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 |
|---|---|---|---|---|---|---|---|---|---|---|
closed_mem m_a | := forall x y, e x y -> in_mem x m_a = in_mem y m_a. | Definition | closed_mem | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
closure_mem m_a : pred T | :=
fun x => ~~ disjoint (mem (connect x)) m_a. | Definition | closure_mem | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect",
"disjoint"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_comp e a | := (n_comp_mem e (mem a)). | Notation | n_comp | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"n_comp_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
closed e a | := (closed_mem e (mem a)). | Notation | closed | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"closed_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
closure e a | := (closure_mem e (mem a)). | Notation | closure | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"closure_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect f | := (connect (coerced_frel f)). | Notation | fconnect | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"coerced_frel",
"connect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
froot f | := (root (coerced_frel f)). | Notation | froot | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"coerced_frel",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
froots f | := (roots (coerced_frel f)). | Notation | froots | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"coerced_frel",
"roots"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcard_mem f | := (n_comp_mem (coerced_frel f)). | Notation | fcard_mem | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"coerced_frel",
"n_comp_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcard f a | := (fcard_mem f (mem a)). | Notation | fcard | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fcard_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fclosed f a | := (closed (coerced_frel f) a). | Notation | fclosed | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"closed",
"coerced_frel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fclosure f a | := (closure (coerced_frel f) a). | Notation | fclosure | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"closure",
"coerced_frel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connect_sub e e' :
subrel e (connect e') -> subrel (connect e) (connect e'). | Proof.
move=> e'e x _ /connectP[p e_p ->]; elim: p x e_p => //= y p IHp x /andP[exy].
by move/IHp; apply: connect_trans; apply: e'e.
Qed. | Lemma | connect_sub | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect",
"connectP",
"connect_trans",
"e'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
relU_sym e e' :
connect_sym e -> connect_sym e' -> connect_sym (relU e e'). | Proof.
move=> sym_e sym_e'; apply: symmetric_from_pre => x _ /connectP[p e_p ->].
elim: p x e_p => //= y p IHp x /andP[e_xy /IHp{IHp}/connect_trans]; apply.
case/orP: e_xy => /connect1; rewrite (sym_e, sym_e');
by apply: connect_sub y x => x y e_xy; rewrite connect1 //= e_xy ?orbT.
Qed. | Lemma | relU_sym | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect1",
"connectP",
"connect_sub",
"connect_sym",
"connect_trans",
"e'",
"sym_e"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_connect e e' : e =2 e' -> connect e =2 connect e'. | Proof.
move=> eq_e x y; apply/connectP/connectP=> [] [p e_p ->];
by exists p; rewrite // (eq_path eq_e) in e_p *.
Qed. | Lemma | eq_connect | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect",
"connectP",
"e'",
"eq_path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_n_comp e e' : connect e =2 connect e' -> n_comp_mem e =1 n_comp_mem e'. | Proof.
move=> eq_e [a]; apply: eq_card => x /=.
by rewrite !inE /= /roots /root /= (eq_pick (eq_e x)).
Qed. | Lemma | eq_n_comp | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect",
"e'",
"eq_card",
"eq_pick",
"inE",
"n_comp_mem",
"root",
"roots"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_n_comp_r {e} a a' : a =i a' -> n_comp e a = n_comp e a'. | Proof. by move=> eq_a; apply: eq_card => x; rewrite inE /= eq_a. Qed. | Lemma | eq_n_comp_r | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"eq_card",
"inE",
"n_comp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_compC a e : n_comp e T = n_comp e a + n_comp e [predC a]. | Proof.
rewrite /n_comp_mem (eq_card (fun _ => andbT _)) -(cardID a); congr (_ + _).
by apply: eq_card => x; rewrite !inE andbC.
