statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
abelem_homocyclic p G : p.-abelem G -> homocyclic G. | Proof.
move=> abelG; have [_ cGG _] := and3P abelG.
rewrite /homocyclic cGG (@all_pred1_constant _ p) //.
case/abelian_structure: cGG (abelian_type_gt1 G) => b defG <- => b_gt1.
apply/allP=> _ /mapP[x b_x ->] /=; rewrite (abelem_order_p abelG) //.
rewrite -cycle_subG -(bigdprodWY defG) ?sub_gen //.
by rewrite bigcu... | Lemma | abelem_homocyclic | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelem",
"abelem_order_p",
"abelian_structure",
"abelian_type_gt1",
"allP",
"all_pred1_constant",
"apply",
"bigcup_seq",
"bigcup_sup",
"bigdprodWY",
"cGG",
"cycle_subG",
"defG",
"homocyclic",
"mapP",
"map_f",
"order_gt1",
"sub_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homocyclic1 : homocyclic [1 gT]. | Proof. exact: abelem_homocyclic (abelem1 _ 2). Qed. | Lemma | homocyclic1 | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelem1",
"abelem_homocyclic",
"gT",
"homocyclic"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Ohm1_homocyclicP p G : p.-group G -> abelian G ->
reflect ('Ohm_1(G) = 'Mho^(logn p (exponent G)).-1(G)) (homocyclic G). | Proof.
move=> pG cGG; set e := logn p (exponent G); rewrite -subn1.
apply: (iffP idP) => [homoG | ]; first exact: homocyclic_Ohm_Mho.
case: (ltnP 1 e) => [lt1e | ]; first exact: Ohm_Mho_homocyclic.
rewrite -subn_eq0 => /eqP->; rewrite Mho0 => <-.
exact: abelem_homocyclic (Ohm1_abelem pG cGG).
Qed. | Lemma | Ohm1_homocyclicP | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"Mho",
"Mho0",
"Ohm1_abelem",
"Ohm_Mho_homocyclic",
"abelem_homocyclic",
"abelian",
"apply",
"cGG",
"exponent",
"group",
"homocyclic",
"homocyclic_Ohm_Mho",
"logn",
"ltnP",
"pG",
"subn1",
"subn_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelian_type_homocyclic G :
homocyclic G -> abelian_type G = nseq 'r(G) (exponent G). | Proof.
case/andP=> cGG; rewrite -size_abelian_type // /abelian_type.
rewrite -(prednK (cardG_gt0 G)) /=; case: andP => //= _; move: (tag _) => H.
by move/all_pred1P->; rewrite genGid size_nseq.
Qed. | Lemma | abelian_type_homocyclic | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelian_type",
"all_pred1P",
"cGG",
"cardG_gt0",
"exponent",
"genGid",
"homocyclic",
"nseq",
"prednK",
"size_abelian_type",
"size_nseq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelian_type_abelem p G : p.-abelem G -> abelian_type G = nseq 'r(G) p. | Proof.
move=> abelG; rewrite (abelian_type_homocyclic (abelem_homocyclic abelG)).
have [-> | ntG] := eqVneq G 1%G; first by rewrite rank1.
congr nseq; apply/eqP; rewrite eqn_dvd; have [pG _ ->] := and3P abelG.
have [p_pr] := pgroup_pdiv pG ntG; case/Cauchy=> // x Gx <- _.
exact: dvdn_exponent.
