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 |
|---|---|---|---|---|---|---|
purely_inseparableU W : bool :=
all (purely_inseparable_element U) (vbasis W).
|
Definition
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
purely_inseparable
| |
separable_addK x y :
separable_element K x -> separable_element K y -> separable_element K (x + y).
Proof.
move/(separable_elementS (subv_adjoin K y))=> sepKy_x sepKy.
have [z defKz] := Primitive_Element_Theorem x sepKy.
have /(adjoin_separableP _): x + y \in <<K; z>>%VS.
by rewrite -defKz rpredD ?memv_adjoin // subvP_adjoin ?memv_adjoin.
apply; apply: adjoin_separable sepKy (adjoin_separable sepKy_x _).
by rewrite defKz base_separable ?memv_adjoin.
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separable_add
| |
separable_sumI r (P : pred I) (v_ : I -> L) K :
(forall i, P i -> separable_element K (v_ i)) ->
separable_element K (\sum_(i <- r | P i) v_ i).
Proof.
move=> sepKi.
by elim/big_ind: _; [apply/base_separable/mem0v | apply: separable_add |].
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separable_sum
| |
inseparable_addK x y :
purely_inseparable_element K x -> purely_inseparable_element K y ->
purely_inseparable_element K (x + y).
Proof.
have insepP := purely_inseparable_elementP_pchar.
move=> /insepP[n pcharLn Kxn] /insepP[m pcharLm Kym]; apply/insepP.
have pcharLnm: [pchar L].-nat (n * m)%N by rewrite pnatM pcharLn.
by exists (n * m)%N; rewrite ?exprDn_pchar // {2}mulnC !exprM memvD // rpredX.
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
inseparable_add
| |
inseparable_sumI r (P : pred I) (v_ : I -> L) K :
(forall i, P i -> purely_inseparable_element K (v_ i)) ->
purely_inseparable_element K (\sum_(i <- r | P i) v_ i).
Proof.
move=> insepKi.
by elim/big_ind: _; [apply/base_inseparable/mem0v | apply: inseparable_add |].
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
inseparable_sum
| |
separableP{K E} :
reflect (forall y, y \in E -> separable_element K y) (separable K E).
Proof.
apply/(iffP idP)=> [/allP|] sepK_E; last by apply/allP=> x /vbasis_mem/sepK_E.
move=> y /coord_vbasis->; apply/separable_sum=> i _.
have: separable_element K (vbasis E)`_i by apply/sepK_E/memt_nth.
by move/adjoin_separableP; apply; rewrite rpredZ ?memv_adjoin.
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separableP
| |
purely_inseparableP{K E} :
reflect (forall y, y \in E -> purely_inseparable_element K y)
(purely_inseparable K E).
Proof.
apply/(iffP idP)=> [/allP|] sep'K_E; last by apply/allP=> x /vbasis_mem/sep'K_E.
move=> y /coord_vbasis->; apply/inseparable_sum=> i _.
have: purely_inseparable_element K (vbasis E)`_i by apply/sep'K_E/memt_nth.
case/purely_inseparable_elementP_pchar=> n pcharLn K_Ein.
by apply/purely_inseparable_elementP_pchar; exists n; rewrite // exprZn rpredZ.
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
purely_inseparableP
| |
adjoin_separable_eqK x : separable_element K x = separable K <<K; x>>%VS.
Proof. exact: sameP adjoin_separableP separableP. Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
adjoin_separable_eq
| |
separable_inseparable_decompositionE K :
{x | x \in E /\ separable_element K x & purely_inseparable <<K; x>> E}.
Proof.
without loss sKE: K / (K <= E)%VS.
case/(_ _ (capvSr K E)) => x [Ex sepKEx] /purely_inseparableP sep'KExE.
exists x; first by split; last exact/(separable_elementS _ sepKEx)/capvSl.
apply/purely_inseparableP=> y /sep'KExE; apply: sub_inseparable.
exact/adjoinSl/capvSl.
pose E_ i := (vbasis E)`_i; pose fP i := separable_exponent_pchar K (E_ i).
pose f i := E_ i ^+ ex_minn (fP i); pose s := mkseq f (\dim E).
pose K' := <<K & s>>%VS.