Qed. | Lemma | n_compC | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"cardID",
"eq_card",
"inE",
"n_comp",
"n_comp_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_root e e' : e =2 e' -> root e =1 root e'. | Proof. by move=> eq_e x; rewrite /root (eq_pick (eq_connect eq_e x)). Qed. | Lemma | eq_root | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"e'",
"eq_connect",
"eq_pick",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_roots e e' : e =2 e' -> roots e =1 roots e'. | Proof. by move=> eq_e x; rewrite /roots (eq_root eq_e). Qed. | Lemma | eq_roots | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"e'",
"eq_root",
"roots"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connect_rev e : connect [rel x y | e y x] =2 [rel x y | connect e y x]. | Proof.
suff crev e': subrel (connect [rel x y | e' y x]) [rel x y | connect e' y x].
by move=> x y; apply/idP/idP; apply: crev.
move=> x y /connectP[p e_p p_y]; apply/connectP.
exists (rev (belast x p)); first by rewrite p_y rev_path.
by rewrite -(last_cons x) -rev_rcons p_y -lastI rev_cons last_rcons.
Qed. | Lemma | connect_rev | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"belast",
"connect",
"connectP",
"e'",
"lastI",
"last_cons",
"last_rcons",
"rel",
"rev",
"rev_cons",
"rev_path",
"rev_rcons"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sym_connect_sym e : symmetric e -> connect_sym e. | Proof. by move=> sym_e x y; rewrite (eq_connect sym_e) connect_rev. Qed. | Lemma | sym_connect_sym | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect_rev",
"connect_sym",
"eq_connect",
"sym_e"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sym_e : connect_sym e. | Hypothesis | sym_e | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
same_connect_rev : connect e =2 connect [rel x y | e y x]. | Proof. by move=> x y; rewrite sym_e connect_rev. Qed. | Lemma | same_connect_rev | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect",
"connect_rev",
"rel",
"sym_e"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intro_closed a : (forall x y, e x y -> x \in a -> y \in a) -> closed e a. | Proof.
move=> cl_a x y e_xy; apply/idP/idP=> [|a_y]; first exact: cl_a.
have{x e_xy} /connectP[p e_p ->]: connect e y x by rewrite sym_e connect1.
by elim: p y a_y e_p => //= y p IHp x a_x /andP[/cl_a/(_ a_x)]; apply: IHp.
Qed. | Lemma | intro_closed | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"cl_a",
"closed",
"connect",
"connect1",
"connectP",
"sym_e"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
closed_connect a :
closed e a -> forall x y, connect e x y -> (x \in a) = (y \in a). | Proof.
move=> cl_a x _ /connectP[p e_p ->].
by elim: p x e_p => //= y p IHp x /andP[/cl_a->]; apply: IHp.
Qed. | Lemma | closed_connect | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"cl_a",
"closed",
"connect",
"connectP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connect_closed x : closed e (connect e x). | Proof. by move=> y z /connect1/same_connect_r; apply. Qed. | Lemma | connect_closed | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"closed",
"connect",
"connect1",
"same_connect_r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predC_closed a : closed e a -> closed e [predC a]. | Proof. by move=> cl_a x y /cl_a /[!inE] ->. Qed. | Lemma | predC_closed | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"cl_a",
"closed",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
closure_closed a : closed e (closure e a). | Proof.
apply: intro_closed => x y /connect1 e_xy; congr (~~ _).
by apply: eq_disjoint; apply: same_connect.
Qed. | Lemma | closure_closed | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"closed",
"closure",
"connect1",
"eq_disjoint",
"intro_closed",
"same_connect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_closure a : {subset a <= closure e a}. | Proof. by move=> x a_x; apply/existsP; exists x; rewrite !inE connect0. Qed. | Lemma | mem_closure | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"closure",
"connect0",
"existsP",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_closure a : a \subset closure e a. | Proof. by apply/subsetP; apply: mem_closure. Qed. | Lemma | subset_closure | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"closure",
"mem_closure",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_comp_closure2 x y :
n_comp e (closure e (pred2 x y)) = (~~ connect e x y).+1. | Proof.
rewrite -(root_connect sym_e) -card2; apply: eq_card => z.
apply/idP/idP=> [/andP[/eqP {2}<- /pred0Pn[t /andP[/= ezt exyt]]] |].
by case/pred2P: exyt => <-; rewrite (rootP sym_e ezt) !inE eqxx ?orbT.
by case/pred2P=> ->; rewrite !inE roots_root //; apply/existsP;
[exists x | exists y]; rewrite !inE eqxx ?orb... | Lemma | n_comp_closure2 | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"card2",
"closure",
"connect",
"connect_root",
"eq_card",
"eqxx",
"existsP",
"inE",
"n_comp",
"pred0Pn",
"pred2",
"pred2P",
"rootP",
"root_connect",
"roots_root",
"sym_e"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_comp_connect x : n_comp e (connect e x) = 1. | Proof.
rewrite -(card1 (root e x)); apply: eq_card => y.
apply/andP/eqP => [[/eqP r_y /rootP-> //] | ->] /=.
by rewrite inE connect_root roots_root.
