fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
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intro_closeda : (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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | intro_closed | |
closed_connecta :
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | closed_connect | |
connect_closedx : closed e (connect e x).
Proof. by move=> y z /connect1/same_connect_r; apply. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | connect_closed | |
predC_closeda : closed e a -> closed e [predC a].
Proof. by move=> cl_a x y /cl_a /[!inE] ->. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | predC_closed | |
closure_closeda : closed e (closure e a).
Proof.
apply: intro_closed => x y /connect1 e_xy; congr (~~ _).
by apply: eq_disjoint; apply: same_connect.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | closure_closed | |
mem_closurea : {subset a <= closure e a}.
Proof. by move=> x a_x; apply/existsP; exists x; rewrite !inE connect0. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | mem_closure | |
subset_closurea : a \subset closure e a.
Proof. by apply/subsetP; apply: mem_closure. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | subset_closure | |
n_comp_closure2x 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 ?orbT sym_e connect_root.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | n_comp_closure2 | |
n_comp_connectx : 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | n_comp_connect | |
orderx := #|fconnect f x|. | Definition | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | order | |
orbitx := traject f x (order x). | Definition | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | orbit | |
findexx y := index y (orbit x). | Definition | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | findex | |
finvx := iter (order x).-1 f x. | Definition | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv | |
fconnect_itern 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect_iter | |
fconnect1x : fconnect f x (f x).
Proof. exact: (fconnect_iter 1). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect1 | |
fconnect_finvx : fconnect f x (finv x).
Proof. exact: fconnect_iter. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect_finv | |
orderSpredx : (order x).-1.+1 = order x.
Proof. by rewrite /order (cardD1 x) [_ x _]connect0. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | orderSpred | |
size_orbitx : size (orbit x) = order x.
Proof. exact: size_traject. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | size_orbit | |
looping_orderx : 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | looping_order | |
fconnect_orbitx 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect_orbit | |
in_orbitx : x \in orbit x. Proof. by rewrite -fconnect_orbit. Qed.
Hint Resolve in_orbit : core. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | in_orbit | |
order_gt0x : order x > 0. Proof. by rewrite -orderSpred. Qed.
Hint Resolve order_gt0 : core. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | order_gt0 | |
orbit_uniqx : 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 -orderSpred -/n ltnS leq_eqVlt in le_jn.
by apply/trajectP; case/predU1P: le_jn => [->|]; [exists i | exists j].
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | orbit_uniq | |
findex_maxx y : fconnect f x y -> findex x y < order x.
Proof. by rewrite [_ y]fconnect_orbit -index_mem size_orbit. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | findex_max | |
findex_iterx 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | findex_iter | |
iter_findexx 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | iter_findex | |
findex0x : findex x x = 0.
Proof. by rewrite /findex /orbit -orderSpred /= eqxx. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | findex0 | |
findex_eq0x y : (findex x y == 0) = (x == y).
Proof. by rewrite /findex /orbit -orderSpred /=; case: (x == y). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | findex_eq0 | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect_invariant | |
mem_orbitx : {homo f : y / y \in orbit x}.
Proof.
by move=> y; rewrite -!fconnect_orbit => /connect_trans->//; apply: fconnect1.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | mem_orbit | |
image_orbitx : {subset image f (orbit x) <= orbit x}.
Proof.
by move=> _ /mapP[y yin ->]; apply: mem_orbit; rewrite ?mem_enum in yin.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | image_orbit | |
finv_in: {homo finv : x / x \in S}.
Proof. by move=> x xS; rewrite iter_in. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv_in | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | f_finv_in | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv_f_in | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv_inj_in | |
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.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect_sym_in | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | iter_order_in | |
iter_finv_inn :
{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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | iter_finv_in | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | cycle_orbit_in | |
fpath_finv_inp 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_cons last_rcons.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fpath_finv_in | |
fpath_finv_f_inp : {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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fpath_finv_f_in | |
fpath_f_finv_inp 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fpath_f_finv_in | |
injectivePcyclex :
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | injectivePcycle | |
f_finv: cancel finv f. Proof. exact: (in1T (f_finv_in _ (in2W _))). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | f_finv | |
finv_f: cancel f finv. Proof. exact: (in1T (finv_f_in _ (in2W _))). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv_f | |
finv_bij: bijective finv.
