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 |
|---|---|---|---|---|---|---|
mul_cardGG H : (#|G| * #|H| = #|G * H|%g * #|G :&: H|)%N.
Proof. by rewrite -LagrangeMr -(LagrangeI G H) -mulnA mulnC. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
mul_cardG
| |
dvdn_cardMgG H : #|G * H| %| #|G| * #|H|.
Proof. by rewrite mul_cardG dvdn_mulr. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
dvdn_cardMg
| |
cardMg_divnG H : #|G * H| = (#|G| * #|H|) %/ #|G :&: H|.
Proof. by rewrite mul_cardG mulnK ?cardG_gt0. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cardMg_divn
| |
cardIg_divnG H : #|G :&: H| = (#|G| * #|H|) %/ #|G * H|.
Proof. by rewrite mul_cardG mulKn // (cardD1 (1 * 1)) mem_mulg. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cardIg_divn
| |
TI_cardMgG H : G :&: H = 1 -> #|G * H| = (#|G| * #|H|)%N.
Proof. by move=> tiGH; rewrite mul_cardG tiGH cards1 muln1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
TI_cardMg
| |
cardMg_TIG H : #|G| * #|H| <= #|G * H| -> G :&: H = 1.
Proof.
move=> leGH; apply: card_le1_trivg.
rewrite -(@leq_pmul2l #|G * H|); first by rewrite -mul_cardG muln1.
by apply: leq_trans leGH; rewrite muln_gt0 !cardG_gt0.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cardMg_TI
| |
coprime_TIgG H : coprime #|G| #|H| -> G :&: H = 1.
Proof.
move=> coGH; apply/eqP; rewrite trivg_card1 -dvdn1 -{}(eqnP coGH).
by rewrite dvdn_gcd /= {2}setIC !cardSg ?subsetIl.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
coprime_TIg
| |
prime_TIgG H : prime #|G| -> ~~ (G \subset H) -> G :&: H = 1.
Proof.
case/primeP=> _ /(_ _ (cardSg (subsetIl G H))).
rewrite (sameP setIidPl eqP) eqEcard subsetIl => /pred2P[/card1_trivg|] //= ->.
by case/negP.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
prime_TIg
| |
prime_meetGG H : prime #|G| -> G :&: H != 1 -> G \subset H.
Proof. by move=> prG; apply: contraR; move/prime_TIg->. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
prime_meetG
| |
coprime_cardMgG H : coprime #|G| #|H| -> #|G * H| = (#|G| * #|H|)%N.
Proof. by move=> coGH; rewrite TI_cardMg ?coprime_TIg. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
coprime_cardMg
| |
coprime_index_mulGG H K :
H \subset G -> K \subset G -> coprime #|G : H| #|G : K| -> H * K = G.
Proof.
move=> sHG sKG co_iG_HK; apply/eqP; rewrite eqEcard mul_subG //=.
rewrite -(@leq_pmul2r #|H :&: K|) ?cardG_gt0 // -mul_cardG.
rewrite -(Lagrange sHG) -(LagrangeI K H) mulnAC setIC -mulnA.
rewrite !leq_pmul2l ?cardG_gt0 // dvdn_leq // -(Gauss_dvdr _ co_iG_HK).
by rewrite -(indexgI K) Lagrange_index ?indexgS ?subsetIl ?subsetIr.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
coprime_index_mulG
| |
subset_genA : A \subset <<A>>.
Proof. rewrite [@generated]unlock; exact/bigcapsP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
subset_gen
| |
sub_genA B : A \subset B -> A \subset <<B>>.
Proof. by move/subset_trans=> -> //; apply: subset_gen. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
sub_gen
| |
mem_genx A : x \in A -> x \in <<A>>.
Proof. exact: subsetP (subset_gen A) x. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
mem_gen
| |
generatedPx A : reflect (forall G, A \subset G -> x \in G) (x \in <<A>>).