Qed. | Lemma | n_comp_connect | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"card1",
"connect",
"connect_root",
"eq_card",
"inE",
"n_comp",
"root",
"rootP",
"roots_root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order x | := #|fconnect f x|. | Definition | order | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fconnect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orbit x | := traject f x (order x). | Definition | orbit | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"order",
"traject"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
findex x y | := index y (orbit x). | Definition | findex | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"index",
"orbit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finv x | := iter (order x).-1 f x. | Definition | finv | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"iter",
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect_iter n x : fconnect f x (iter n f x). | Proof.
apply/connectP.
by exists (traject f (f x) n); [apply: fpath_traject | rewrite last_traject].
Qed. | Lemma | fconnect_iter | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connectP",
"fconnect",
"fpath_traject",
"iter",
"last_traject",
"traject"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect1 x : fconnect f x (f x). | Proof. exact: (fconnect_iter 1). Qed. | Lemma | fconnect1 | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fconnect",
"fconnect_iter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect_finv x : fconnect f x (finv x). | Proof. exact: fconnect_iter. Qed. | Lemma | fconnect_finv | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fconnect",
"fconnect_iter",
"finv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orderSpred x : (order x).-1.+1 = order x. | Proof. by rewrite /order (cardD1 x) [_ x _]connect0. Qed. | Lemma | orderSpred | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"cardD1",
"connect0",
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_orbit x : size (orbit x) = order x. | Proof. exact: size_traject. Qed. | Lemma | size_orbit | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"orbit",
"order",
"size",
"size_traject"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
looping_order x : looping f x (order x). | Proof.
apply: contraFT (ltnn (order x)); rewrite -looping_uniq => /card_uniqP.
rewrite size_traject => <-; apply: subset_leq_card.
by apply/subsetP=> _ /trajectP[i _ ->]; apply: fconnect_iter.
Qed. | Lemma | looping_order | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"card_uniqP",
"fconnect_iter",
"looping",
"looping_uniq",
"ltnn",
"order",
"size_traject",
"subsetP",
"subset_leq_card",
"trajectP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect_orbit x y : fconnect f x y = (y \in orbit x). | Proof.
apply/idP/idP=> [/connectP[_ /fpathP[m ->] ->] | /trajectP[i _ ->]].
by rewrite last_traject; apply/loopingP/looping_order.
exact: fconnect_iter.
Qed. | Lemma | fconnect_orbit | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connectP",
"fconnect",
"fconnect_iter",
"fpathP",
"last_traject",
"loopingP",
"looping_order",
"orbit",
"trajectP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_orbit x : x \in orbit x. | Proof. by rewrite -fconnect_orbit. Qed. | Lemma | in_orbit | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fconnect_orbit",
"orbit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_gt0 x : order x > 0. | Proof. by rewrite -orderSpred. Qed. | Lemma | order_gt0 | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"order",
"orderSpred"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orbit_uniq x : uniq (orbit x). | Proof.
rewrite /orbit -orderSpred looping_uniq; set n := (order x).-1.
apply: contraFN (ltnn n) => /trajectP[i lt_i_n eq_fnx_fix].
rewrite orderSpred -(size_traject f x n).
apply: (leq_trans (subset_leq_card _) (card_size _)); apply/subsetP=> z.
rewrite inE fconnect_orbit => /trajectP[j le_jn ->{z}].
rewrite -orderSpre... | Lemma | orbit_uniq | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"card_size",
"fconnect_orbit",
"inE",
"leq_eqVlt",
"leq_trans",
"looping_uniq",
"ltnS",
"ltnn",
"orbit",
"order",
"orderSpred",
"predU1P",
"size_traject",
"subsetP",
"subset_leq_card",
"trajectP",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
findex_max x y : fconnect f x y -> findex x y < order x. | Proof. by rewrite [_ y]fconnect_orbit -index_mem size_orbit. Qed. | Lemma | findex_max | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fconnect",
"fconnect_orbit",
"findex",
"index_mem",
"order",
"size_orbit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
findex_iter x i : i < order x -> findex x (iter i f x) = i. | Proof.
move=> lt_ix; rewrite -(nth_traject f lt_ix) /findex index_uniq ?orbit_uniq //.
by rewrite size_orbit.