have sepKs: all (separable_element K) s.
by rewrite all_map /f; apply/allP=> i _ /=; case: ex_minnP => m /andP[].
have [x sepKx defKx]: {x | x \in E /\ separable_element K x & K' = <<K; x>>%VS}.
have: all [in E] s.
rewrite all_map; apply/allP=> i; rewrite mem_iota => ltis /=.
by rewrite rpredX // vbasis_mem // memt_nth.
rewrite {}/K'; elim/last_ind: s sepKs => [|s t IHs].
by exists 0; [rewrite base_separable mem0v | rewrite adjoin_nil addv0].
rewrite adjoin_rcons !all_rcons => /andP[sepKt sepKs] /andP[/= Et Es].
have{IHs sepKs Es} [y [Ey sepKy] ->{s}] := IHs sepKs Es.
have /sig_eqW[x defKx] := Primitive_Element_Theorem t sepKy.
exists x; [split | exact: defKx].
suffices: (<<K; x>> <= E)%VS by case/FadjoinP.
by rewrite -defKx !(sameP FadjoinP andP) sKE Ey Et.
apply/adjoin_separableP=> z; rewrite -defKx => Kyt_z.
apply: adjoin_separable sepKy _; apply: adjoin_separableP Kyt_z.
exact: separable_elementS (subv_adjoin K y
...
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separable_inseparable_decomposition
| |
separable_generatorK E : L :=
s2val (locked (separable_inseparable_decomposition E K)).
|
Definition
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separable_generator
| |
separable_generator_memE K : separable_generator K E \in E.
Proof. by rewrite /separable_generator; case: (locked _) => ? []. Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separable_generator_mem
| |
separable_generatorPE K : separable_element K (separable_generator K E).
Proof. by rewrite /separable_generator; case: (locked _) => ? []. Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separable_generatorP
| |
separable_generator_maximalE K :
purely_inseparable <<K; separable_generator K E>> E.
Proof. by rewrite /separable_generator; case: (locked _). Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separable_generator_maximal
| |
sub_adjoin_separable_generatorE K :
separable K E -> (E <= <<K; separable_generator K E>>)%VS.
Proof.
move/separableP=> sepK_E; apply/subvP=> v Ev.
rewrite -separable_inseparable_element.
have /purely_inseparableP-> // := separable_generator_maximal E K.
by rewrite (separable_elementS _ (sepK_E _ Ev)) // subv_adjoin.
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
sub_adjoin_separable_generator
| |
eq_adjoin_separable_generatorE K :
separable K E -> (K <= E)%VS ->
E = <<K; separable_generator K E>>%VS :> {vspace _}.
Proof.
move=> sepK_E sKE; apply/eqP; rewrite eqEsubv sub_adjoin_separable_generator //.
by apply/FadjoinP/andP; rewrite sKE separable_generator_mem.
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
eq_adjoin_separable_generator
| |
separable_reflK : separable K K.
Proof. exact/separableP/base_separable. Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separable_refl
| |
separable_transM K E : separable K M -> separable M E -> separable K E.
Proof.
move/sub_adjoin_separable_generator.
set x := separable_generator K M => sMKx /separableP sepM_E.
apply/separableP => w /sepM_E/(separable_elementS sMKx).
case/strong_Primitive_Element_Theorem => _ _ -> //.
exact: separable_generatorP.
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separable_trans
| |
separableSK1 K2 E2 E1 :
(K1 <= K2)%VS -> (E2 <= E1)%VS -> separable K1 E1 -> separable K2 E2.
Proof.
move=> sK12 /subvP sE21 /separableP sepK1_E1.
by apply/separableP=> y /sE21/sepK1_E1/(separable_elementS sK12).
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separableS
| |
separableSlK M E : (K <= M)%VS -> separable K E -> separable M E.
Proof. by move/separableS; apply. Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separableSl
| |
separableSrK M E : (M <= E)%VS -> separable K E -> separable K M.
Proof. exact: separableS. Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separableSr
| |
separable_Fadjoin_seqK rs :
all (separable_element K) rs -> separable K <<K & rs>>.
