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 |
|---|---|---|---|---|---|---|
perm_ontos : codom s =i predT.
Proof. by apply/subset_cardP; rewrite ?card_codom ?subset_predT. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_onto
| |
perm_one:= perm (@inj_id T).
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_one
| |
perm_invKs : cancel (fun x => iinv (perm_onto s x)) s.
Proof. by move=> x /=; rewrite f_iinv. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_invK
| |
perm_invs := perm (can_inj (perm_invK s)).
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_inv
| |
perm_muls t := perm (inj_comp (@perm_inj t) (@perm_inj s)).
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_mul
| |
perm_oneP: left_id perm_one perm_mul.
Proof. by move=> s; apply/permP => x; rewrite permE /= permE. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_oneP
| |
perm_invP: left_inverse perm_one perm_inv perm_mul.
Proof. by move=> s; apply/permP=> x; rewrite !permE /= permE f_iinv. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_invP
| |
perm_mulP: associative perm_mul.
Proof. by move=> s t u; apply/permP=> x; do !rewrite permE /=. Qed.
HB.instance Definition _ := Finite_isGroup.Build (perm_type T)
perm_mulP perm_oneP perm_invP.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_mulP
| |
perm1x : (1 : {perm T}) x = x.
Proof. by rewrite permE. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm1
| |
permMs t x : (s * t) x = t (s x).
Proof. by rewrite permE. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
permM
| |
permKs : cancel s s^-1.
Proof. by move=> x; rewrite -permM mulgV perm1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
permK
| |
permKVs : cancel s^-1 s.
Proof. by have:= permK s^-1; rewrite invgK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
permKV
| |
permJs t x : (s ^ t) (t x) = t (s x).
Proof. by rewrite !permM permK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
permJ
| |
permXs x n : (s ^+ n) x = iter n s x.
Proof. by elim: n => [|n /= <-]; rewrite ?perm1 // -permM expgSr. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
permX
| |
permX_fixs x n : s x = x -> (s ^+ n) x = x.
Proof.
move=> Hs; elim: n => [|n IHn]; first by rewrite expg0 perm1.
by rewrite expgS permM Hs.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
permX_fix
| |
im_permVs S : s^-1 @: S = s @^-1: S.
Proof. exact: can2_imset_pre (permKV s) (permK s). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
im_permV
| |
preim_permVs S : s^-1 @^-1: S = s @: S.
Proof. by rewrite -im_permV invgK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
preim_permV
| |
perm_onS : pred {perm T} := fun s => [pred x | s x != x] \subset S.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_on
| |
perm_closedS s x : perm_on S s -> (s x \in S) = (x \in S).
Proof.
move/subsetP=> s_on_S; have [-> // | nfix_s_x] := eqVneq (s x) x.
by rewrite !s_on_S // inE /= ?(inj_eq perm_inj).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_closed
| |
perm_on1H : perm_on H 1.
Proof. by apply/subsetP=> x; rewrite inE /= perm1 eqxx. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_on1
| |
perm_onMH s t : perm_on H s -> perm_on H t -> perm_on H (s * t).
Proof.
move/subsetP=> sH /subsetP tH; apply/subsetP => x; rewrite inE /= permM.
by have [-> /tH | /sH] := eqVneq (s x) x.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_onM
| |
perm_onVH s : perm_on H s -> perm_on H s^-1.
Proof.
move=> /subsetP sH; apply/subsetP => i /[!inE] sVi; apply: sH; rewrite inE.
by apply: contra_neq sVi => si_id; rewrite -[in LHS]si_id permK.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_onV
| |
out_permS u x : perm_on S u -> x \notin S -> u x = x.
Proof. by move=> uS; apply: contraNeq (subsetP uS x). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
out_perm
| |
im_perm_onu S : perm_on S u -> u @: S = S.
Proof.
move=> Su; rewrite -preim_permV; apply/setP=> x.
by rewrite !inE -(perm_closed _ Su) permKV.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
im_perm_on
| |
perm_on_idu S : perm_on S u -> #|S| <= 1 -> u = 1%g.