Qed. | Lemma | findex_iter | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"findex",
"index_uniq",
"iter",
"nth_traject",
"orbit_uniq",
"order",
"size_orbit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_findex x y : fconnect f x y -> iter (findex x y) f x = y. | Proof.
rewrite [_ y]fconnect_orbit => fxy; pose i := index y (orbit x).
have lt_ix: i < order x by rewrite -size_orbit index_mem.
by rewrite -(nth_traject f lt_ix) nth_index.
Qed. | Lemma | iter_findex | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fconnect",
"fconnect_orbit",
"findex",
"index",
"index_mem",
"iter",
"nth_index",
"nth_traject",
"orbit",
"order",
"size_orbit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
findex0 x : findex x x = 0. | Proof. by rewrite /findex /orbit -orderSpred /= eqxx. Qed. | Lemma | findex0 | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"eqxx",
"findex",
"orbit",
"orderSpred"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
findex_eq0 x y : (findex x y == 0) = (x == y). | Proof. by rewrite /findex /orbit -orderSpred /=; case: (x == y). Qed. | Lemma | findex_eq0 | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"findex",
"orbit",
"orderSpred"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect_invariant (T' : eqType) (k : T -> T') :
invariant f k =1 xpredT -> forall x y, fconnect f x y -> k x = k y. | Proof.
move=> eq_k_f x y /iter_findex <-; elim: {y}(findex x y) => //= n ->.
by rewrite (eqP (eq_k_f _)).
Qed. | Lemma | fconnect_invariant | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"T'",
"fconnect",
"findex",
"invariant",
"iter_findex"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_orbit x : {homo f : y / y \in orbit x}. | Proof.
by move=> y; rewrite -!fconnect_orbit => /connect_trans->//; apply: fconnect1.
Qed. | Lemma | mem_orbit | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect_trans",
"fconnect1",
"fconnect_orbit",
"orbit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
image_orbit x : {subset image f (orbit x) <= orbit x}. | Proof.
by move=> _ /mapP[y yin ->]; apply: mem_orbit; rewrite ?mem_enum in yin.
Qed. | Lemma | image_orbit | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"image",
"mapP",
"mem_enum",
"mem_orbit",
"orbit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
f_in : {homo f : x / x \in S}. | Hypothesis | f_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
injf : {in S &, injective f}. | Hypothesis | injf | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
finv_in : {homo finv : x / x \in S}. | Proof. by move=> x xS; rewrite iter_in. Qed. | Lemma | finv_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"finv",
"iter_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
f_finv_in : {in S, cancel finv f}. | Proof.
move=> x xS; move: (looping_order x) (orbit_uniq x).
rewrite /looping /orbit -orderSpred looping_uniq /= /looping; set n := _.-1.
case/predU1P=> // /trajectP[i lt_i_n]; rewrite -iterSr.
by move=> /injf ->; rewrite ?(iter_in _ f_in) //; case/trajectP; exists i.
Qed. | Lemma | f_finv_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"f_in",
"finv",
"injf",
"iterSr",
"iter_in",
"looping",
"looping_order",
"looping_uniq",
"orbit",
"orbit_uniq",
"orderSpred",
"predU1P",
"trajectP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finv_f_in : {in S, cancel f finv}. | Proof. by move=> x xS; apply/injf; rewrite ?iter_in ?f_finv_in ?f_in. Qed. | Lemma | finv_f_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"f_finv_in",
"f_in",
"finv",
"injf",
"iter_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finv_inj_in : {in S &, injective finv}. | Proof. by move=> x y xS yS q; rewrite -(f_finv_in xS) q f_finv_in. Qed. | Lemma | finv_inj_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"f_finv_in",
"finv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect_sym_in : {in S &, forall x y, fconnect f x y = fconnect f y x}. | Proof.
suff Sf : {in S &, forall x y, fconnect f x y -> fconnect f y x}.
by move=> *; apply/idP/idP=> /Sf->.
move=> x _ xS _ /connectP [p f_p ->]; elim: p => //= y p IHp in x xS f_p *.
case/andP: f_p => /eqP <- /(IHp _ (f_in xS)) /connect_trans -> //.
by apply: (connect_trans (fconnect_finv _)); rewrite finv_f_in.