Proof. rewrite [@generated]unlock; exact: bigcapP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
generatedP
| |
gen_subGA G : (<<A>> \subset G) = (A \subset G).
Proof.
apply/idP/idP=> [|sAG]; first exact: subset_trans (subset_gen A).
by apply/subsetP=> x /generatedP; apply.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
gen_subG
| |
genGidG : <<G>> = G.
Proof. by apply/eqP; rewrite eqEsubset gen_subG subset_gen andbT. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
genGid
| |
genGidGG : <<G>>%G = G.
Proof. by apply: val_inj; apply: genGid. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
genGidG
| |
gen_set_idA : group_set A -> <<A>> = A.
Proof. by move=> gA; apply: (genGid (group gA)). Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
gen_set_id
| |
genSA B : A \subset B -> <<A>> \subset <<B>>.
Proof. by move=> sAB; rewrite gen_subG sub_gen. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
genS
| |
gen0: <<set0>> = 1 :> {set gT}.
Proof. by apply/eqP; rewrite eqEsubset sub1G gen_subG sub0set. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
gen0
| |
gen_expgsA : {n | <<A>> = (1 |: A) ^+ n}.
Proof.
set B := (1 |: A); pose N := #|gT|.
have BsubG n : B ^+ n \subset <<A>>.
by elim: n => [|n IHn]; rewrite ?expgS ?mul_subG ?subUset ?sub1G ?subset_gen.
have B_1 n : 1 \in B ^+ n.
by elim: n => [|n IHn]; rewrite ?set11 // expgS mulUg mul1g inE IHn.
case: (pickP (fun i : 'I_N => B ^+ i.+1 \subset B ^+ i)) => [n fixBn | no_fix].
exists n; apply/eqP; rewrite eqEsubset BsubG andbT.
rewrite -[B ^+ n]gen_set_id ?genS ?subsetUr //.
by apply: subset_trans fixBn; rewrite expgS mulUg subsetU ?mulg_subl ?orbT.
rewrite /group_set B_1 /=.
elim: {2}(n : nat) => [|m IHm]; first by rewrite mulg1.
by apply: subset_trans fixBn; rewrite !expgSr mulgA mulSg.
suffices: N < #|B ^+ N| by rewrite ltnNge max_card.
have [] := ubnPgeq N; elim=> [|n IHn] lt_nN; first by rewrite cards1.
apply: leq_ltn_trans (IHn (ltnW lt_nN)) (proper_card _).
by rewrite /proper (no_fix (Ordinal lt_nN)) expgS mulUg mul1g subsetUl.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
gen_expgs
| |
gen_prodgPA x :
reflect (exists n, exists2 c, forall i : 'I_n, c i \in A & x = \prod_i c i)
(x \in <<A>>).
Proof.
apply: (iffP idP) => [|[n [c Ac ->]]]; last first.
by apply: group_prod => i _; rewrite mem_gen ?Ac.
have [n ->] := gen_expgs A; rewrite /natexp Monoid.iteropE /=.
rewrite -[n]card_ord -big_const => /prodsgP[/= c Ac def_x].
have{Ac def_x} ->: x = \prod_(i | c i \in A) c i.
rewrite big_mkcond {x}def_x; apply: eq_bigr => i _.
by case/setU1P: (Ac i isT) => -> //; rewrite if_same.
have [e <- [_ /= mem_e] _] := big_enumP [preim c of A].
pose t := in_tuple e; rewrite -[e]/(val t) big_tuple.
by exists (size e), (c \o tnth t) => // i; rewrite -mem_e mem_tnth.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
gen_prodgP
| |
genDA B : A \subset <<A :\: B>> -> <<A :\: B>> = <<A>>.
Proof.
by move=> sAB; apply/eqP; rewrite eqEsubset genS (subsetDl, gen_subG).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
genD
| |
genVA : <<A^-1>> = <<A>>.