Qed. | Lemma | abelian_type_abelem | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"Cauchy",
"abelem",
"abelem_homocyclic",
"abelian_type",
"abelian_type_homocyclic",
"apply",
"dvdn_exponent",
"eqVneq",
"eqn_dvd",
"nseq",
"pG",
"p_pr",
"pgroup_pdiv",
"rank1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_card_abelian G :
abelian G -> #|G| <= exponent G ^ 'r(G) ?= iff homocyclic G. | Proof.
move=> cGG; have [b defG def_tG] := abelian_structure cGG.
have Gb: all [in G] b.
apply/allP=> x b_x; rewrite -(bigdprodWY defG); have [b1 b2] := splitPr b_x.
by rewrite big_cat big_cons /= mem_gen // setUCA inE cycle_id.
have ->: homocyclic G = all (pred1 (exponent G)) (abelian_type G).
rewrite /homocycli... | Lemma | max_card_abelian | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelian",
"abelian_structure",
"abelian_type",
"all",
"allP",
"apply",
"big_cat",
"big_cons",
"bigdprodWY",
"bigdprod_card",
"cGG",
"cycle_id",
"defG",
"dvdn_exponent",
"dvdn_leq",
"eqn0Ngt",
"eqxx",
"expnS",
"expn_eq0",
"exponent",
"exponent_gt0",
"genGid",
"homocyclic"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_homocyclic G : homocyclic G -> #|G| = (exponent G ^ 'r(G))%N. | Proof.
by move=> homG; have [cGG _] := andP homG; apply/eqP; rewrite max_card_abelian.
Qed. | Lemma | card_homocyclic | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"apply",
"cGG",
"exponent",
"homocyclic",
"max_card_abelian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelian_type_dprod_homocyclic p K H G :
K \x H = G -> p.-group G -> homocyclic G ->
abelian_type K = nseq 'r(K) (exponent G)
/\ abelian_type H = nseq 'r(H) (exponent G). | Proof.
move=> defG pG homG; have [cGG _] := andP homG.
have /mulG_sub[sKG sHG]: K * H = G by case/dprodP: defG.
have [cKK cHH] := (abelianS sKG cGG, abelianS sHG cGG).
suffices: all (pred1 (exponent G)) (abelian_type K ++ abelian_type H).
rewrite all_cat => /andP[/all_pred1P-> /all_pred1P->].
by rewrite !size_abeli... | Lemma | abelian_type_dprod_homocyclic | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelianS",
"abelian_structure",
"abelian_type",
"abelian_type_gt1",
"abelian_type_homocyclic",
"abelian_type_pgroup",
"all",
"all_cat",
"all_map",
"all_pred1P",
"all_predC",
"apply",
"big_cat",
"cGG",
"defG",
"dprodP",
"eq_all",
"eq_all_r",
"exponent",
"group",
"has_pred1",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_homocyclic p K H G :
K \x H = G -> p.-group G -> homocyclic G -> homocyclic K /\ homocyclic H. | Proof.
move=> defG pG homG; have [cGG _] := andP homG.
have /mulG_sub[sKG sHG]: K * H = G by case/dprodP: defG.
have [abtK abtH] := abelian_type_dprod_homocyclic defG pG homG.
by rewrite /homocyclic !(abelianS _ cGG) // abtK abtH !constant_nseq.
Qed. | Lemma | dprod_homocyclic | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelianS",
"abelian_type_dprod_homocyclic",
"cGG",
"constant_nseq",
"defG",
"dprodP",
"group",
"homocyclic",
"mulG_sub",
"pG",
"sHG",
"sKG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exponent_dprod_homocyclic p K H G :
K \x H = G -> p.-group G -> homocyclic G -> K :!=: 1 ->
exponent K = exponent G. | Proof.
move=> defG pG homG ntK; have [homK _] := dprod_homocyclic defG pG homG.
have [] := abelian_type_dprod_homocyclic defG pG homG.
by rewrite abelian_type_homocyclic // -['r(K)]prednK ?rank_gt0 => [|[]].