Qe... | Lemma | fconnect_sym_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connectP",
"connect_trans",
"f_in",
"fconnect",
"fconnect_finv",
"finv_f_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_order_in : {in S, forall x, iter (order x) f x = x}. | Proof. by move=> x xS; rewrite -orderSpred iterS; apply: f_finv_in. Qed. | Lemma | iter_order_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"f_finv_in",
"iter",
"iterS",
"order",
"orderSpred"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_finv_in n :
{in S, forall x, n <= order x -> iter n finv x = iter (order x - n) f x}. | Proof.
move=> x xS; rewrite -[x in LHS]iter_order_in => // /subnKC {1}<-.
move: (_ - n) => m; rewrite iterD; elim: n => // n {2}<-.
by rewrite iterSr /= finv_f_in // -iterD iter_in.
Qed. | Lemma | iter_finv_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"finv",
"finv_f_in",
"iter",
"iterD",
"iterSr",
"iter_in",
"iter_order_in",
"order",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cycle_orbit_in : {in S, forall x, (fcycle f) (orbit x)}. | Proof.
move=> x xS; rewrite /orbit -orderSpred (cycle_path x) /= last_traject.
by rewrite -/(finv x) fpath_traject f_finv_in ?eqxx.
Qed. | Lemma | cycle_orbit_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"cycle_path",
"eqxx",
"f_finv_in",
"fcycle",
"finv",
"fpath_traject",
"last_traject",
"orbit",
"orderSpred"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fpath_finv_in p x :
(x \in S) && (fpath finv x p) =
(last x p \in S) && (fpath f (last x p) (rev (belast x p))). | Proof.
elim: p x => //= y p IHp x; rewrite rev_cons rcons_path.
transitivity [&& y \in S, f y == x & fpath finv y p].
apply/and3P/and3P => -[xS /eqP<- fxp]; split;
by rewrite ?f_finv_in ?finv_f_in ?finv_in ?f_in.
rewrite andbCA {}IHp !andbA [RHS]andbC -andbA; congr [&& _, _ & _].
by case: p => //= z p; rewrite rev_... | Lemma | fpath_finv_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"belast",
"f_finv_in",
"f_in",
"finv",
"finv_f_in",
"finv_in",
"fpath",
"last",
"last_rcons",
"rcons_path",
"rev",
"rev_cons",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fpath_finv_f_in p : {in S, forall x,
fpath finv x p -> fpath f (last x p) (rev (belast x p))}. | Proof. by move=> x xS /(conj xS)/andP; rewrite fpath_finv_in => /andP[]. Qed. | Lemma | fpath_finv_f_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"belast",
"conj",
"finv",
"fpath",
"fpath_finv_in",
"last",
"rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fpath_f_finv_in p x : last x p \in S ->
fpath f (last x p) (rev (belast x p)) -> fpath finv x p. | Proof. by move=> lS /(conj lS)/andP; rewrite -fpath_finv_in => /andP[]. Qed. | Lemma | fpath_f_finv_in | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"belast",
"conj",
"finv",
"fpath",
"fpath_finv_in",
"last",
"rev"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injectivePcycle x :
reflect {in orbit x &, injective f} (fcycle f (orbit x)). | Proof.
apply: (iffP idP) => [/inj_cycle//|/cycle_orbit_in].
by apply; [apply: mem_orbit|apply: in_orbit].