Proof.
rewrite leq_eqVlt ltnS leqn0 => pSu S10; apply/permP => t; rewrite perm1.
case/orP : S10; last first.
by move/eqP/cards0_eq => S0; apply: (out_perm pSu); rewrite S0 inE.
move=> /cards1P[x Sx].
have [-> | ntx] := eqVneq t x; last by apply: (out_perm pSu); rewrite Sx inE.
by apply/eqP; have := perm_closed x pSu; rewrite Sx !inE => ->.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_on_id
| |
perm_onC(S1 S2 : {set T}) (u1 u2 : {perm T}) :
perm_on S1 u1 -> perm_on S2 u2 ->
[disjoint S1 & S2] ->
commute u1 u2.
Proof.
move=> pS1 pS2 S12; apply/permP => t; rewrite !permM.
case/boolP : (t \in S1) => tS1.
have /[!disjoint_subset] /subsetP {}S12 := S12.
by rewrite !(out_perm pS2) //; apply: S12; rewrite // perm_closed.
case/boolP : (t \in S2) => tS2.
have /[1!disjoint_sym] /[!disjoint_subset] /subsetP {}S12 := S12.
by rewrite !(out_perm pS1) //; apply: S12; rewrite // perm_closed.
by rewrite (out_perm pS1) // (out_perm pS2) // (out_perm pS1).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
perm_onC
| |
imset_perm1(S : {set T}) : [set (1 : {perm T}) x | x in S] = S.
Proof. apply: im_perm_on; exact: perm_on1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
imset_perm1
| |
tperm_proofx y : involutive [fun z => z with x |-> y, y |-> x].
Proof.
move=> z /=; case: (z =P x) => [-> | ne_zx]; first by rewrite eqxx; case: eqP.
by case: (z =P y) => [->| ne_zy]; [rewrite eqxx | do 2?case: eqP].
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tperm_proof
| |
tpermx y := perm (can_inj (tperm_proof x y)).
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tperm
| |
tperm_specx y z : T -> Type :=
| TpermFirst of z = x : tperm_spec x y z y
| TpermSecond of z = y : tperm_spec x y z x
| TpermNone of z <> x & z <> y : tperm_spec x y z z.
|
Variant
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tperm_spec
| |
tpermPx y z : tperm_spec x y z (tperm x y z).
Proof. by rewrite permE /=; do 2?[case: eqP => /=]; constructor; auto. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tpermP
| |
tpermLx y : tperm x y x = y.
Proof. by case: tpermP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tpermL
| |
tpermRx y : tperm x y y = x.
Proof. by case: tpermP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tpermR
| |
tpermDx y z : x != z -> y != z -> tperm x y z = z.
Proof. by case: tpermP => // ->; rewrite eqxx. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tpermD
| |
tpermCx y : tperm x y = tperm y x.
Proof. by apply/permP => z; do 2![case: tpermP => //] => ->. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tpermC
| |
tperm1x : tperm x x = 1.
Proof. by apply/permP => z; rewrite perm1; case: tpermP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tperm1
| |
tpermKx y : involutive (tperm x y).
Proof. by move=> z; rewrite !permE tperm_proof. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tpermK
| |
tpermKgx y : involutive (mul (tperm x y)).
Proof. by move=> s; apply/permP=> z; rewrite !permM tpermK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tpermKg
| |
tpermVx y : (tperm x y)^-1 = tperm x y.
Proof. by set t := tperm x y; rewrite -{2}(mulgK t t) -mulgA tpermKg. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tpermV
| |
tperm2x y : tperm x y * tperm x y = 1.
Proof. by rewrite -{1}tpermV mulVg. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tperm2
| |
tperm_onx y : perm_on [set x; y] (tperm x y).
Proof.
by apply/subsetP => z /[!inE]; case: tpermP => [->|->|]; rewrite eqxx // orbT.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tperm_on
| |
card_permA : #|perm_on A| = (#|A|)`!.
Proof.
pose ffA := {ffun {x | x \in A} -> T}.
rewrite -ffactnn -{2}(card_sig [in A]) /= -card_inj_ffuns_on.
pose fT (f : ffA) := [ffun x => oapp f x (insub x)].
pose pfT f := insubd (1 : {perm T}) (fT f).
pose fA s : ffA := [ffun u => s (val u)].
rewrite -!sum1dep_card -sum1_card (reindex_onto fA pfT) => [|f].
apply: eq_bigl => p; rewrite andbC; apply/idP/and3P=> [onA | []]; first split.