Qed. | Lemma | exponent_dprod_homocyclic | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelian_type_dprod_homocyclic",
"abelian_type_homocyclic",
"defG",
"dprod_homocyclic",
"exponent",
"group",
"homocyclic",
"pG",
"prednK",
"rank_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isog_abelian_type G H : isog G H -> abelian_type G = abelian_type H. | Proof.
pose lnO p n gT (A : {set gT}) := logn p #|'Ohm_n.+1(A) : 'Ohm_n(A)|.
pose lni i p gT (A : {set gT}) := \max_(e < logn p #|A| | i < lnO p e _ A) e.+1.
suffices{G} nth_abty gT (G : {group gT}) i:
abelian G -> i < size (abelian_type G) ->
nth 1%N (abelian_type G) i = (\prod_(p < #|G|.+1) p ^ lni i p _ G)%N.
... | Lemma | isog_abelian_type | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"Ohm_leq",
"abelian",
"abelian_structure",
"abelian_type",
"abelian_type_dvdn_sorted",
"abelian_type_gt1",
"addnC",
"allP",
"all_count",
"all_rcons",
"apply",
"big_cat",
"big_cons",
"big_mkcond",
"big_mkord",
"bigdprodWY",
"bigmax_leqP",
"bigmax_sup",
"cGG",
"cardG_gt0",
"car... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_abelian_type_isog G H :
abelian G -> abelian H -> isog G H = (abelian_type G == abelian_type H). | Proof.
move=> cGG cHH; apply/idP/eqP; first exact: isog_abelian_type.
have{cGG} [bG defG <-] := abelian_structure cGG.
have{cHH} [bH defH <-] := abelian_structure cHH.
elim: bG bH G H defG defH => [|x bG IHb] [|y bH] // G H.
rewrite !big_nil => <- <- _.
by rewrite isog_cyclic_card ?cyclic1 ?cards1.
rewrite !big_con... | Lemma | eq_abelian_type_isog | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"G'",
"abelian",
"abelian_structure",
"abelian_type",
"apply",
"big_cons",
"big_nil",
"cGG",
"cards1",
"cycle_cyclic",
"cyclic1",
"defG",
"dprodP",
"eqb",
"isog",
"isog_abelian_type",
"isog_cyclic_card",
"isog_dprod",
"orderE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isog_abelem_card p G H :
p.-abelem G -> isog G H = p.-abelem H && (#|H| == #|G|). | Proof.
move=> abelG; apply/idP/andP=> [isoGH | [abelH eqGH]].
by rewrite -(isog_abelem isoGH) (card_isog isoGH).
rewrite eq_abelian_type_isog ?(@abelem_abelian _ p) //.
by rewrite !(@abelian_type_abelem _ p) ?(@rank_abelem _ p) // (eqP eqGH).
Qed. | Lemma | isog_abelem_card | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelH",
"abelem",
"abelem_abelian",
"abelian_type_abelem",
"apply",
"card_isog",
"eqGH",
"eq_abelian_type_isog",
"isoGH",
"isog",
"isog_abelem",
"rank_abelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_rank_abelian G : abelian G -> 'r(f @* G) <= 'r(G). | Proof.
move=> cGG; have sHG := subsetIr D G; apply: leq_trans (rankS sHG).
rewrite -!grank_abelian ?morphim_abelian ?(abelianS sHG) //=.
by rewrite -morphimIdom morphim_grank ?subsetIl.
Qed. | Lemma | morphim_rank_abelian | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelian",
"abelianS",
"apply",
"cGG",
"grank_abelian",
"leq_trans",
"morphimIdom",
"morphim_abelian",
"morphim_grank",
"rankS",
"sHG",
"subsetIl",
"subsetIr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_p_rank_abelian p G : abelian G -> 'r_p(f @* G) <= 'r_p(G). | Proof.
move=> cGG; have sHG := subsetIr D G; apply: leq_trans (p_rankS p sHG).
have cHH := abelianS sHG cGG; rewrite -morphimIdom /=; set H := D :&: G.
have sylP := nilpotent_pcore_Hall p (abelian_nil cHH).
have sPH := pHall_sub sylP.
have sPD: 'O_p(H) \subset D by rewrite (subset_trans sPH) ?subsetIl.