Qed. | Lemma | injectivePcycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"cycle_orbit_in",
"fcycle",
"in_orbit",
"inj_cycle",
"mem_orbit",
"orbit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injf : injective f. | Hypothesis | injf | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
f_finv : cancel finv f. | Proof. exact: (in1T (f_finv_in _ (in2W _))). Qed. | Lemma | f_finv | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"f_finv_in",
"finv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finv_f : cancel f finv. | Proof. exact: (in1T (finv_f_in _ (in2W _))). Qed. | Lemma | finv_f | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"finv",
"finv_f_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finv_bij : bijective finv. | Proof. by exists f; [apply: f_finv|apply: finv_f]. Qed. | Lemma | finv_bij | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"f_finv",
"finv",
"finv_f"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finv_inj : injective finv. | Proof. exact: (can_inj f_finv). Qed. | Lemma | finv_inj | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"f_finv",
"finv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect_sym x y : fconnect f x y = fconnect f y x. | Proof. exact: (in2T (fconnect_sym_in _ (in2W _))). Qed. | Lemma | fconnect_sym | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fconnect",
"fconnect_sym_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
symf | := fconnect_sym. | Let | symf | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"fconnect_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_order x : iter (order x) f x = x. | Proof. exact: (in1T (iter_order_in _ (in2W _))). Qed. | Lemma | iter_order | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"iter",
"iter_order_in",
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iter_finv n x : n <= order x -> iter n finv x = iter (order x - n) f x. | Proof. exact: (in1T (@iter_finv_in _ _ (in2W _) _)). Qed. | Lemma | iter_finv | 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 x : fcycle f (orbit x). | Proof. exact: (in1T (cycle_orbit_in _ (in2W _))). Qed. | Lemma | cycle_orbit | 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 x p : fpath finv x p = fpath f (last x p) (rev (belast x p)). | Proof. exact: (@fpath_finv_in T _ (in2W _)). Qed. | Lemma | fpath_finv | 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 | |
same_fconnect_finv : fconnect finv =2 fconnect f. | Proof.
move=> x y; rewrite (same_connect_rev symf); apply: {x y}eq_connect => x y /=.
by rewrite (canF_eq finv_f) eq_sym.
Qed. | Lemma | same_fconnect_finv | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"canF_eq",
"eq_connect",
"eq_sym",
"fconnect",
"finv",
"finv_f",
"same_connect_rev",
"symf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcard_finv : fcard_mem finv =1 fcard_mem f. | Proof. exact: eq_n_comp same_fconnect_finv. Qed. | Lemma | fcard_finv | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"eq_n_comp",
"fcard_mem",
"finv",
"same_fconnect_finv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
order_set n : pred T | := [pred x | order x == n]. | Definition | order_set | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"order"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcard_order_set n (a : {pred T}) :
a \subset order_set n -> fclosed f a -> fcard f a * n = #|a|. | Proof.
move=> a_n cl_a; rewrite /n_comp_mem; set b := [predI froots f & a].
suff <-: #|preim (froot f) b| = #|b| * n.
apply: eq_card => x; rewrite !inE (roots_root fconnect_sym).
exact/esym/(closed_connect cl_a)/connect_root.
have{cl_a a_n} (x): b x -> froot f x = x /\ order x = n.
by case/andP=> /eqP-> /(subsetP... | Lemma | fcard_order_set | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"cardD1",
"cardID",
"cl_a",
"closed_connect",
"connect_root",
"eqSS",
"eq_card",
"eq_card0",
"eqnP",
"eqxx",
"fcard",
"fclosed",
"fconnect",
"fconnect_sym",
"froot",
"froots",
"inE",
"last",
"mulSn",
"n_comp_mem",
"order",
"order_set",
"pickP",
"pred0P",
"r... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fclosed1 (a : {pred T}) :
fclosed f a -> forall x, (x \in a) = (f x \in a). | Proof. by move=> cl_a x; apply: cl_a (eqxx _). Qed. | Lemma | fclosed1 | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"cl_a",
"eqxx",
"fclosed"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
same_fconnect1 x : fconnect f x =1 fconnect f (f x). | Proof. by apply: same_connect1 => /=. Qed. | Lemma | same_fconnect1 | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"fconnect",
"same_connect1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
same_fconnect1_r x y : fconnect f x y = fconnect f x (f y). | Proof. by apply: same_connect1r x => /=. Qed. | Lemma | same_fconnect1_r | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"fconnect",
"same_connect1r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcard_gt0P (a : {pred T}) :
fclosed f a -> reflect (exists x, x \in a) (0 < fcard f a). | Proof.
move=> clfA; apply: (iffP card_gt0P) => [[x /andP[]]|[x xA]]; first by exists x.
exists (froot f x); rewrite inE roots_root /=; first exact: fconnect_sym.
by rewrite -(closed_connect clfA (connect_root _ x)).
