_connect",
"eq_rf",
"fconnect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_fcard : fcard_mem f =1 fcard_mem f'. | Proof. exact: eq_n_comp eq_fconnect. Qed. | Lemma | eq_fcard | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"eq_fconnect",
"eq_n_comp",
"fcard_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_finv : finv f =1 finv f'. | Proof.
by move=> x; rewrite /finv /order (eq_card (@eq_fconnect x)) (eq_iter eq_f).
Qed. | Lemma | eq_finv | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"eq_card",
"eq_f",
"eq_fconnect",
"eq_iter",
"finv",
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_froot : froot f =1 froot f'. | Proof. exact: eq_root eq_rf. Qed. | Lemma | eq_froot | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"eq_rf",
"eq_root",
"froot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_froots : froots f =1 froots f'. | Proof. exact: eq_roots eq_rf. Qed. | Lemma | eq_froots | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"eq_rf",
"eq_roots",
"froots"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finv_inv : finv (finv f) =1 f. | Proof. exact: (finv_eq_can (f_finv injf)). Qed. | Lemma | finv_inv | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"f_finv",
"finv",
"finv_eq_can",
"injf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_finv : order (finv f) =1 order f. | Proof. by move=> x; apply: eq_card (@same_fconnect_finv _ _ injf x). Qed. | Lemma | order_finv | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"eq_card",
"finv",
"injf",
"order",
"same_fconnect_finv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_set_finv n : order_set (finv f) n =i order_set f n. | Proof. by move=> x; rewrite !inE order_finv. Qed. | Lemma | order_set_finv | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"finv",
"inE",
"order_finv",
"order_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(sym_e : connect_sym e) (sym_e' : connect_sym e'). | Hypotheses | sym_e | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect_sym",
"e'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
rel_adjunction_mem m_a | := RelAdjunction {
rel_unit x : in_mem x m_a -> {x' : T' | connect e x (h x')};
rel_functor x' y' :
in_mem (h x') m_a -> connect e' x' y' = connect e (h x') (h y')
}. | Record | rel_adjunction_mem | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"T'",
"connect",
"e'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cl_a : closed e a. | Hypothesis | cl_a | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"closed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
rel_adjunction | := (rel_adjunction_mem (mem a)). | Notation | rel_adjunction | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"rel_adjunction_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intro_adjunction (h' : forall x, x \in a -> T') :
(forall x a_x,
[/\ connect e x (h (h' x a_x))
& forall y a_y, e x y -> connect e' (h' x a_x) (h' y a_y)]) ->
(forall x' a_x,
[/\ connect e' x' (h' (h x') a_x)
& forall y', e' x' y' -> connect e (h x') (h y')]) ->
rel_adjunction. | Proof.
move=> Aee' Ae'e; split=> [y a_y | x' z' a_x].
by exists (h' y a_y); case/Aee': (a_y).
apply/idP/idP=> [/connectP[p e'p ->{z'}] | /connectP[p e_p p_z']].
elim: p x' a_x e'p => //= y' p IHp x' a_x.
case: (Ae'e x' a_x) => _ Ae'x /andP[/Ae'x e_xy /IHp e_yz] {Ae'x}.
by apply: connect_trans (e_yz _); rewrite ... | Lemma | intro_adjunction | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"T'",
"apply",
"cl_a",
"closed_connect",
"connect",
"connectP",
"connect_trans",
"e'",
"rel_adjunction",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
strict_adjunction :
injective h -> a \subset codom h -> rel_base h e e' [predC a] ->
rel_adjunction. | Proof.
move=> /= injh h_a a_ee'; pose h' x Hx := iinv (subsetP h_a x Hx).
apply: (@intro_adjunction h') => [x a_x | x' a_x].
rewrite f_iinv connect0; split=> // y a_y e_xy.
by rewrite connect1 // -a_ee' !f_iinv ?negbK.
rewrite [h' _ _]iinv_f //; split=> // y' e'xy.
by rewrite connect1 // a_ee' ?negbK.
Qed. | Lemma | strict_adjunction | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"codom",
"connect0",
"connect1",
"e'",
"f_iinv",
"iinv",
"iinv_f",
"intro_adjunction",
"rel_adjunction",
"rel_base",
"split",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ccl_a | := closed_connect cl_a. | Let | ccl_a | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"cl_a",
"closed_connect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
adjunction_closed : rel_adjunction -> closed e' [preim h of a]. | Proof.
case=> _ Ae'e; apply: intro_closed => // x' y' /connect1 e'xy a_x.
by rewrite Ae'e // in e'xy; rewrite !inE -(ccl_a e'xy).