Proof.
elim/last_ind: rs => [|s x IHs] in K *.
by rewrite adjoin_nil subfield_closed separable_refl.
rewrite all_rcons adjoin_rcons => /andP[sepKx /IHs/separable_trans-> //].
by rewrite -adjoin_separable_eq (separable_elementS _ sepKx) ?subv_adjoin_seq.
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
separable_Fadjoin_seq
| |
purely_inseparable_reflK : purely_inseparable K K.
Proof. by apply/purely_inseparableP; apply: base_inseparable. Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
purely_inseparable_refl
| |
purely_inseparable_transM K E :
purely_inseparable K M -> purely_inseparable M E -> purely_inseparable K E.
Proof.
have insepP := purely_inseparableP => /insepP insepK_M /insepP insepM_E.
have insepPe := purely_inseparable_elementP_pchar.
apply/insepP=> x /insepM_E/insepPe[n pcharLn /insepK_M/insepPe[m pcharLm Kxnm]].
by apply/insepPe; exists (n * m)%N; rewrite ?exprM // pnatM pcharLn pcharLm.
Qed.
|
Lemma
|
field
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset prime",
"From mathcomp Require Import binomial ssralg poly polydiv fingroup perm",
"From mathcomp Require Import morphism quotient gproduct finalg zmodp cyclic",
"From mathcomp Require Import matrix mxalgebra mxpoly polyXY vector falgebra",
"From mathcomp Require Import fieldext"
] |
field/separable.v
|
purely_inseparable_trans
| |
act_morphto x := forall a b, to x (a * b) = to (to x a) b.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
act_morph
| |
is_actionto :=
left_injective to /\ forall x, {in D &, act_morph to x}.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
is_action
| |
action:= Action {act :> rT -> aT -> rT; _ : is_action act}.
|
Record
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
action
| |
clone_actionto :=
let: Action _ toP := to return {type of Action for to} -> action in
fun k => k toP.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
clone_action
| |
act_domaT D rT of @action aT D rT := D.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
act_dom
| |
is_total_action: is_action setT to.
Proof.
split=> [a | x a b _ _] /=; last by rewrite toM.
by apply: can_inj (to^~ a^-1) _ => x; rewrite -toM ?mulgV.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
is_total_action
| |
TotalAction:= Action is_total_action.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
TotalAction
| |
morph_actrT rT' (to : action D rT) (to' : action D' rT') f fA :=
forall x a, f (to x a) = to' (f x) (fA a).
Variable rT : finType.
Implicit Type to : action D rT.
Implicit Type A : {set aT}.
Implicit Type S : {set rT}.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
morph_act
| |
actmto a := if a \in D then to^~ a else id.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actm
| |
setactto S a := [set to x a | x in S].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
setact
| |
orbitto A x := to x @: A.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbit
| |
amoveto A x y := [set a in A | to x a == y].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
amove
| |
afixto A := [set x | A \subset [set a | to x a == x]].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
afix
| |
astabS to := D :&: [set a | S \subset [set x | to x a == x]].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astab
| |
astabsS to := D :&: [set a | S \subset to^~ a @^-1: S].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabs
| |
acts_onA S to := {in A, forall a x, (to x a \in S) = (x \in S)}.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
acts_on
| |
atransA S to := S \in orbit to A @: S.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
atrans
| |
faithfulA S to := A :&: astab S to \subset [1].
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
faithful
| |
act_inj: left_injective to. Proof. by case: to => ? []. Qed.
Arguments act_inj : clear implicits.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
act_inj
| |
actMinx : {in D &, act_morph to x}.
Proof. by case: to => ? []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actMin
| |
actmEfuna : a \in D -> actm to a = to^~ a.
Proof. by rewrite /actm => ->. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actmEfun
| |
actmEa : a \in D -> actm to a =1 to^~ a.
Proof. by move=> Da; rewrite actmEfun. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actmE
| |
setactES a : to^* S a = [set to x a | x in S].
Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
setactE
| |
mem_setactS a x : x \in S -> to x a \in to^* S a.
Proof. exact: imset_f. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
mem_setact
| |
card_setactS a : #|to^* S a| = #|S|.
Proof. by apply: card_imset; apply: act_inj. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
card_setact
| |
setact_is_action: is_action D to^*.