Qed. | Lemma | fcard_gt0P | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"card_gt0P",
"closed_connect",
"connect_root",
"fcard",
"fclosed",
"fconnect_sym",
"froot",
"inE",
"roots_root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcard_gt1P (A : {pred T}) :
fclosed f A ->
reflect (exists2 x, x \in A & exists2 y, y \in A & ~~ fconnect f x y)
(1 < fcard f A). | Proof.
move=> clAf; apply: (iffP card_gt1P) => [|[x xA [y yA not_xfy]]].
move=> [x [y [/andP [/= rfx xA] /andP[/= rfy yA] xDy]]].
by exists x; try exists y; rewrite // -root_connect // (eqP rfx) (eqP rfy).
exists (froot f x), (froot f y); rewrite !inE !roots_root ?root_connect //=.
by split => //; rewrite -(closed_... | Lemma | fcard_gt1P | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"card_gt1P",
"closed_connect",
"connect_root",
"fcard",
"fclosed",
"fconnect",
"froot",
"inE",
"root_connect",
"roots_root",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fconnect_cycle y : fconnect f x y = (y \in p). | Proof.
have [i q def_p] := rot_to p_x; rewrite -(mem_rot i p) def_p.
have{i def_p} /andP[/eqP q_x f_q]: (f (last x q) == x) && fpath f x q.
by have:= f_p; rewrite -(rot_cycle i) def_p (cycle_path x).
apply/idP/idP=> [/connectP[_ /fpathP[j ->] ->] | ]; last exact: path_connect.
case/fpathP: f_q q_x => n ->; rewrite !l... | Lemma | fconnect_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connectP",
"cycle_path",
"def_p",
"fconnect",
"fpath",
"fpathP",
"iterS",
"last",
"last_traject",
"looping",
"loopingP",
"mem_head",
"mem_rot",
"path_connect",
"rot_cycle",
"rot_to"
] | fconnect_cycle does not dependent on Up | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
order_le_cycle : order x <= size p. | Proof.
apply: leq_trans (card_size _); apply/subset_leq_card/subsetP=> y.
by rewrite !(fconnect_cycle, inE) ?eqxx.
Qed. | Lemma | order_le_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"card_size",
"eqxx",
"fconnect_cycle",
"inE",
"leq_trans",
"order",
"size",
"subsetP",
"subset_leq_card"
] | order_le_cycle does not dependent on Up | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
order_cycle : order x = size p. | Proof. by rewrite -(card_uniqP Up); apply: (eq_card fconnect_cycle). Qed. | Lemma | order_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"Up",
"apply",
"card_uniqP",
"eq_card",
"fconnect_cycle",
"order",
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orbitE : orbit x = rot (index x p) p. | Proof.
set i := index _ _; rewrite /orbit order_cycle -(size_rot i) rot_index// -/i.
set q := _ ++ _; suffices /fpathP[j ->]: fpath f x q by rewrite /= size_traject.
by move: f_p; rewrite -(rot_cycle i) rot_index// (cycle_path x); case/andP.
Qed. | Lemma | orbitE | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"cycle_path",
"fpath",
"fpathP",
"index",
"orbit",
"order_cycle",
"rot",
"rot_cycle",
"rot_index",
"size_rot",
"size_traject"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orbit_rot_cycle : {i : nat | orbit x = rot i p}. | Proof. by rewrite orbitE; exists (index x p). Qed. | Lemma | orbit_rot_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"index",
"nat",
"orbit",
"orbitE",
"rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
f_inj | := inj_cycle f_p. | Let | f_inj | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"inj_cycle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homo_f | := mem_fcycle f_p. | Let | homo_f | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"mem_fcycle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finv_cycle : {homo finv : x / x \in p}. | Proof. exact: finv_in. Qed. | Lemma | finv_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"finv",
"finv_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
f_finv_cycle : {in p, cancel finv f}. | Proof. exact: f_finv_in. Qed. | Lemma | f_finv_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"f_finv_in",
"finv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finv_f_cycle : {in p, cancel f finv}. | Proof. exact: finv_f_in. Qed. | Lemma | finv_f_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"finv",
"finv_f_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finv_inj_cycle : {in p &, injective finv}. | Proof. exact: finv_inj_in. Qed. | Lemma | finv_inj_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"finv",
"finv_inj_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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