rewrite -(p_rank... | Lemma | morphim_p_rank_abelian | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelian",
"abelianS",
"abelian_nil",
"apply",
"cGG",
"leq_trans",
"morphimIdom",
"morphim_pHall",
"morphim_pgroup",
"morphim_rank_abelian",
"nilpotent_pcore_Hall",
"pHall_sub",
"p_rankS",
"p_rank_Sylow",
"pcore_pgroup",
"rank_pgroup",
"sHG",
"subsetIl",
"subsetIr",
"subset_tra... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isog_homocyclic G H : G \isog H -> homocyclic G = homocyclic H. | Proof.
move=> isoGH.
by rewrite /homocyclic (isog_abelian isoGH) (isog_abelian_type isoGH).
Qed. | Lemma | isog_homocyclic | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"homocyclic",
"isoGH",
"isog",
"isog_abelian",
"isog_abelian_type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cGG : abelian G. | Hypothesis | cGG | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
quotient_rank_abelian : 'r(G / H) <= 'r(G). | Proof. exact: morphim_rank_abelian. Qed. | Lemma | quotient_rank_abelian | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"morphim_rank_abelian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_p_rank_abelian : 'r_p(G / H) <= 'r_p(G). | Proof. exact: morphim_p_rank_abelian. Qed. | Lemma | quotient_p_rank_abelian | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"morphim_p_rank_abelian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fin_lmod_pchar_abelem p (R : nzRingType) (V : finLmodType R):
p \in [pchar R]%R -> p.-abelem [set: V]. | Proof.
case/andP=> p_pr /eqP-pR0; apply/abelemP=> //.
by split=> [|v _]; rewrite ?zmod_abelian // zmodXgE -scaler_nat pR0 scale0r.
Qed. | Lemma | fin_lmod_pchar_abelem | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelem",
"abelemP",
"apply",
"p_pr",
"pchar",
"scale0r",
"scaler_nat",
"split",
"zmodXgE",
"zmod_abelian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fin_Fp_lmod_abelem p (V : finLmodType 'F_p) :
prime p -> p.-abelem [set: V]. | Proof. by move/pchar_Fp/fin_lmod_pchar_abelem->. Qed. | Lemma | fin_Fp_lmod_abelem | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelem",
"fin_lmod_pchar_abelem",
"pchar_Fp",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fin_ring_pchar_abelem p (R : finNzRingType) :
p \in [pchar R]%R -> p.-abelem [set: R]. | Proof. exact: fin_lmod_pchar_abelem R^o. Qed. | Lemma | fin_ring_pchar_abelem | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"abelem",
"fin_lmod_pchar_abelem",
"pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fin_lmod_char_abelem | := (fin_lmod_pchar_abelem) (only parsing). | Notation | fin_lmod_char_abelem | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"fin_lmod_pchar_abelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fin_ring_char_abelem | := (fin_ring_pchar_abelem) (only parsing). | Notation | fin_ring_char_abelem | solvable | solvable/abelian.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"choice",
"div",
"fintype",
"finfun",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"action",
"quotient",
"gfunctor",
"gproduct",
"ssralg",
"co... | [
"fin_ring_pchar_abelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sym : {set {perm T}} | := setT. | Definition | Sym | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"setT"
] | Definitions of the alternate groups and some Properties * | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Sym_group | := Eval hnf in [group of Sym]. | Canonical | Sym_group | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Sym",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'Sym_T" | := Sym. | Notation | 'Sym_T | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sign_morph | := @Morphism _ _ 'Sym_T _ (in2W (@odd_permM _)). | Canonical | sign_morph | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"odd_permM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Alt | := 'ker (@odd_perm T). | Definition | Alt | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"ker",
"odd_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Alt_group | := Eval hnf in [group of Alt]. | Canonical | Alt_group | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Alt",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"'Alt_T" | := Alt. | Notation | 'Alt_T | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Alt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Alt_even p : (p \in 'Alt_T) = ~~ p. | Proof. by rewrite !inE /=; case: odd_perm. Qed. | Lemma | Alt_even | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"inE",
"odd_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Alt_subset : 'Alt_T \subset 'Sym_T. | Proof. exact: subsetT. Qed. | Lemma | Alt_subset | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Alt_normal : 'Alt_T <| 'Sym_T. | Proof. exact: ker_normal. Qed. | Lemma | Alt_normal | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"ker_normal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Alt_norm : 'Sym_T \subset 'N('Alt_T). | Proof. by case/andP: Alt_normal. Qed. | Lemma | Alt_norm | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Alt_normal"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Alt_index : 1 < n -> #|'Sym_T : 'Alt_T| = 2. | Proof.
