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
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|---|---|---|---|---|---|---|
class_support_set1lA x : class_support [set x] A = x ^: A.
Proof. exact: imset2_set1l. 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
|
class_support_set1l
| |
class_support_set1rA x : class_support A [set x] = A :^ x.
Proof. exact: imset2_set1r. 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
|
class_support_set1r
| |
classMx A B : x ^: (A * B) = class_support (x ^: A) B.
Proof. by rewrite -!class_support_set1l class_supportM. 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
|
classM
| |
class_lcosetx y A : x ^: (y *: A) = (x ^ y) ^: A.
Proof. by rewrite classM class_set1 class_support_set1l. 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
|
class_lcoset
| |
class_rcosetx A y : x ^: (A :* y) = (x ^: A) :^ y.
Proof. by rewrite -class_support_set1r classM. 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
|
class_rcoset
| |
conjugatesSA B C : B \subset C -> A :^: B \subset A :^: C.
Proof. exact: imsetS. 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
|
conjugatesS
| |
conjugates_set1A x : A :^: [set x] = [set A :^ x].
Proof. exact: imset_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
|
conjugates_set1
| |
conjugates_conjA x B : (A :^ x) :^: B = A :^: (x *: B).
Proof.
rewrite /conjugates [x *: B]imset2_set1l -imset_comp.
by apply: eq_imset => y /=; rewrite conjsgM.
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
|
conjugates_conj
| |
class_supportElA B : class_support A B = \bigcup_(x in A) x ^: B.
Proof. exact: curry_imset2l. 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
|
class_supportEl
| |
class_supportErA B : class_support A B = \bigcup_(x in B) A :^ x.
Proof. exact: curry_imset2r. 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
|
class_supportEr
| |
group_setA := (1 \in A) && (A * A \subset A).
|
Definition
|
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
| |
group_setPA :
reflect (1 \in A /\ {in A & A, forall x y, x * y \in A}) (group_set A).
Proof.
apply: (iffP andP) => [] [A1 AM]; split=> {A1}//.
by move=> x y Ax Ay; apply: (subsetP AM); rewrite mem_mulg.
by apply/subsetP=> _ /mulsgP[x y Ax Ay ->]; apply: AM.
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_setP
| |
group_type: Type := Group {
gval :> GroupSet.sort gT;
_ : group_set gval
}.
|
Structure
|
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_type
| |
group_of: predArgType := group_type.
Local Notation groupT := group_of.
Identity Coercion type_of_group : group_of >-> group_type.
HB.instance Definition _ := [isSub for gval].
#[hnf] HB.instance Definition _ := [Finite of group_type by <:].
|
Definition
|
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_of
| |
Definition_ := SubFinite.copy groupT group_type.
|
HB.instance
|
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
|
Definition
| |
group(A : {set gT}) gA : groupT := @Group A gA.
|
Definition
|
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
| |
clone_groupG :=
let: Group _ gP := G return {type of Group for G} -> groupT in fun k => k gP.
|
Definition
|
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
|
clone_group
| |
group_inj: injective gval. Proof. exact: val_inj. 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_inj
| |
groupP(G : groupT) : group_set G. Proof. by case: G. 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
|
groupP
| |
congr_group(H K : groupT) : H = K -> H :=: K.
Proof. exact: congr1. 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
|
congr_group
| |
isgroupPA : reflect (exists G : groupT, A = G) (group_set A).
Proof. by apply: (iffP idP) => [gA | [[B gB] -> //]]; exists (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
|
isgroupP
| |
group_set_one: group_set 1.
Proof. by rewrite /group_set set11 mulg1 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
|
group_set_one
| |
one_group:= group group_set_one.
|
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
|
one_group
| |
set1_group:= @group [set 1] group_set_one.
|
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
|
set1_group
| |
group_setT: group_set (setTfor gT).