- apply/eqP; suffices fTAp: fT (fA p) = pval p.
by apply/permP=> x; rewrite -!pvalE insubdK fTAp //; apply: (valP p).
apply/ffunP=> x; rewrite ffunE pvalE.
by case: insubP => [u _ <- | /out_perm->] //=; rewrite ffunE.
- by apply/forallP=> [[x Ax]]; rewrite ffunE /= perm_closed.
- by apply/injectiveP=> u v; rewrite !ffunE => /perm_inj; apply: val_inj.
move/eqP=> <- _ _; apply/subsetP=> x; rewrite !inE -pvalE val_insubd fun_if.
by rewrite if_arg ffunE; case: insubP; rewrite // pvalE perm1 if_same eqxx.
case/andP=> /forallP-onA /injectiveP-f_inj.
apply/ffunP=> u; rewrite ffunE -pvalE insubdK; first by rewrite ffunE valK.
apply/injectiveP=> {u} x y; rewrite !ffunE.
case: insubP => [u _ <-|]; case: insubP => [v _ <-|] //=; first by move/f_inj->.
by move=> Ay' def_y; rewrite -def_y [_ \in A]onA in Ay'.
by move=> Ax' def_x; rewrite def_x [_ \in A]onA in Ax'.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
card_perm
| |
reindex_perms := (reindex_inj (@perm_inj _ s)).
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
reindex_perm
| |
inj_tperm(T T' : finType) (f : T -> T') x y z :
injective f -> f (tperm x y z) = tperm (f x) (f y) (f z).
Proof. by move=> injf; rewrite !permE /= !(inj_eq injf) !(fun_if f). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
inj_tperm
| |
tpermJx y s : (tperm x y) ^ s = tperm (s x) (s y).
Proof.
by apply/permP => z; rewrite -(permKV s z) permJ; apply/inj_tperm/perm_inj.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tpermJ
| |
tpermJ_tpermx y z :
x != z -> y != z -> tperm x z ^ tperm x y = tperm y z.
Proof. by move=> nxz nyz; rewrite tpermJ tpermL [tperm _ _ z]tpermD. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tpermJ_tperm
| |
tuple_permP{T : eqType} {n} {s : seq T} {t : n.-tuple T} :
reflect (exists p : 'S_n, s = [tuple tnth t (p i) | i < n]) (perm_eq s t).
Proof.
apply: (iffP idP) => [|[p ->]]; last first.
rewrite /= (map_comp (tnth t)) -{1}(map_tnth_enum t) perm_map //.
apply: uniq_perm => [||i]; rewrite ?enum_uniq //.
by apply/injectiveP; apply: perm_inj.
by rewrite mem_enum -[i](permKV p) image_f.
case: n => [|n] in t *; last have x0 := tnth t ord0.
rewrite tuple0 => /perm_small_eq-> //.
by exists 1; rewrite [mktuple _]tuple0.
case/(perm_iotaP x0); rewrite size_tuple => Is eqIst ->{s}.
have uniqIs: uniq Is by rewrite (perm_uniq eqIst) iota_uniq.
have szIs: size Is == n.+1 by rewrite (perm_size eqIst) !size_tuple.
have pP i : tnth (Tuple szIs) i < n.+1.
by rewrite -[_ < _](mem_iota 0) -(perm_mem eqIst) mem_tnth.
have inj_p: injective (fun i => Ordinal (pP i)).
by apply/injectiveP/(@map_uniq _ _ val); rewrite -map_comp map_tnth_enum.
exists (perm inj_p); rewrite -[Is]/(tval (Tuple szIs)); congr (tval _).
by apply: eq_from_tnth => i; rewrite tnth_map tnth_mktuple permE (tnth_nth x0).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
tuple_permP
| |
aperm(T : finType) x (s : {perm T}) := s x.
HB.lock
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
aperm
| |
porbit(T : finType) (s : {perm T}) x := aperm x @: <[s]>.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbit
| |
porbit_unlockable:= Unlockable porbit.unlock.