Proof.
split=> [a R S eqRS | a b Da Db S]; last first.
by rewrite /setact /= -imset_comp; apply: eq_imset => x; apply: actMin.
apply/setP=> x; apply/idP/idP=> /(mem_setact a).
by rewrite eqRS => /imsetP[y Sy /act_inj->].
by rewrite -eqRS => /imsetP[y Sy /act_inj->].
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
setact_is_action
| |
set_action:= Action setact_is_action.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
set_action
| |
orbitEA x : orbit to A x = to x @: A. Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbitE
| |
orbitPA x y :
reflect (exists2 a, a \in A & to x a = y) (y \in orbit to A x).
Proof. by apply: (iffP imsetP) => [] [a]; exists a. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbitP
| |
mem_orbitA x a : a \in A -> to x a \in orbit to A x.
Proof. exact: imset_f. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
mem_orbit
| |
afixPA x : reflect (forall a, a \in A -> to x a = x) (x \in 'Fix_to(A)).
Proof.
rewrite inE; apply: (iffP subsetP) => [xfix a /xfix | xfix a Aa].
by rewrite inE => /eqP.
by rewrite inE xfix.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
afixP
| |
afixSA B : A \subset B -> 'Fix_to(B) \subset 'Fix_to(A).
Proof. by move=> sAB; apply/subsetP=> u /[!inE]; apply: subset_trans. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
afixS
| |
afixUA B : 'Fix_to(A :|: B) = 'Fix_to(A) :&: 'Fix_to(B).
Proof. by apply/setP=> x; rewrite !inE subUset. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
afixU
| |
afix1Pa x : reflect (to x a = x) (x \in 'Fix_to[a]).
Proof. by rewrite inE sub1set inE; apply: eqP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
afix1P
| |
astabIdomS : 'C_D(S | to) = 'C(S | to).
Proof. by rewrite setIA setIid. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabIdom
| |
astab_domS : {subset 'C(S | to) <= D}.
Proof. by move=> a /setIP[]. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astab_dom
| |
astab_actS a x : a \in 'C(S | to) -> x \in S -> to x a = x.
Proof.
rewrite 2!inE => /andP[_ cSa] Sx; apply/eqP.
by have /[1!inE] := subsetP cSa x Sx.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astab_act
| |
astabSS1 S2 : S1 \subset S2 -> 'C(S2 | to) \subset 'C(S1 | to).
Proof.
by move=> sS12; apply/subsetP=> x /[!inE] /andP[->]; apply: subset_trans.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabS
| |
astabsIdomS : 'N_D(S | to) = 'N(S | to).
Proof. by rewrite setIA setIid. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabsIdom
| |
astabs_domS : {subset 'N(S | to) <= D}.
Proof. by move=> a /setIdP[]. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabs_dom
| |
astabs_actS a x : a \in 'N(S | to) -> (to x a \in S) = (x \in S).
Proof.
rewrite 2!inE subEproper properEcard => /andP[_].
rewrite (card_preimset _ (act_inj _)) ltnn andbF orbF => /eqP{2}->.
by rewrite inE.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabs_act
| |
astab_subS : 'C(S | to) \subset 'N(S | to).
Proof.
apply/subsetP=> a cSa; rewrite !inE (astab_dom cSa).
by apply/subsetP=> x Sx; rewrite inE (astab_act cSa).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astab_sub
| |
astabsCS : 'N(~: S | to) = 'N(S | to).
Proof.
apply/setP=> a; apply/idP/idP=> nSa; rewrite !inE (astabs_dom nSa).
by rewrite -setCS -preimsetC; apply/subsetP=> x; rewrite inE astabs_act.
by rewrite preimsetC setCS; apply/subsetP=> x; rewrite inE astabs_act.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabsC
| |
astabsIS T : 'N(S | to) :&: 'N(T | to) \subset 'N(S :&: T | to).
Proof.
apply/subsetP=> a; rewrite !inE -!andbA preimsetI => /and4P[-> nSa _ nTa] /=.
by rewrite setISS.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabsI
| |
astabs_setactS a : a \in 'N(S | to) -> to^* S a = S.
Proof.
move=> nSa; apply/eqP; rewrite eqEcard card_setact leqnn andbT.
by apply/subsetP=> _ /imsetP[x Sx ->]; rewrite astabs_act.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabs_setact
| |
astab1_setS : 'C[S | set_action] = 'N(S | to).