Proof.
apply/eqP; rewrite eqEsubset !gen_subG -!(invSg _ <<_>>) invgK.
by rewrite !invGid !subset_gen.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
genV
| |
genJA z : <<A :^z>> = <<A>> :^ z.
Proof.
by apply/eqP; rewrite eqEsubset sub_conjg !gen_subG conjSg -?sub_conjg !sub_gen.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
genJ
| |
conjYgA B z : (A <*> B) :^z = A :^ z <*> B :^ z.
Proof. by rewrite -genJ conjUg. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
conjYg
| |
genD1A x : x \in <<A :\ x>> -> <<A :\ x>> = <<A>>.
Proof.
move=> gA'x; apply/eqP; rewrite eqEsubset genS; last by rewrite subsetDl.
rewrite gen_subG; apply/subsetP=> y Ay.
by case: (y =P x) => [-> //|]; move/eqP=> nyx; rewrite mem_gen // !inE nyx.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
genD1
| |
genD1idA : <<A^#>> = <<A>>.
Proof. by rewrite genD1 ?group1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
genD1id
| |
joingT:= (@joing gT) (only parsing).
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joingT
| |
joinGT:= (@joinG gT) (only parsing).
|
Notation
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joinGT
| |
joingEA B : A <*> B = <<A :|: B>>. Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joingE
| |
joinGEG H : (G * H)%G = (G <*> H)%G. Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joinGE
| |
joingC: commutative joingT.
Proof. by move=> A B; rewrite /joing setUC. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joingC
| |
joing_idrA B : A <*> <<B>> = A <*> B.
Proof.
apply/eqP; rewrite eqEsubset gen_subG subUset gen_subG /=.
by rewrite -subUset subset_gen genS // setUS // subset_gen.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joing_idr
| |
joing_idlA B : <<A>> <*> B = A <*> B.
Proof. by rewrite -!(joingC B) joing_idr. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joing_idl
| |
joing_sublA B : A \subset A <*> B.
Proof. by rewrite sub_gen ?subsetUl. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joing_subl
| |
joing_subrA B : B \subset A <*> B.
Proof. by rewrite sub_gen ?subsetUr. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joing_subr
| |
join_subGA B G : (A <*> B \subset G) = (A \subset G) && (B \subset G).
Proof. by rewrite gen_subG subUset. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
join_subG
| |
joing_idPlG A : reflect (G <*> A = G) (A \subset G).
Proof.
apply: (iffP idP) => [sHG | <-]; last by rewrite joing_subr.
by rewrite joingE (setUidPl sHG) genGid.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joing_idPl
| |
joing_idPrA G : reflect (A <*> G = G) (A \subset G).
Proof. by rewrite joingC; apply: joing_idPl. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joing_idPr
| |
joing_subPA B G :
reflect (A \subset G /\ B \subset G) (A <*> B \subset G).
Proof. by rewrite join_subG; apply: andP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joing_subP
| |
joing_subA B C : A <*> B = C -> A \subset C /\ B \subset C.
Proof. by move <-; apply/joing_subP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joing_sub
| |
genDUA B C : A \subset C -> <<C :\: A>> = <<B>> -> <<A :|: B>> = <<C>>.
Proof.
move=> sAC; rewrite -joingE -joing_idr => <- {B}; rewrite joing_idr.
by congr <<_>>; rewrite setDE setUIr setUCr setIT; apply/setUidPr.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
genDU
| |
joingA: associative joingT.
Proof. by move=> A B C; rewrite joing_idl joing_idr /joing setUA. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joingA
| |
joing1GG : 1 <*> G = G.
Proof. by rewrite -gen0 joing_idl /joing set0U genGid. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joing1G
| |
joingG1G : G <*> 1 = G.
Proof. by rewrite joingC joing1G. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joingG1
| |
genM_joinG H : <<G * H>> = G <*> H.