Proof. by exists f; [apply: f_finv|apply: finv_f]. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv_bij | |
finv_inj: injective finv. Proof. exact: (can_inj f_finv). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv_inj | |
fconnect_symx y : fconnect f x y = fconnect f y x.
Proof. exact: (in2T (fconnect_sym_in _ (in2W _))). Qed.
Let symf := fconnect_sym. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect_sym | |
iter_orderx : iter (order x) f x = x.
Proof. exact: (in1T (iter_order_in _ (in2W _))). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | iter_order | |
iter_finvn x : n <= order x -> iter n finv x = iter (order x - n) f x.
Proof. exact: (in1T (@iter_finv_in _ _ (in2W _) _)). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | iter_finv | |
cycle_orbitx : fcycle f (orbit x).
Proof. exact: (in1T (cycle_orbit_in _ (in2W _))). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | cycle_orbit | |
fpath_finvx p : fpath finv x p = fpath f (last x p) (rev (belast x p)).
Proof. exact: (@fpath_finv_in T _ (in2W _)). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fpath_finv | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | same_fconnect_finv | |
fcard_finv: fcard_mem finv =1 fcard_mem f.
Proof. exact: eq_n_comp same_fconnect_finv. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fcard_finv | |
order_setn : pred T := [pred x | order x == n]. | Definition | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | order_set | |
fcard_order_setn (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 a_n)/eqnP->.
elim: {a b}#|b| {1 3 4}b (eqxx #|b|) => [|m IHm] b def_m f_b.
by rewrite eq_card0 // => x; apply: (pred0P def_m).
have [x b_x | b0] := pickP b; last by rewrite (eq_card0 b0) in def_m.
have [r_x ox_n] := f_b x b_x; rewrite (cardD1 x) [x \in b]b_x eqSS in def_m.
rewrite mulSn -{1}ox_n -(IHm _ def_m) => [|_ /andP[_ /f_b //]].
rewrite -(cardID (fconnect f x)); congr (_ + _); apply: eq_card => y.
by apply: andb_idl => /= fxy; rewrite !inE -(rootP symf fxy) r_x.
by congr (~~ _ && _); rewrite /= /in_mem /= symf -(root_connect symf) r_x.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fcard_order_set | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fclosed1 | |
same_fconnect1x : fconnect f x =1 fconnect f (f x).
Proof. by apply: same_connect1 => /=. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | same_fconnect1 | |
same_fconnect1_rx y : fconnect f x y = fconnect f x (f y).
Proof. by apply: same_connect1r x => /=. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | same_fconnect1_r | |
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 /=; last exact: fconnect_sym.
by rewrite -(closed_connect clfA (connect_root _ x)).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fcard_gt0P | |
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_connect clAf (connect_root _ _)).
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fcard_gt1P | |
fconnect_cycley : 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 !last_traject -iterS => def_x.
by apply: (@loopingP _ f x n.+1); rewrite /looping def_x /= mem_head.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect_cycle | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | order_le_cycle | |
order_cycle: order x = size p.
Proof. by rewrite -(card_uniqP Up); apply: (eq_card fconnect_cycle). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | order_cycle | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | orbitE | |
orbit_rot_cycle: {i : nat | orbit x = rot i p}.
Proof. by rewrite orbitE; exists (index x p). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | orbit_rot_cycle | |
finv_cycle: {homo finv : x / x \in p}. Proof. exact: finv_in. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv_cycle | |
f_finv_cycle: {in p, cancel finv f}. Proof. exact: f_finv_in. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | f_finv_cycle | |
finv_f_cycle: {in p, cancel f finv}. Proof. exact: finv_f_in. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv_f_cycle | |
finv_inj_cycle: {in p &, injective finv}. Proof. exact: finv_inj_in. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv_inj_cycle | |
iter_finv_cyclen :
{in p, forall x, n <= order x -> iter n finv x = iter (order x - n) f x}.