Proof.
apply/setP=> a; apply/idP/idP=> nSa.
case/setIdP: nSa => Da; rewrite !inE Da sub1set inE => /eqP defS.
by apply/subsetP=> x Sx; rewrite inE -defS mem_setact.
by rewrite !inE (astabs_dom nSa) sub1set inE /= astabs_setact.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astab1_set
| |
astabs_set1x : 'N([set x] | to) = 'C[x | to].
Proof.
apply/eqP; rewrite eqEsubset astab_sub andbC setIS //.
by apply/subsetP=> a; rewrite ?(inE,sub1set).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabs_set1
| |
acts_domA S : [acts A, on S | to] -> A \subset D.
Proof. by move=> nSA; rewrite (subset_trans nSA) ?subsetIl. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
acts_dom
| |
acts_actA S : [acts A, on S | to] -> {acts A, on S | to}.
Proof. by move=> nAS a Aa x; rewrite astabs_act ?(subsetP nAS). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
acts_act
| |
astabCinA S :
A \subset D -> (A \subset 'C(S | to)) = (S \subset 'Fix_to(A)).
Proof.
move=> sAD; apply/subsetP/subsetP=> [sAC x xS | sSF a aA].
by apply/afixP=> a aA; apply: astab_act (sAC _ aA) xS.
rewrite !inE (subsetP sAD _ aA); apply/subsetP=> x xS.
by move/afixP/(_ _ aA): (sSF _ xS) => /[1!inE] ->.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabCin
| |
astabU: 'C(S :|: T | to) = 'C(S | to) :&: 'C(T | to).
Proof. by apply/setP=> a; rewrite !inE subUset; case: (a \in D). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabU
| |
astabsU: 'N(S | to) :&: 'N(T | to) \subset 'N(S :|: T | to).
Proof.
by rewrite -(astabsC S) -(astabsC T) -(astabsC (S :|: T)) setCU astabsI.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabsU
| |
astabsD: 'N(S | to) :&: 'N(T | to) \subset 'N(S :\: T| to).
Proof. by rewrite setDE -(astabsC T) astabsI. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
astabsD
| |
actsI: [acts A, on S :&: T | to].
Proof. by apply: subset_trans (astabsI S T); rewrite subsetI AactS. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actsI
| |
actsU: [acts A, on S :|: T | to].
Proof. by apply: subset_trans astabsU; rewrite subsetI AactS. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actsU
| |
actsD: [acts A, on S :\: T | to].
Proof. by apply: subset_trans astabsD; rewrite subsetI AactS. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actsD
| |
acts_in_orbitA S x y :
[acts A, on S | to] -> y \in orbit to A x -> x \in S -> y \in S.
Proof.
by move=> nSA/imsetP[a Aa ->{y}] Sx; rewrite (astabs_act _ (subsetP nSA a Aa)).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
acts_in_orbit
| |
subset_faithfulA B S :
B \subset A -> [faithful A, on S | to] -> [faithful B, on S | to].
Proof. by move=> sAB; apply: subset_trans; apply: setSI. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
subset_faithful
| |
reindex_astabsa F : a \in 'N(S | to) ->
\big[op/idx]_(i in S) F i = \big[op/idx]_(i in S) F (to i a).
Proof.
move=> nSa; rewrite (reindex_inj (act_inj a)); apply: eq_bigl => x.
exact: astabs_act.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
reindex_astabs
| |
reindex_actsA a F : [acts A, on S | to] -> a \in A ->
\big[op/idx]_(i in S) F i = \big[op/idx]_(i in S) F (to i a).
Proof. by move=> nSA /(subsetP nSA); apply: reindex_astabs. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
reindex_acts
| |
act1x : to x 1 = x.
Proof. by apply: (act_inj to 1); rewrite -actMin ?mulg1. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
act1
| |
actKin: {in D, right_loop inv to}.
Proof. by move=> a Da /= x; rewrite -actMin ?groupV // mulgV act1. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actKin
| |
actKVin: {in D, rev_right_loop inv to}.