Proof.
apply/eqP; rewrite eqEsubset gen_subG /= -{1}[G <*> H]mulGid.
rewrite genS; last by rewrite subUset mulG_subl mulG_subr.
by rewrite mulgSS ?(sub_gen, subsetUl, subsetUr).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
genM_join
| |
mulG_subGG H K : (G * H \subset K) = (G \subset K) && (H \subset K).
Proof. by rewrite -gen_subG genM_join join_subG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
mulG_subG
| |
mulGsubPK H G : reflect (K \subset G /\ H \subset G) (K * H \subset G).
Proof. by rewrite mulG_subG; apply: andP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
mulGsubP
| |
mulG_subK H A : K * H = A -> K \subset A /\ H \subset A.
Proof. by move <-; rewrite mulG_subl mulG_subr. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
mulG_sub
| |
trivMgG H : (G * H == 1) = (G :==: 1) && (H :==: 1).
Proof.
by rewrite !eqEsubset -{2}[1]mulGid mulgSS ?sub1G // !andbT mulG_subG.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
trivMg
| |
comm_joingEG H : commute G H -> G <*> H = G * H.
Proof.
by move/comm_group_setP=> gGH; rewrite -genM_join; apply: (genGid (group gGH)).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
comm_joingE
| |
joinGC: commutative joinGT.
Proof. by move=> G H; apply: val_inj; apply: joingC. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joinGC
| |
joinGA: associative joinGT.
Proof. by move=> G H K; apply: val_inj; apply: joingA. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joinGA
| |
join1G: left_id 1%G joinGT.
Proof. by move=> G; apply: val_inj; apply: joing1G. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
join1G
| |
joinG1: right_id 1%G joinGT.
Proof. by move=> G; apply: val_inj; apply: joingG1. Qed.
HB.instance Definition _ := Monoid.isComLaw.Build {group gT} 1%G joinGT
joinGA joinGC join1G.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
joinG1
| |
bigprodGEgenI r (P : pred I) (F : I -> {set gT}) :
(\prod_(i <- r | P i) <<F i>>)%G :=: << \bigcup_(i <- r | P i) F i >>.
Proof.
elim/big_rec2: _ => /= [|i A _ _ ->]; first by rewrite gen0.
by rewrite joing_idl joing_idr.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
bigprodGEgen
| |
bigprodGEI r (P : pred I) (F : I -> {group gT}) :
(\prod_(i <- r | P i) F i)%G :=: << \bigcup_(i <- r | P i) F i >>.
Proof.
rewrite -bigprodGEgen /=; apply: congr_group.
by apply: eq_bigr => i _; rewrite genGidG.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
bigprodGE
| |
mem_commgA B x y : x \in A -> y \in B -> [~ x, y] \in [~: A, B].
Proof. by move=> Ax By; rewrite mem_gen ?imset2_f. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
mem_commg
| |
commSgA B C : A \subset B -> [~: A, C] \subset [~: B, C].
Proof. by move=> sAC; rewrite genS ?imset2S. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
commSg
| |
commgSA B C : B \subset C -> [~: A, B] \subset [~: A, C].
Proof. by move=> sBC; rewrite genS ?imset2S. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
commgS
| |
commgSSA B C D :
A \subset B -> C \subset D -> [~: A, C] \subset [~: B, D].
Proof. by move=> sAB sCD; rewrite genS ?imset2S. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
commgSS
| |
der1_subGG : [~: G, G] \subset G.
Proof.
by rewrite gen_subG; apply/subsetP=> _ /imset2P[x y Gx Gy ->]; apply: groupR.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
der1_subG
| |
comm_subGA B G : A \subset G -> B \subset G -> [~: A, B] \subset G.
Proof.
by move=> sAG sBG; apply: subset_trans (der1_subG G); apply: commgSS.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
comm_subG
| |
commGCA B : [~: A, B] = [~: B, A].