Proof. by apply/group_setP; split=> [|x y _ _]; rewrite 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
|
group_setT
| |
setT_group:= group group_setT.
|
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
|
setT_group
| |
generated(gT : finGroupType) (A : {set gT}) :=
\bigcap_(G : {group gT} | A \subset G) G.
|
Definition
|
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
|
generated
| |
generated_unlockable:= Unlockable generated.unlock.
|
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
|
generated_unlockable
| |
gcore(gT : finGroupType) (A B : {set gT}) := \bigcap_(x in B) A :^ x.
|
Definition
|
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
|
gcore
| |
joing(gT : finGroupType) (A B : {set gT}) := generated (A :|: B).
|
Definition
|
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
| |
commutator(gT : finGroupType) (A B : {set gT}) := generated (commg_set A B).
|
Definition
|
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
|
commutator
| |
cycle(gT : finGroupType) (x : gT) := generated [set x].
|
Definition
|
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
| |
order(gT : finGroupType) (x : gT) := #|cycle x|.
Arguments commutator _ _%_g _%_g.
Arguments joing _ _%_g _%_g.
Arguments generated _ _%_g.
|
Definition
|
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
| |
gsortgT := (FinStarMonoid.arg_sort gT%type) (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
|
gsort
| |
sT:= {set gT}.
Implicit Types A B C D : sT.
Implicit Types x y z : gT.
Implicit Types G H K : {group gT}.
|
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
|
sT
| |
valG: val G = 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
|
valG
| |
group1: 1 \in G. Proof. by case/group_setP: (valP G). Qed.
#[local] Hint Resolve group1 : core.
|
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
|
group1
| |
group1_contrax : x \notin G -> x != 1.
Proof. by apply: contraNneq => ->. 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
|
group1_contra
| |
sub1G: [1 gT] \subset G. Proof. by rewrite 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
|
sub1G
| |
subG1: (G \subset [1]) = (G :==: 1).
Proof. by rewrite eqEsubset sub1G 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
|
subG1
| |
setI1g: 1 :&: G = 1. Proof. exact: (setIidPl sub1G). 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
|
setI1g
| |
setIg1: G :&: 1 = 1. Proof. exact: (setIidPr sub1G). 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
|
setIg1
| |
subG1_contraH : G \subset H -> G :!=: 1 -> H :!=: 1.
Proof. by move=> sGH; rewrite -subG1; apply: contraNneq => <-. 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
|
subG1_contra
| |
repr_group: repr G = 1. Proof. by rewrite /repr 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
|
repr_group
| |
cardG_gt0: 0 < #|G|.
Proof. by rewrite lt0n; apply/existsP; exists (1 : gT). 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
|
cardG_gt0
| |
indexg_gt0A : 0 < #|G : A|.
Proof.
rewrite lt0n; apply/existsP; exists A.
by rewrite -{2}[A]mulg1 -rcosetE; apply: imset_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
|
indexg_gt0
| |
trivgP: reflect (G :=: 1) (G \subset [1]).
Proof. by rewrite subG1; apply: eqP. 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
|
trivgP
| |
trivGP: reflect (G = 1%G) (G \subset [1]).
Proof. by rewrite subG1; apply: eqP. 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
|
trivGP
| |
proper1G: ([1] \proper G) = (G :!=: 1).
Proof. by rewrite properEneq sub1G andbT eq_sym. 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
|
proper1G
| |
in_one_groupx : (x \in 1%G) = (x == 1).
Proof. by rewrite -[x \in _]/(x \in [set 1]) !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
|
in_one_group
| |
inE:= (in_one_group, inE).
|
Definition
|
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
|
inE
| |
trivgPn: reflect (exists2 x, x \in G & x != 1) (G :!=: 1).
Proof.
rewrite -subG1.
by apply: (iffP subsetPn) => [] [x Gx x1]; exists x; rewrite ?inE in x1 *.
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
|
trivgPn
| |
trivg_card_le1: (G :==: 1) = (#|G| <= 1).
Proof. by rewrite eq_sym eqEcard cards1 sub1G. 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
|
trivg_card_le1
| |
trivg_card1: (G :==: 1) = (#|G| == 1%N).
Proof. by rewrite trivg_card_le1 eqn_leq cardG_gt0 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
|
trivg_card1
| |
cardG_gt1: (#|G| > 1) = (G :!=: 1).
Proof. by rewrite trivg_card_le1 ltnNge. 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
|
cardG_gt1
| |
card_le1_trivg: #|G| <= 1 -> G :=: 1.
Proof. by rewrite -trivg_card_le1; move/eqP. 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
|
card_le1_trivg
| |
card1_trivg: #|G| = 1%N -> G :=: 1.
Proof. by move=> G1; rewrite card_le1_trivg ?G1. 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
|
card1_trivg
| |
mulG_sublA : A \subset A * G.