Qed. | Lemma | adjunction_closed | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"ccl_a",
"closed",
"connect1",
"e'",
"inE",
"intro_closed",
"rel_adjunction"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
adjunction_n_comp :
rel_adjunction -> n_comp e a = n_comp e' [preim h of a]. | Proof.
case=> Aee' Ae'e.
have inj_h: {in predI (roots e') [preim h of a] &, injective (root e \o h)}.
move=> x' y' /andP[/eqP r_x' /= a_x'] /andP[/eqP r_y' _] /(rootP sym_e).
by rewrite -Ae'e // => /(rootP sym_e'); rewrite r_x' r_y'.
rewrite /n_comp_mem -(card_in_image inj_h); apply: eq_card => x.
apply/andP/imageP... | Lemma | adjunction_n_comp | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"card_in_image",
"ccl_a",
"connect",
"connect_root",
"e'",
"eq_card",
"imageP",
"inE",
"last",
"n_comp",
"n_comp_mem",
"rel_adjunction",
"root",
"rootP",
"roots",
"roots_root",
"sym_e"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rel_adjunction h e e' a | := (rel_adjunction_mem h e e' (mem a)). | Notation | rel_adjunction | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"e'",
"rel_adjunction_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"@ 'rel_adjunction' T T' h e e' a" | :=
(@rel_adjunction_mem T T' h e e' (mem a))
(at level 10, T, T', h, e, e', a at level 8, only parsing) : type_scope. | Notation | @ 'rel_adjunction' T T' h e e' a | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"T'",
"e'",
"rel_adjunction_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fun_adjunction h f f' a | := (rel_adjunction h (frel f) (frel f') a). | Notation | fun_adjunction | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"frel",
"rel_adjunction"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"@ 'fun_adjunction' T T' h f f' a" | :=
(@rel_adjunction T T' h (frel f) (frel f') a)
(at level 10, T, T', h, f, f', a at level 8, only parsing) : type_scope. | Notation | @ 'fun_adjunction' T T' h f f' a | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"T'",
"frel",
"rel_adjunction"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_type : predArgType | := FinSet of {ffun pred T}. | Inductive | set_type | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finfun_of_set A | := let: FinSet f := A in f. | Definition | finfun_of_set | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_of | := set_type. | Definition | set_of | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set_type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_isSub | := Eval hnf in [isNew for finfun_of_set]. | Definition | set_isSub | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"finfun_of_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'set' T }" | := (set_of T) (format "{ 'set' T }") : type_scope. | Notation | { 'set' T } | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"set_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A :=: B" | := (A = B :> {set _})
(at level 70, no associativity, only parsing) : set_scope. | Notation | A :=: B | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | notation to make this technicality less obstructive. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"A :<>: B" | := (A <> B :> {set _})
(at level 70, no associativity, only parsing) : set_scope. | Notation | A :<>: B | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A :==: B" | := (A == B :> {set _})
(at level 70, no associativity, only parsing) : set_scope. | Notation | A :==: B | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A :!=: B" | := (A != B :> {set _})
(at level 70, no associativity, only parsing) : set_scope. | Notation | A :!=: B | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A :=P: B" | := (A =P B :> {set _})
(at level 70, no associativity, only parsing) : set_scope. | Notation | A :=P: B | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finset_unlock | := Unlockable finset.unlock. | Canonical | finset_unlock | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred_of_set_unlock | := Unlockable pred_of_set.unlock. | Canonical | pred_of_set_unlock | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"pred_of_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' x : T | P ]" | := (finset (fun x : T => P%B))
(x at level 99, only parsing) : set_scope. | Notation | [ 'set' x : T | P ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' x | P ]" | := [set x : _ | P]
(P at level 99, format "[ 'set' x | P ]") : set_scope. | Notation | [ 'set' x | P ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' x 'in' A ]" | := [set x | x \in A]
(format "[ 'set' x 'in' A ]") : set_scope. | Notation | [ 'set' x 'in' A ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' x : T 'in' A ]" | := [set x : T | x \in A]
(only parsing) : set_scope. | Notation | [ 'set' x : T 'in' A ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' x : T | P & Q ]" | := [set x : T | P && Q]
(only parsing) : set_scope. | Notation | [ 'set' x : T | P & Q ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' x | P & Q ]" | := [set x | P && Q ]
(P at level 99, format "[ 'set' x | P & Q ]") : set_scope. | Notation | [ 'set' x | P & Q ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' x : T 'in' A | P ]" | := [set x : T | x \in A & P]