Proof. exact: iter_finv_in. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | iter_finv_cycle | |
cycle_orbit_cycle: {in p, forall x, fcycle f (orbit x)}.
Proof. exact: cycle_orbit_in. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | cycle_orbit_cycle | |
fpath_finv_cycleq 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fpath_finv_cycle | |
fpath_finv_f_cycleq : {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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fpath_finv_f_cycle | |
fpath_f_finv_cycleq 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fpath_f_finv_cycle | |
prevEx : 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | prevE | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fcycle_rconsE | |
fcycle_consE: x :: p = traject f x (size p).+1.
Proof. by have := fcycle_rconsE; rewrite trajectSr => /rcons_inj[/= <-]. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fcycle_consE | |
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 (mem_fcycle f_p) (inj_cycle f_p)) ?mem_head//.
set m := (_ - _) => cycle_cat.
have [||k->] := IHn m; last by exists k.+1.
by rewrite ltn_subrL (leq_trans _ n_ge) ?order_gt0.
move: cycle_cat; rewrite -orderSpred/= rcons_cat rcons_cons -cat_rcons.
by rewrite cat_path last_rcons => /andP[].
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fcycle_consEflatten | |
undup_cycle_cons: undup (x :: p) = orbit x.
Proof.
by have [n {1}->] := fcycle_consEflatten; rewrite undup_flatten_nseq ?undup_id.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | undup_cycle_cons | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fcycleEflatten | |
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.
Let p_undup_uniq := undup_uniq p.
Let f_inj := inj_cycle f_p.
Let homo_f := mem_fcycle f_p. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fcycle_undup | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | in_orbit_cycle | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | eq_order_cycle | |
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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | iter_order_cycle | |
fconnect_eqVfx 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect_eqVf | |
orbitPcycle{x} : [<-> | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | orbitPcycle | |
order_id_cyclex : fcycle f (orbit x) -> order (f x) = order x.
Proof. by move/(orbitPcycle 0 1). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | order_id_cycle | |
order_spec_cyclex : 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | order_spec_cycle | |
orderPcyclex : 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 rewrite cardU1 inE (contraNN (all_iffLR orbitPcycle 2 0)).
by move=> y; rewrite !inE fconnect_eqVf eq_sym.
Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | orderPcycle | |
fconnect_fx : fconnect f (f x) x = fcycle f (orbit x).
Proof. by apply/idP/idP => /(orbitPcycle 0 2). Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect_f | |
fconnect_findexx 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 | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect_findex | |
fconnect_id(x : T) : fconnect id x =1 xpred1 x.
Proof. by move=> y; rewrite (@fconnect_cycle _ _ [:: x]) //= ?inE ?eqxx. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fconnect_id | |
order_id(x : T) : order id x = 1.
Proof. by rewrite /order (eq_card (fconnect_id x)) card1. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | order_id | |
orbit_id(x : T) : orbit id x = [:: x].
Proof. by rewrite /orbit order_id. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | orbit_id | |
froots_id(x : T) : froots id x.
Proof. by rewrite /roots -fconnect_id connect_root. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | froots_id | |
froot_id(x : T) : froot id x = x.
Proof. by apply/eqP; apply: froots_id. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | froot_id | |
fcard_id(a : {pred T}) : fcard id a = #|a|.
Proof. by apply: eq_card => x; rewrite inE froots_id. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | fcard_id | |
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.
Hypothesis eq_f : f =1 f'.
Let eq_rf := eq_frel eq_f. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | finv_eq_can | |
eq_fconnect: fconnect f =2 fconnect f'.
Proof. exact: eq_connect eq_rf. Qed. | Lemma | boot | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import seq path fintype"
] | boot/fingraph.v | eq_fconnect |
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