move=> lt1n; rewrite -card_quotient ?Alt_norm //=.
have : ('Sym_T / 'Alt_T) \isog (@odd_perm T @* 'Sym_T) by apply: first_isog.
case/isogP=> g /injmP/card_in_imset <-.
rewrite /morphim setIid=> ->; rewrite -card_bool; apply: eq_card => b.
apply/imsetP; case: b => /=; last first.
by exists (1 : {perm T}); [rewr... | Lemma | Alt_index | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Alt_norm",
"apply",
"cardD1",
"card_bool",
"card_in_imset",
"card_quotient",
"eq_card",
"eq_card0",
"eq_sym",
"existsP",
"first_isog",
"imsetP",
"inE",
"injmP",
"isog",
"isogP",
"last",
"lt0n",
"ltnS",
"morphim",
"odd_perm",
"odd_perm1",
"odd_tperm",
"pickP",
"setIid... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_Sym : #|'Sym_T| = n`!. | Proof.
rewrite -[n]cardsE -card_perm; apply: eq_card => p.
by apply/idP/subsetP=> [? ?|]; rewrite !inE.
Qed. | Lemma | card_Sym | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"apply",
"card_perm",
"cardsE",
"eq_card",
"inE",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_Alt : 1 < n -> (2 * #|'Alt_T|)%N = n`!. | Proof.
by move/Alt_index <-; rewrite mulnC (Lagrange Alt_subset) card_Sym.
Qed. | Lemma | card_Alt | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Alt_index",
"Alt_subset",
"Lagrange",
"card_Sym",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sym_trans : [transitive^n 'Sym_T, on setT | 'P]. | Proof.
apply/imsetP; pose t1 := [tuple of enum T].
have dt1: t1 \in n.-dtuple(setT) by rewrite inE enum_uniq; apply/subsetP.
exists t1 => //; apply/setP=> t; apply/idP/imsetP=> [|[a _ ->{t}]]; last first.
by apply: n_act_dtuple => //; apply/astabsP=> x; rewrite !inE.
case/dtuple_onP=> injt _; have injf := inj_comp in... | Lemma | Sym_trans | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"aperm",
"apply",
"astabsP",
"dtuple_onP",
"enum",
"enum_default",
"enum_rank_inj",
"enum_uniq",
"enum_valK",
"eq_from_tnth",
"imsetP",
"inE",
"injf",
"last",
"n_act_dtuple",
"on",
"permE",
"setP",
"setT",
"subsetP",
"tnth",
"tnth_map",
"tnth_nth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Alt_trans : [transitive^n.-2 'Alt_T, on setT | 'P]. | Proof.
case n_m2: n Sym_trans => [|[|m]] /= tr_m2; try exact: ntransitive0.
have tr_m := ntransitive_weak (leqW (leqnSn m)) tr_m2.
case/imsetP: tr_m2; case/tupleP=> x; case/tupleP=> y t.
rewrite !dtuple_on_add 2![x \in _]inE inE negb_or /= -!andbA.
case/and4P=> nxy ntx nty dt _; apply/imsetP; exists t => //; apply/setP... | Lemma | Alt_trans | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Sym_trans",
"actM",
"aperm",
"apply",
"astabsP",
"atransP2",
"dtuple_on_add",
"eq_in_map",
"imsetP",
"inE",
"last",
"leqW",
"leqnSn",
"n_act_dtuple",
"ntransitive0",
"ntransitive_weak",
"odd_perm",
"odd_permM",
"odd_tperm",
"on",
"setP",
"setT",
"tperm",
"tpermP",
"t... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aperm_faithful (A : {group {perm T}}) : [faithful A, on setT | 'P]. | Proof.
by apply/faithfulP=> /= p _ np1; apply/eqP/perm_act1P=> y; rewrite np1 ?inE.