(only parsing) : set_scope. | Notation | [ 'set' x : T 'in' A | P ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' x 'in' A | P ]" | := [set x | x \in A & P]
(format "[ 'set' x 'in' A | P ]") : set_scope. | Notation | [ 'set' x 'in' A | P ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' x 'in' A | P & Q ]" | := [set x in A | P && Q]
(format "[ 'set' x 'in' A | P & Q ]") : set_scope. | Notation | [ 'set' x 'in' A | P & Q ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' x : T 'in' A | P & Q ]" | := [set x : T in A | P && Q]
(only parsing) : set_scope. | Notation | [ 'set' x : T 'in' A | P & Q ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' :: s ]" | := (finset [in pred_of_seq s])
(format "[ 'set' :: s ]") : set_scope. | Notation | [ 'set' :: s ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"pred_of_seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred_of_set: set_type >-> fin_pred_sort. | Coercion | pred_of_set | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"fin_pred_sort",
"set_type"
] | collective predicates and as arguments of the \pi(_) notation. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set_predType T | := @PredType _ (unkeyed (set_type T)) (@pred_of_set T). | Canonical | set_predType | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"pred_of_set",
"set_type"
] | coercion to resolve mem A to @mem (predPredType T) (pred_of_set A). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
in_set pA x : (x \in finset pA) = pA x. | Proof. by rewrite [@finset]unlock unlock [x \in _]ffunE. Qed. | Lemma | in_set | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"ffunE",
"pA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setP A B : A =i B <-> A = B. | Proof.
by split=> [eqAB|-> //]; apply/val_inj/ffunP=> x; have:= eqAB x; rewrite unlock.
Qed. | Lemma | setP | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"ffunP",
"split",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
set0 | := [set x : T | false]. | Definition | set0 | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setTfor | := [set x : T | true]. | Definition | setTfor | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_setT x : x \in setTfor. | Proof. by rewrite in_set. Qed. | Lemma | in_setT | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"in_set",
"setTfor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqsVneq A B : eq_xor_neq A B (B == A) (A == B). | Proof. exact: eqVneq. Qed. | Lemma | eqsVneq | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"eqVneq",
"eq_xor_neq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_finset (pA pB : pred T) : pA =1 pB -> finset pA = finset pB. | Proof. by move=> eq_p; apply/setP => x; rewrite !(in_set, inE) eq_p. Qed. | Lemma | eq_finset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"apply",
"inE",
"in_set",
"pA",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' : T ]" | := (setTfor T) (format "[ 'set' : T ]") : set_scope. | Notation | [ 'set' : T ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setTfor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setT | := [set: _] (only parsing). | Notation | setT | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setU A B | := [set x | (x \in A) || (x \in B)]. | Definition | setU | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setI A B | := [set x in A | x \in B]. | Definition | setI | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setC A | := [set x | x \notin A]. | Definition | setC | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setD A B | := [set x | x \notin B & x \in A]. | Definition | setD | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ssetI P D | := [set A in P | A \subset D]. | Definition | ssetI | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
powerset D | := [set A : {set T} | A \subset D]. | Definition | powerset | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' a ]" | := (set1 a)
(a at level 99, format "[ 'set' a ]") : set_scope. | Notation | [ 'set' a ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' a : T ]" | := [set (a : T)]
(a at level 99, format "[ 'set' a : T ]") : set_scope. | Notation | [ 'set' a : T ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A :|: B" | := (setU A B) : set_scope. | Notation | A :|: B | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"a |: A" | := ([set a] :|: A) : set_scope. | Notation | a |: A | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'set' a1 ; a2 ; .. ; an ]" | := (setU .. (a1 |: [set a2]) .. [set an])
(format "[ 'set' a1 ; a2 ; .. ; an ]") : set_scope. | Notation | [ 'set' a1 ; a2 ; .. ; an ] | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"a1",
"a2",
"setU"
] | This is left-associative due to historical limitations of the .. Notation. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"A :&: B" | := (setI A B) : set_scope. | Notation | A :&: B | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"~: A" | := (setC A) (at level 35, right associativity) : set_scope. | Notation | ~: A | boot | boot/finset.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"Monoid"
] | [
"setC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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