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbit_unlockable
| |
porbits(T : finType) (s : {perm T}) := porbit s @: T.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbits
| |
odd_perm(s : perm_type T) := odd #|T| (+) odd #|porbits s|.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
odd_perm
| |
apermEx s : aperm x s = s x. Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
apermE
| |
mem_porbits i x : (s ^+ i) x \in porbit s x.
Proof. by rewrite [@porbit]unlock (imset_f (aperm x)) ?mem_cycle. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
mem_porbit
| |
porbit_ids x : x \in porbit s x.
Proof. by rewrite -{1}[x]perm1 (mem_porbit s 0). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbit_id
| |
card_porbit_neq0s x : #|porbit s x| != 0.
Proof.
by rewrite -lt0n card_gt0; apply/set0Pn; exists x; exact: porbit_id.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
card_porbit_neq0
| |
uniq_traject_porbits x : uniq (traject s x #|porbit s x|).
Proof.
case def_n: #|_| => // [n]; rewrite looping_uniq.
apply: contraL (card_size (traject s x n)) => /loopingP t_sx.
rewrite -ltnNge size_traject -def_n ?subset_leq_card // porbit.unlock.
by apply/subsetP=> _ /imsetP[_ /cycleP[i ->] ->]; rewrite /aperm permX t_sx.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
uniq_traject_porbit
| |
porbit_trajects x : porbit s x =i traject s x #|porbit s x|.
Proof.
apply: fsym; apply/subset_cardP.
by rewrite (card_uniqP _) ?size_traject ?uniq_traject_porbit.
by apply/subsetP=> _ /trajectP[i _ ->]; rewrite -permX mem_porbit.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbit_traject
| |
iter_porbits x : iter #|porbit s x| s x = x.
Proof.
case def_n: #|_| (uniq_traject_porbit s x) => [//|n] Ut.
have: looping s x n.+1.
by rewrite -def_n -[looping _ _ _]porbit_traject -permX mem_porbit.
rewrite /looping => /trajectP[[|i] //= lt_i_n /perm_inj eq_i_n_sx].
move: lt_i_n; rewrite ltnS ltn_neqAle andbC => /andP[le_i_n /negP[]].
by rewrite -(nth_uniq x _ _ Ut) ?size_traject ?nth_traject // eq_i_n_sx.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
iter_porbit
| |
eq_porbit_mems x y : (porbit s x == porbit s y) = (x \in porbit s y).
Proof.
apply/eqP/idP; first by move<-; exact: porbit_id.
rewrite porbit.unlock => /imsetP[si s_si ->].
apply/setP => z; apply/imsetP/imsetP=> [] [sj s_sj ->].
by exists (si * sj); rewrite ?groupM /aperm ?permM.
exists (si^-1 * sj); first by rewrite groupM ?groupV.
by rewrite /aperm permM permK.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
eq_porbit_mem
| |
porbit_syms x y : (x \in porbit s y) = (y \in porbit s x).
Proof. by rewrite -!eq_porbit_mem eq_sym. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbit_sym
| |
porbit_perms i x : porbit s ((s ^+ i) x) = porbit s x.
Proof. by apply/eqP; rewrite eq_porbit_mem mem_porbit. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbit_perm
| |
porbitPmins x y :
y \in porbit s x -> exists2 i, i < #[s] & y = (s ^+ i) x.
Proof.
by rewrite porbit.unlock=> /imsetP [z /cyclePmin[ i Hi ->{z}] ->{y}]; exists i.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbitPmin
| |
porbitPs x y :
reflect (exists i, y = (s ^+ i) x) (y \in porbit s x).
Proof.
apply (iffP idP) => [/porbitPmin [i _ ->]| [i ->]]; last exact: mem_porbit.
by exists i.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbitP
| |
porbitVs : porbit s^-1 =1 porbit s.
Proof.
move=> x; apply/setP => y; rewrite porbit_sym.
by apply/porbitP/porbitP => -[i ->]; exists i; rewrite expgVn ?permK ?permKV.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbitV
| |
porbitsVs : porbits s^-1 = porbits s.
Proof.
rewrite /porbits; apply/setP => y.
by apply/imsetP/imsetP => -[x _ ->{y}]; exists x; rewrite // porbitV.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbitsV
| |
porbit_setPs t x : porbit s x =i porbit t x <-> porbit s x = porbit t x.