Proof.
rewrite -[[~: A, B]]genV; congr <<_>>; apply/setP=> z; rewrite inE.
by apply/imset2P/imset2P=> [] [x y Ax Ay]; last rewrite -{1}(invgK z);
rewrite -invg_comm => /invg_inj->; exists y x.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
commGC
| |
conjsRgA B x : [~: A, B] :^ x = [~: A :^ x, B :^ x].
Proof.
wlog suffices: A B x / [~: A, B] :^ x \subset [~: A :^ x, B :^ x].
move=> subJ; apply/eqP; rewrite eqEsubset subJ /= -sub_conjgV.
by rewrite -{2}(conjsgK x A) -{2}(conjsgK x B).
rewrite -genJ gen_subG; apply/subsetP=> _ /imsetP[_ /imset2P[y z Ay Bz ->] ->].
by rewrite conjRg mem_commg ?memJ_conjg.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
conjsRg
| |
cycle1: <[1]> = [1 gT].
Proof. exact: genGid. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cycle1
| |
order1: #[1 : gT] = 1%N.
Proof. by rewrite /order cycle1 cards1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
order1
| |
cycle_idx : x \in <[x]>.
Proof. by rewrite mem_gen // set11. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cycle_id
| |
mem_cyclex i : x ^+ i \in <[x]>.
Proof. by rewrite groupX // cycle_id. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
mem_cycle
| |
cycle_subGx G : (<[x]> \subset G) = (x \in G).
Proof. by rewrite gen_subG sub1set. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cycle_subG
| |
cycle_eq1x : (<[x]> == 1) = (x == 1).
Proof. by rewrite eqEsubset sub1G andbT cycle_subG inE. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cycle_eq1
| |
orderEx : #[x] = #|<[x]>|. Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
orderE
| |
order_eq1x : (#[x] == 1%N) = (x == 1).
Proof. by rewrite -trivg_card1 cycle_eq1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
order_eq1
| |
order_gt1x : (#[x] > 1) = (x != 1).
Proof. by rewrite ltnNge -trivg_card_le1 cycle_eq1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
order_gt1
| |
cycle_trajectx : <[x]> =i traject (mul x) 1 #[x].
Proof.
set t := _ 1; apply: fsym; apply/subset_cardP; last first.
by apply/subsetP=> _ /trajectP[i _ ->]; rewrite -iteropE mem_cycle.
rewrite (card_uniqP _) ?size_traject //; case def_n: #[_] => // [n].
rewrite looping_uniq; apply: contraL (card_size (t n)) => /loopingP t_xi.
rewrite -ltnNge size_traject -def_n ?subset_leq_card //.
rewrite -(eq_subset_r (in_set _)) {}/t; set G := finset _.
rewrite -[x]mulg1 -[G]gen_set_id ?genS ?sub1set ?inE ?(t_xi 1%N)//.
apply/group_setP; split=> [|y z]; rewrite !inE ?(t_xi 0) //.
by do 2!case/trajectP=> ? _ ->; rewrite -!iteropE -expgD [x ^+ _]iteropE.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cycle_traject
| |
cycle2gx : #[x] = 2 -> <[x]> = [set 1; x].
Proof. by move=> ox; apply/setP=> y; rewrite cycle_traject ox !inE mulg1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cycle2g
| |
cyclePminx y : y \in <[x]> -> {i | i < #[x] & y = x ^+ i}.
Proof.
rewrite cycle_traject; set tx := traject _ _ #[x] => tx_y; pose i := index y tx.
have lt_i_x : i < #[x] by rewrite -index_mem size_traject in tx_y.
by exists i; rewrite // [x ^+ i]iteropE /= -(nth_traject _ lt_i_x) nth_index.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cyclePmin
| |
cyclePx y : reflect (exists i, y = x ^+ i) (y \in <[x]>).
Proof.
by apply: (iffP idP) => [/cyclePmin[i _]|[i ->]]; [exists i | apply: mem_cycle].
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cycleP
| |
expg_orderx : x ^+ #[x] = 1.