Proof. exact: mulg_subl 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
|
mulG_subl
| |
mulG_subrA : A \subset (G * A).
Proof. exact: mulg_subr 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
|
mulG_subr
| |
mulGid: G * G = G.
Proof.
by apply/eqP; rewrite eqEsubset mulG_subr andbT; case/andP: (valP G).
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
|
mulGid
| |
mulGSA B : (G * A \subset G * B) = (A \subset G * B).
Proof.
apply/idP/idP; first exact: subset_trans (mulG_subr A).
by move/(mulgS G); rewrite mulgA mulGid.
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
|
mulGS
| |
mulSGA B : (A * G \subset B * G) = (A \subset B * G).
Proof.
apply/idP/idP; first exact: subset_trans (mulG_subl A).
by move/(mulSg G); rewrite -mulgA mulGid.
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
|
mulSG
| |
mul_subGA B : A \subset G -> B \subset G -> A * B \subset G.
Proof. by move=> sAG sBG; rewrite -mulGid mulgSS. 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_subG
| |
prod_subG(I : Type) (r : seq I) (P : {pred I}) (F : I -> {set gT}) :
(forall i, P i -> F i \subset G) -> \prod_(i <- r | P i) F i \subset G.
Proof.
move=> subFG; elim/big_rec: _ => [|/= i A /subFG]; first by rewrite sub1set.
exact: mul_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
|
prod_subG
| |
groupMx y : x \in G -> y \in G -> x * y \in G.
Proof. by case/group_setP: (valP G) x y. 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
|
groupM
| |
groupXx n : x \in G -> x ^+ n \in G.
Proof. by move=> Gx; elim: n => [|n IHn]; rewrite ?group1 // expgS groupM. 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
|
groupX
| |
groupVrx : x \in G -> x^-1 \in G.
Proof.
move=> Gx; rewrite -(mul1g x^-1) -mem_rcoset ((G :* x =P G) _) //.
by rewrite eqEcard card_rcoset leqnn mul_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
|
groupVr
| |
groupVlx : x^-1 \in G -> x \in G.
Proof. by move/groupVr; rewrite invgK. 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
|
groupVl
| |
groupVx : (x^-1 \in G) = (x \in G).
Proof. by apply/idP/idP; [apply: groupVl | apply: groupVr]. 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
|
groupV
| |
groupMlx y : x \in G -> (x * y \in G) = (y \in G).
Proof.
move=> Gx; apply/idP/idP=> [Gxy|]; last exact: groupM.
by rewrite -(mulKg x y) groupM ?groupVr.
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
|
groupMl
| |
groupMrx y : x \in G -> (y * x \in G) = (y \in G).
Proof. by move=> Gx; rewrite -[_ \in G]groupV invMg groupMl groupV. 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
|
groupMr
| |
in_group:= (group1, groupV, (groupMl, groupX)).
|
Definition
|
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
|
in_group
| |
groupJx y : x \in G -> y \in G -> x ^ y \in G.
Proof. by move=> Gx Gy; rewrite !in_group. 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
|
groupJ
| |
groupJrx y : y \in G -> (x ^ y \in G) = (x \in G).
Proof. by move=> Gy; rewrite groupMl (groupMr, groupV). 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
|
groupJr
| |
groupRx y : x \in G -> y \in G -> [~ x, y] \in G.
Proof. by move=> Gx Gy; rewrite !in_group. 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
|
groupR
| |
group_prodI r (P : pred I) F :
(forall i, P i -> F i \in G) -> \prod_(i <- r | P i) F i \in G.
Proof. by move=> G_P; elim/big_ind: _ => //; apply: groupM. 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_prod
| |
invGid: G^-1 = G. Proof. by apply/setP=> x; rewrite inE groupV. 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
|
invGid
| |
inv_subGA : (A^-1 \subset G) = (A \subset G).
Proof. by rewrite -{1}invGid invSg. 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
|
inv_subG
| |
invg_lcosetx : (x *: G)^-1 = G :* x^-1.
Proof. by rewrite invMg invGid invg_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
|
invg_lcoset
| |
invg_rcosetx : (G :* x)^-1 = x^-1 *: G.
Proof. by rewrite invMg invGid invg_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
|
invg_rcoset
| |
memV_lcosetVx y : (y^-1 \in x^-1 *: G) = (y \in G :* x).