Proof. by rewrite porbit.unlock; exact: setP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbit_setP
| |
porbits_mul_tperms x y : let t := tperm x y in
#|porbits (t * s)| + (x \notin porbit s y).*2 = #|porbits s| + (x != y).
Proof.
pose xf a b u := seq.find (pred2 a b) (traject u (u a) #|porbit u a|).
have xf_size a b u: xf a b u <= #|porbit u a|.
by rewrite (leq_trans (find_size _ _)) ?size_traject.
have lt_xf a b u n : n < xf a b u -> ~~ pred2 a b ((u ^+ n.+1) a).
move=> lt_n; apply: contraFN (before_find (u a) lt_n).
by rewrite permX iterSr nth_traject // (leq_trans lt_n).
pose t a b u := tperm a b * u.
have tC a b u : t a b u = t b a u by rewrite /t tpermC.
have tK a b: involutive (t a b) by move=> u; apply: tpermKg.
have tXC a b u n: n <= xf a b u -> (t a b u ^+ n.+1) b = (u ^+ n.+1) a.
elim: n => [|n IHn] lt_n_f; first by rewrite permM tpermR.
rewrite !(expgSr _ n.+1) !permM {}IHn 1?ltnW //; congr (u _).
by case/lt_xf/norP: lt_n_f => ne_a ne_b; rewrite tpermD // eq_sym.
have eq_xf a b u: pred2 a b ((u ^+ (xf a b u).+1) a).
have ua_a: a \in porbit u (u a) by rewrite porbit_sym (mem_porbit _ 1).
have has_f: has (pred2 a b) (traject u (u a) #|porbit u (u a)|).
by apply/hasP; exists a; rewrite /= ?eqxx -?porbit_traject.
have:= nth_find (u a) has_f; rewrite has_find size_traject in has_f.
rewrite -eq_porbit_mem in ua_a.
by rewrite nth_traject // -iterSr -permX -(eqP ua_a).
have xfC a b u: xf b a (t a b u) = xf a b u.
without loss lt_a: a b u / xf b a (t a b u) < xf a b u.
move=> IHab; set m := xf b a _; set n := xf a b u.
by case: (ltngtP m n) => // ltx; [a
...
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
porbits_mul_tperm
| |
odd_perm1: odd_perm 1 = false.
Proof.
rewrite /odd_perm card_imset ?addbb // => x y; move/eqP; rewrite eq_porbit_mem.
by rewrite porbit.unlock cycle1 imset_set1 /aperm perm1 inE=> /eqP.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
odd_perm1
| |
odd_mul_tpermx y s : odd_perm (tperm x y * s) = (x != y) (+) odd_perm s.
Proof.
rewrite addbC -addbA -[~~ _]oddb -oddD -porbits_mul_tperm.
by rewrite oddD odd_double addbF.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
odd_mul_tperm
| |
odd_tpermx y : odd_perm (tperm x y) = (x != y).
Proof. by rewrite -[_ y]mulg1 odd_mul_tperm odd_perm1 addbF. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
odd_tperm
| |
dpair(eT : eqType) := [pred t | t.1 != t.2 :> eT].
Arguments dpair {eT}.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
dpair
| |
prod_tpermPs :
{ts : seq (T * T) | s = \prod_(t <- ts) tperm t.1 t.2 & all dpair ts}.
Proof.
have [n] := ubnP #|[pred x | s x != x]|; elim: n s => // n IHn s /ltnSE-le_s_n.
case: (pickP (fun x => s x != x)) => [x s_x | s_id]; last first.
exists nil; rewrite // big_nil; apply/permP=> x.
by apply/eqP/idPn; rewrite perm1 s_id.
have [|ts def_s ne_ts] := IHn (tperm x (s^-1 x) * s); last first.
exists ((x, s^-1 x) :: ts); last by rewrite /= -(canF_eq (permK _)) s_x.
by rewrite big_cons -def_s mulgA tperm2 mul1g.
rewrite (cardD1 x) !inE s_x in le_s_n; apply: leq_ltn_trans le_s_n.
apply: subset_leq_card; apply/subsetP=> y.
rewrite !inE permM permE /= -(canF_eq (permK _)).
have [-> | ne_yx] := eqVneq y x; first by rewrite permKV eqxx.
by case: (s y =P x) => // -> _; rewrite eq_sym.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
prod_tpermP
| |
odd_perm_prodts :
all dpair ts -> odd_perm (\prod_(t <- ts) tperm t.1 t.2) = odd (size ts).