Qed. | Lemma | aperm_faithful | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"apply",
"faithful",
"faithfulP",
"group",
"inE",
"on",
"perm_act1P",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Sym_' T" | := (Sym T)
(at level 8, T at level 2, format "''Sym_' T") : group_scope. | Notation | ''Sym_' T | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Sym_' T" | := (Sym_group T) : Group_scope. | Notation | ''Sym_' T | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Sym_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Alt_' T" | := (Alt T)
(at level 8, T at level 2, format "''Alt_' T") : group_scope. | Notation | ''Alt_' T | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Alt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''Alt_' T" | := (Alt_group T) : Group_scope. | Notation | ''Alt_' T | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Alt_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trivial_Alt_2 (T : finType) : #|T| <= 2 -> 'Alt_T = 1. | Proof.
rewrite leq_eqVlt => /predU1P[] oT.
by apply: card_le1_trivg; rewrite -leq_double -mul2n card_Alt oT.
suffices Sym1: 'Sym_T = 1 by apply/trivgP; rewrite -Sym1 subsetT.
by apply: card1_trivg; rewrite card_Sym; case: #|T| oT; do 2?case.
Qed. | Lemma | trivial_Alt_2 | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"apply",
"card1_trivg",
"card_Alt",
"card_Sym",
"card_le1_trivg",
"leq_double",
"leq_eqVlt",
"mul2n",
"predU1P",
"subsetT",
"trivgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simple_Alt_3 (T : finType) : #|T| = 3 -> simple 'Alt_T. | Proof.
move=> T3; have{T3} oA: #|'Alt_T| = 3.
by apply: double_inj; rewrite -mul2n card_Alt T3.
apply/simpleP; split=> [|K]; [by rewrite trivg_card1 oA | case/andP=> sKH _].
have:= cardSg sKH; rewrite oA dvdn_divisors // !inE orbC /= -oA.
case/pred2P=> eqK; [right | left]; apply/eqP.
by rewrite eqEcard sKH eqK leqn... | Lemma | simple_Alt_3 | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"apply",
"cardSg",
"card_Alt",
"cards1",
"double_inj",
"dvdn_divisors",
"eqEcard",
"eq_sym",
"inE",
"leqnn",
"mul2n",
"pred2P",
"simple",
"simpleP",
"split",
"sub1G",
"trivg_card1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
not_simple_Alt_4 (T : finType) : #|T| = 4 -> ~~ simple 'Alt_T. | Proof.
move=> oT; set A := 'Alt_T.
have oA: #|A| = 12 by apply: double_inj; rewrite -mul2n card_Alt oT.
suffices [p]: exists p, [/\ prime p, 1 < #|A|`_p < #|A| & #|'Syl_p(A)| == 1%N].
case=> p_pr pA_int; rewrite /A; case/normal_sylowP=> P; case/pHallP.
rewrite /= -/A => sPA pP nPA; apply/simpleP=> [] [_]; rewrite -... | Lemma | not_simple_Alt_4 | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Px",
"Sylow_exists",
"apply",
"card_Alt",
"card_Syl_dvd",
"card_Syl_mod",
"cards1",
"cardsD1",
"cardsID",
"cycle_id",
"cycle_subG",
"divisors",
"double_inj",
"dvdn1",
"dvdn_divisors",
"eqEcard",
"eqEsubset",
"eq_bigl",
"eq_bigr",
"exists_inP",
"expg1",
"filter",
"group",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simple_Alt5_base (T : finType) : #|T| = 5 -> simple 'Alt_T. | Proof.
move=> oT.
have F1: #|'Alt_T| = 60 by apply: double_inj; rewrite -mul2n card_Alt oT.
have FF (H : {group {perm T}}): H <| 'Alt_T -> H :<>: 1 -> 20 %| #|H|.