Proof. by rewrite -invg_rcoset memV_invg. 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
|
memV_lcosetV
| |
memV_rcosetVx y : (y^-1 \in G :* x^-1) = (y \in x *: G).
Proof. by rewrite -invg_lcoset memV_invg. 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
|
memV_rcosetV
| |
mulSgGidA x : x \in A -> A \subset G -> A * G = G.
Proof.
move=> Ax sAG; apply/eqP; rewrite eqEsubset -{2}mulGid mulSg //=.
apply/subsetP=> y Gy; rewrite -(mulKVg x y) mem_mulg // groupMr // groupV.
exact: (subsetP sAG).
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
|
mulSgGid
| |
mulGSgidA x : x \in A -> A \subset G -> G * A = G.
Proof.
rewrite -memV_invg -invSg invGid => Ax sAG.
by apply: invg_inj; rewrite invMg invGid (mulSgGid Ax).
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
|
mulGSgid
| |
lcoset_reflx : x \in x *: G.
Proof. by rewrite mem_lcoset mulVg 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
|
lcoset_refl
| |
lcoset_symx y : (x \in y *: G) = (y \in x *: G).
Proof. by rewrite !mem_lcoset -groupV invMg invgK. 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
|
lcoset_sym
| |
lcoset_eqP{x y} : reflect (x *: G = y *: G) (x \in y *: G).
Proof.
suffices <-: (x *: G == y *: G) = (x \in y *: G) by apply: eqP.
by rewrite eqEsubset !mulSG !sub1set lcoset_sym andbb.
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
|
lcoset_eqP
| |
lcoset_translx y z : x \in y *: G -> (x \in z *: G) = (y \in z *: G).
Proof. by move=> Gyx; rewrite -2!(lcoset_sym z) (lcoset_eqP Gyx). 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
|
lcoset_transl
| |
lcoset_transx y z : x \in y *: G -> y \in z *: G -> x \in z *: G.
Proof. by move/lcoset_transl->. 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
|
lcoset_trans
| |
lcoset_idx : x \in G -> x *: G = G.
Proof. by move=> Gx; rewrite (lcoset_eqP (_ : x \in 1 *: G)) 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
|
lcoset_id
| |
rcoset_reflx : x \in G :* x.
Proof. by rewrite mem_rcoset mulgV 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
|
rcoset_refl
| |
rcoset_symx y : (x \in G :* y) = (y \in G :* x).
Proof. by rewrite -!memV_lcosetV lcoset_sym. 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
|
rcoset_sym
| |
rcoset_eqP{x y} : reflect (G :* x = G :* y) (x \in G :* y).
Proof.
suffices <-: (G :* x == G :* y) = (x \in G :* y) by apply: eqP.
by rewrite eqEsubset !mulGS !sub1set rcoset_sym andbb.
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
|
rcoset_eqP
| |
rcoset_translx y z : x \in G :* y -> (x \in G :* z) = (y \in G :* z).
Proof. by move=> Gyx; rewrite -2!(rcoset_sym z) (rcoset_eqP Gyx). 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
|
rcoset_transl
| |
rcoset_transx y z : x \in G :* y -> y \in G :* z -> x \in G :* z.
Proof. by move/rcoset_transl->. 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
|
rcoset_trans
| |
rcoset_idx : x \in G -> G :* x = G.
Proof. by move=> Gx; rewrite (rcoset_eqP (_ : x \in G :* 1)) 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
|
rcoset_id
| |
rcoset_repr_specx : gT -> Type :=
RcosetReprSpec g : g \in G -> rcoset_repr_spec x (g * x).
|
Variant
|
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
|
rcoset_repr_spec
| |
mem_repr_rcosetx : repr (G :* x) \in G :* x.
Proof. exact: mem_repr (rcoset_refl 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_repr_rcoset
| |
repr_rcosetPx : rcoset_repr_spec x (repr (G :* x)).
Proof.
by rewrite -[repr _](mulgKV x); split; rewrite -mem_rcoset mem_repr_rcoset.
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
|
repr_rcosetP
| |
rcoset_reprx : G :* (repr (G :* x)) = G :* x.
Proof. exact/rcoset_eqP/mem_repr_rcoset. 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
|
rcoset_repr
|
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