Proof. by move=> a Da /= x; rewrite -{2}(invgK a) actKin ?groupV. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actKVin
| |
setactVinS a : a \in D -> to^* S a^-1 = to^~ a @^-1: S.
Proof.
by move=> Da; apply: can2_imset_pre; [apply: actKVin | apply: actKin].
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
setactVin
| |
actXinx a i : a \in D -> to x (a ^+ i) = iter i (to^~ a) x.
Proof.
move=> Da; elim: i => /= [|i <-]; first by rewrite act1.
by rewrite expgSr actMin ?groupX.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
actXin
| |
afix1: 'Fix_to(1) = setT.
Proof. by apply/setP=> x; rewrite !inE sub1set inE act1 eqxx. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
afix1
| |
afixD1G : 'Fix_to(G^#) = 'Fix_to(G).
Proof. by rewrite -{2}(setD1K (group1 G)) afixU afix1 setTI. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
afixD1
| |
orbit_reflG x : x \in orbit to G x.
Proof. by rewrite -{1}[x]act1 mem_orbit. Qed.
Local Notation orbit_rel A := (fun x y => x \in orbit to A y).
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbit_refl
| |
contra_orbitG x y : x \notin orbit to G y -> x != y.
Proof. by apply: contraNneq => ->; apply: orbit_refl. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
contra_orbit
| |
orbit_in_symG : G \subset D -> symmetric (orbit_rel G).
Proof.
move=> sGD; apply: symmetric_from_pre => x y /imsetP[a Ga].
by move/(canLR (actKin (subsetP sGD a Ga))) <-; rewrite mem_orbit ?groupV.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbit_in_sym
| |
orbit_in_transG : G \subset D -> transitive (orbit_rel G).
Proof.
move=> sGD _ _ z /imsetP[a Ga ->] /imsetP[b Gb ->].
by rewrite -actMin ?mem_orbit ?groupM // (subsetP sGD).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbit_in_trans
| |
orbit_in_eqPG x y :
G \subset D -> reflect (orbit to G x = orbit to G y) (x \in orbit to G y).
Proof.
move=> sGD; apply: (iffP idP) => [yGx|<-]; last exact: orbit_refl.
by apply/setP=> z; apply/idP/idP=> /orbit_in_trans-> //; rewrite orbit_in_sym.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbit_in_eqP
| |
orbit_in_translG x y z :
G \subset D -> y \in orbit to G x ->
(y \in orbit to G z) = (x \in orbit to G z).
Proof.
by move=> sGD Gxy; rewrite !(orbit_in_sym sGD _ z) (orbit_in_eqP y x sGD Gxy).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbit_in_transl
| |
orbit_act_inx a G :
G \subset D -> a \in G -> orbit to G (to x a) = orbit to G x.
Proof. by move=> sGD /mem_orbit/orbit_in_eqP->. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbit_act_in
| |
orbit_actr_inx a G y :
G \subset D -> a \in G -> (to y a \in orbit to G x) = (y \in orbit to G x).
Proof. by move=> sGD /mem_orbit/orbit_in_transl->. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbit_actr_in
| |
orbit_inv_inA x y :
A \subset D -> (y \in orbit to A^-1 x) = (x \in orbit to A y).
Proof.
move/subsetP=> sAD; apply/imsetP/imsetP=> [] [a Aa ->].
by exists a^-1; rewrite -?mem_invg ?actKin // -groupV sAD -?mem_invg.
by exists a^-1; rewrite ?memV_invg ?actKin // sAD.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbit_inv_in
| |
orbit_lcoset_inA a x :
A \subset D -> a \in D ->
orbit to (a *: A) x = orbit to A (to x a).
Proof.
move/subsetP=> sAD Da; apply/setP=> y; apply/imsetP/imsetP=> [] [b Ab ->{y}].
by exists (a^-1 * b); rewrite -?actMin ?mulKVg // ?sAD -?mem_lcoset.
by exists (a * b); rewrite ?mem_mulg ?set11 ?actMin // sAD.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun ssrnotations eqtype",
"From mathcomp Require Import ssrnat div seq prime fintype bigop finset",
"From mathcomp Require Import fingroup morphism perm automorphism quotient"
] |
fingroup/action.v
|
orbit_lcoset_in
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.