Proof.
elim: ts => [_|t ts IHts] /=; first by rewrite big_nil odd_perm1.
by case/andP=> dt12 dts; rewrite big_cons odd_mul_tperm dt12 IHts.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
odd_perm_prod
| |
odd_permM: {morph odd_perm : s1 s2 / s1 * s2 >-> s1 (+) s2}.
Proof.
move=> s1 s2; case: (prod_tpermP s1) => ts1 ->{s1} dts1.
case: (prod_tpermP s2) => ts2 ->{s2} dts2.
by rewrite -big_cat !odd_perm_prod ?all_cat ?dts1 // size_cat oddD.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
odd_permM
| |
odd_permVs : odd_perm s^-1 = odd_perm s.
Proof. by rewrite -{2}(mulgK s s) !odd_permM -addbA addKb. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
odd_permV
| |
odd_permJs1 s2 : odd_perm (s1 ^ s2) = odd_perm s1.
Proof. by rewrite !odd_permM odd_permV addbC addbK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
odd_permJ
| |
gen_tpermx : <<[set tperm x y | y in T]>>%g = [set: {perm T}].
Proof.
apply/eqP; rewrite eqEsubset subsetT/=; apply/subsetP => s _.
have [ts -> _] := prod_tpermP s; rewrite group_prod// => -[/= y z] _.
have [<-|Nyz] := eqVneq y z; first by rewrite tperm1 group1.
have [<-|Nxz] := eqVneq x z; first by rewrite tpermC mem_gen ?imset_f.
by rewrite -(tpermJ_tperm Nxz Nyz) groupJ ?mem_gen ?imset_f.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
gen_tperm
| |
odd_perm: perm_type >-> bool.
Arguments dpair {eT}.
Prenex Implicits porbit dpair porbits aperm.
|
Coercion
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
odd_perm
| |
Sym: {set {perm T}} := [set s | perm_on S s].
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
Sym
| |
Sym_group_set: group_set Sym.
Proof.
apply/group_setP; split => [|s t] /[!inE]; [exact: perm_on1 | exact: perm_onM].
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
Sym_group_set
| |
Sym_group: {group {perm T}} := Group Sym_group_set.
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
Sym_group
| |
card_Sym: #|Sym| = #|S|`!.
Proof. by rewrite cardsE /= card_perm. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
card_Sym
| |
card_Sn: #|'S_(n)| = n`!.
Proof.
rewrite (eq_card (B := perm_on [set : 'I_n])).
by rewrite card_perm /= cardsE /= card_ord.
move=> p; rewrite inE unfold_in /perm_on /=.
by apply/esym/subsetP => i _; rewrite in_set.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
card_Sn
| |
lift_perm_funi j s k :=
if unlift i k is Some k' then lift j (s k') else j.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
lift_perm_fun
| |
lift_permKi j s :
cancel (lift_perm_fun i j s) (lift_perm_fun j i s^-1).
Proof.
rewrite /lift_perm_fun => k.
by case: (unliftP i k) => [j'|] ->; rewrite (liftK, unlift_none) ?permK.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
lift_permK
| |
lift_permi j s := perm (can_inj (lift_permK i j s)).
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
lift_perm
| |
lift_perm_idi j s : lift_perm i j s i = j.
Proof. by rewrite permE /lift_perm_fun unlift_none. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
lift_perm_id
| |
lift_perm_lifti j s k' :
lift_perm i j s (lift i k') = lift j (s k') :> 'I_n.+1.
Proof. by rewrite permE /lift_perm_fun liftK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
lift_perm_lift
| |
lift_permMi j k s t :
lift_perm i j s * lift_perm j k t = lift_perm i k (s * t).
Proof.
apply/permP=> i1; case: (unliftP i i1) => [i2|] ->{i1}.
by rewrite !(permM, lift_perm_lift).
by rewrite permM !lift_perm_id.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
lift_permM
| |
lift_perm1i : lift_perm i i 1 = 1.