Proof.
have: uniq (traject (mul x) 1 #[x]).
by apply/card_uniqP; rewrite size_traject -(eq_card (cycle_traject x)).
case/cyclePmin: (mem_cycle x #[x]) => [] [//|i] ltix.
rewrite -(subnKC ltix) addSnnS /= expgD; move: (_ - _) => j x_j1.
case/andP=> /trajectP[]; exists j; first exact: leq_addl.
by apply: (mulgI (x ^+ i.+1)); rewrite -iterSr iterS -iteropE -expgS mulg1.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
expg_order
| |
expg_modp k x : x ^+ p = 1 -> x ^+ (k %% p) = x ^+ k.
Proof.
move=> xp.
by rewrite {2}(divn_eq k p) expgD mulnC expgM xp expg1n mul1g.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
expg_mod
| |
expg_mod_orderx i : x ^+ (i %% #[x]) = x ^+ i.
Proof. by rewrite expg_mod // expg_order. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
expg_mod_order
| |
invg_expgx : x^-1 = x ^+ #[x].-1.
Proof. by apply/eqP; rewrite eq_invg_mul -expgS prednK ?expg_order. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
invg_expg
| |
invg2idx : #[x] = 2 -> x^-1 = x.
Proof. by move=> ox; rewrite invg_expg ox. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
invg2id
| |
cycleXx i : <[x ^+ i]> \subset <[x]>.
Proof. by rewrite cycle_subG; apply: mem_cycle. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cycleX
| |
cycleVx : <[x^-1]> = <[x]>.
Proof.
by apply/eqP; rewrite eq_sym eqEsubset !cycle_subG groupV -groupV !cycle_id.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cycleV
| |
orderVx : #[x^-1] = #[x].
Proof. by rewrite /order cycleV. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
orderV
| |
cycleJx y : <[x ^ y]> = <[x]> :^ y.
Proof. by rewrite -genJ conjg_set1. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
cycleJ
| |
orderJx y : #[x ^ y] = #[x].
Proof. by rewrite /order cycleJ cardJg. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
orderJ
| |
normPx A : reflect (A :^ x = A) (x \in 'N(A)).
Proof.
suffices ->: (x \in 'N(A)) = (A :^ x == A) by apply: eqP.
by rewrite eqEcard cardJg leqnn andbT inE.
Qed.
Arguments normP {x A}.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normP
| |
group_set_normaliserA : group_set 'N(A).
Proof.
apply/group_setP; split=> [|x y Nx Ny]; rewrite inE ?conjsg1 //.
by rewrite conjsgM !(normP _).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
group_set_normaliser
| |
normaliser_groupA := group (group_set_normaliser A).
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normaliser_group
| |
normsPA B : reflect {in A, normalised B} (A \subset 'N(B)).
Proof.
apply: (iffP subsetP) => nBA x Ax; last by rewrite inE nBA //.
by apply/normP; apply: nBA.
Qed.
Arguments normsP {A B}.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normsP
| |
memJ_normx y A : x \in 'N(A) -> (y ^ x \in A) = (y \in A).
Proof. by move=> Nx; rewrite -{1}(normP Nx) memJ_conjg. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
memJ_norm
| |
norms_cyclex y : (<[y]> \subset 'N(<[x]>)) = (x ^ y \in <[x]>).
Proof. by rewrite cycle_subG inE -cycleJ cycle_subG. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norms_cycle
| |
norm1: 'N(1) = setT :> {set gT}.
Proof. by apply/setP=> x; rewrite !inE conjs1g subxx. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norm1
| |
norms1A : A \subset 'N(1).
Proof. by rewrite norm1 subsetT. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
norms1
| |
normCsA : 'N(~: A) = 'N(A).
Proof. by apply/setP=> x; rewrite -groupV !inE conjCg setCS sub_conjg. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normCs
| |
normGG : G \subset 'N(G).
Proof. by apply/normsP; apply: conjGid. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
normG
|
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