- move=> Hh1 Hh3.
have [x _]: exists x, x \in T by apply/existsP/eqP; rewrite oT.
have F2 := Alt_trans T; rewrite oT /= in F2.
have F3: [transitive 'Al... | Lemma | simple_Alt5_base | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Alt_trans",
"Cauchy",
"F1",
"F2",
"F3",
"F4",
"F5",
"Gauss_dvd",
"Hh",
"S1",
"S2",
"Sylow_exists",
"aperm",
"aperm_faithful",
"apply",
"astab1P",
"astabP",
"atransP",
"atransP2",
"atrans_dvd",
"card",
"card1_trivg",
"cardSg",
"card_Alt",
"card_Syl_dvd",
"card_Syl_m... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
T' | := {y | y != x}. | Notation | T' | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfd_funP (p : {perm T}) (u : T') :
let p1 := if p x == x then p else 1 in p1 (val u) != x. | Proof.
case: (p x =P x) => /= [pxx | _]; last by rewrite perm1 (valP u).
by rewrite -[x in _ != x]pxx (inj_eq perm_inj); apply: (valP u).
Qed. | Lemma | rfd_funP | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"T'",
"apply",
"inj_eq",
"last",
"perm1",
"perm_inj",
"val",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfd_fun p | := [fun u => Sub ((_ : {perm T}) _) (rfd_funP p u) : T']. | Definition | rfd_fun | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Sub",
"T'",
"rfd_funP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfdP p : injective (rfd_fun p). | Proof.
apply: can_inj (rfd_fun p^-1) _ => u; apply: val_inj => /=.
rewrite -(can_eq (permK p)) permKV eq_sym.
by case: eqP => _; rewrite !(perm1, permK).
Qed. | Lemma | rfdP | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"apply",
"can_eq",
"eq_sym",
"perm1",
"permK",
"permKV",
"rfd_fun",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfd p | := perm (@rfdP p). | Definition | rfd | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"rfdP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_T : 2 < #|T|. | Hypothesis | card_T | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
rfd_morph : {in 'C_('Sym_T)[x | 'P] &, {morph rfd : y z / y * z}}. | Proof.
move=> p q; rewrite !setIA !setIid; move/astab1P=> p_x; move/astab1P=> q_x.
apply/permP=> u; apply: val_inj.
by rewrite permE /= !permM !permE /= [p x]p_x [q x]q_x eqxx permM /=.
Qed. | Lemma | rfd_morph | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"apply",
"astab1P",
"eqxx",
"permE",
"permM",
"permP",
"rfd",
"setIA",
"setIid",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfd_morphism | := Morphism rfd_morph. | Canonical | rfd_morphism | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"rfd_morph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rgd_fun (p : {perm T'}) | :=
[fun x1 => if insub x1 is Some u then sval (p u) else x]. | Definition | rgd_fun | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"T'",
"insub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rgdP p : injective (rgd_fun p). | Proof.
apply: can_inj (rgd_fun p^-1) _ => y /=.
case: (insubP _ y) => [u _ val_u|]; first by rewrite valK permK.
by rewrite negbK; move/eqP->; rewrite insubF //= eqxx.