Proof. by apply: (mulgI (lift_perm i i 1)); rewrite lift_permM !mulg1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
lift_perm1
| |
lift_permVi j s : (lift_perm i j s)^-1 = lift_perm j i s^-1.
Proof. by apply/eqP; rewrite eq_invg_mul lift_permM mulgV lift_perm1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
lift_permV
| |
odd_lift_permi j s : lift_perm i j s = odd i (+) odd j (+) s :> bool.
Proof.
rewrite -{1}(mul1g s) -(lift_permM _ j) odd_permM.
congr (_ (+) _); last first.
case: (prod_tpermP s) => ts ->{s} _.
elim: ts => [|t ts IHts] /=; first by rewrite big_nil lift_perm1 !odd_perm1.
rewrite big_cons odd_mul_tperm -(lift_permM _ j) odd_permM {}IHts //.
congr (_ (+) _); transitivity (tperm (lift j t.1) (lift j t.2)); last first.
by rewrite odd_tperm (inj_eq (pcan_inj (liftK j))).
congr odd_perm; apply/permP=> k; case: (unliftP j k) => [k'|] ->.
by rewrite lift_perm_lift inj_tperm //; apply: lift_inj.
by rewrite lift_perm_id tpermD // eq_sym neq_lift.
suff{i j s} odd_lift0 (k : 'I_n.+1): lift_perm ord0 k 1 = odd k :> bool.
rewrite -!odd_lift0 -{2}invg1 -lift_permV odd_permV -odd_permM.
by rewrite lift_permM mulg1.
elim: {k}(k : nat) {1 3}k (erefl (k : nat)) => [|m IHm] k def_k.
by rewrite (_ : k = ord0) ?lift_perm1 ?odd_perm1 //; apply: val_inj.
have le_mn: m < n.+1 by [rewrite -def_k ltnW]; pose j := Ordinal le_mn.
rewrite -(mulg1 1)%g -(lift_permM _ j) odd_permM {}IHm // addbC.
rewrite (_ : _ 1 = tperm j k); first by rewrite odd_tperm neq_ltn/= def_k leqnn.
apply/permP=> i; case: (unliftP j i) => [i'|] ->; last first.
by rewrite lift_perm_id tpermL.
apply: ord_inj; rewrite lift_perm_lift !permE /= eq_sym -if_neg neq_lift.
rewrite fun_if -val_eqE /= def_k /bump ltn_neqAle andbC.
case: leqP => [_ | lt_i'm] /=; last by rewrite -if_neg neq_ltn leqW.
by rewrite add1n eqSS; case:
...
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
odd_lift_perm
| |
permS0: all_equal_to (1 : 'S_0).
Proof. by move=> g; apply/permP; case. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
permS0
| |
permS1: all_equal_to (1 : 'S_1).
Proof. by move=> g; apply/permP => i; rewrite !ord1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
permS1
| |
permS01n : n <= 1 -> all_equal_to (1 : 'S_n).
Proof. by case: n => [|[|]//=] _ g; rewrite (permS0, permS1). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
permS01
| |
cast_permm n (eq_mn : m = n) (s : 'S_m) :=
let: erefl in _ = n := eq_mn return 'S_n in s.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
cast_perm
| |
cast_perm_idn eq_n s : cast_perm eq_n s = s :> 'S_n.
Proof. by apply/permP => i; rewrite /cast_perm /= eq_axiomK. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
cast_perm_id
| |
cast_ord_permEm n eq_m_n (s : 'S_m) i :
@cast_ord m n eq_m_n (s i) = (cast_perm eq_m_n s) (cast_ord eq_m_n i).
Proof. by subst m; rewrite cast_perm_id !cast_ord_id. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
cast_ord_permE
| |
cast_permEm n (eq_m_n : m = n) (s : 'S_m) (i : 'I_n) :
cast_perm eq_m_n s i = cast_ord eq_m_n (s (cast_ord (esym eq_m_n) i)).
Proof. by rewrite cast_ord_permE cast_ordKV. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq path",
"From mathcomp Require Import choice fintype tuple finfun bigop finset binomial",
"From mathcomp Require Import fingroup morphism"
] |
fingroup/perm.v
|
cast_permE
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.