Qed. | Lemma | rgdP | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"apply",
"eqxx",
"insubF",
"insubP",
"permK",
"rgd_fun",
"valK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rgd p | := perm (@rgdP p). | Definition | rgd | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"rgdP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfd_odd (p : {perm T}) : p x = x -> rfd p = p :> bool. | Proof.
have rfd1: rfd 1 = 1.
by apply/permP => u; apply: val_inj; rewrite permE /= if_same !perm1.
have [n] := ubnP #|[set x | p x != x]|; elim: n p => // n IHn p le_p_n px_x.
have [p_id | [x1 Hx1]] := set_0Vmem [set x | p x != x].
suffices ->: p = 1 by rewrite rfd1 !odd_perm1.
by apply/permP => z; apply: contraF... | Lemma | rfd_odd | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Sub",
"T'",
"apply",
"astab1P",
"cardsD1",
"contraFeq",
"contraNneq",
"contraTneq",
"eq_sym",
"eqxx",
"inE",
"in_set0",
"inj_eq",
"last",
"leq_trans",
"ltnS",
"morphM",
"mulgK",
"odd_perm1",
"odd_permM",
"odd_tperm",
"perm1",
"permE",
"permM",
"permP",
"perm_inj",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfd_iso : 'C_('Alt_T)[x | 'P] \isog 'Alt_T'. | Proof.
have rgd_x p: rgd p x = x by rewrite permE /= insubF //= eqxx.
have rfd_rgd p: rfd (rgd p) = p.
apply/permP => [[z Hz]]; apply/val_eqP; rewrite !permE.
by rewrite /= [rgd _ _]permE /= insubF eqxx // permE /= insubT.
have sSd: 'C_('Alt_T)[x | 'P] \subset 'dom rfd.
by apply/subsetP=> p /[!inE]/= /andP[].
app... | Lemma | rfd_iso | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Alt_even",
"Sub",
"T'",
"aperm",
"apply",
"astab1P",
"dom",
"eqxx",
"inE",
"injmP",
"insubF",
"insubT",
"isog",
"isogP",
"last",
"mker",
"morphimP",
"morphim_restrm",
"morphism",
"permE",
"permP",
"restrm",
"rfd",
"rfd_odd",
"rgd",
"set11",
"setIP",
"setIid",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simple_Alt5 (T : finType) : #|T| >= 5 -> simple 'Alt_T. | Proof.
suff F1 n: #|T| = n + 5 -> simple 'Alt_T by move/subnK/esym/F1.
elim: n T => [| n Hrec T Hde]; first exact: simple_Alt5_base.
have oT: 5 < #|T| by rewrite Hde addnC.
apply/simpleP; split=> [|H Hnorm]; last have [Hh1 nH] := andP Hnorm.
rewrite trivg_card1 -[#|_|]half_double -mul2n card_Alt Hde addnC //.
by re... | Lemma | simple_Alt5 | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Alt_trans",
"F1",
"F2",
"F3",
"F4",
"F5",
"Hh",
"S1",
"S2",
"act1",
"addSn",
"addnC",
"aperm",
"aperm_faithful",
"apply",
"astab1P",
"astabP",
"atransP",
"atransP2",
"atrans_dvd",
"can_eq",
"card1_trivg",
"cardC1",
"cardD1",
"cardG_gt0",
"card_Alt",
"card_sig",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gen_tperm_circular_shift (X : finType) x y c : prime #|X| ->
x != y -> #[c]%g = #|X| ->
<<[set tperm x y; c]>>%g = ('Sym_X)%g. | Proof.
move=> Xprime neq_xy ord_c; apply/eqP; rewrite eqEsubset subsetT/=.
have c_gt1 : (1 < #[c]%g)%N by rewrite ord_c prime_gt1.
have cppSS : #[c]%g.-2.+2 = #|X| by rewrite ?prednK ?ltn_predRL.
pose f (i : 'Z_#[c]%g) : X := Zpm i x.
have [g fK gK] : bijective f.
apply: inj_card_bij; rewrite ?cppSS ?card_ord// /f /Z... | Lemma | gen_tperm_circular_shift | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Zpm",
"act",
"addrC",
"addrI",
"addrNK",
"apply",
"can_eq",
"card_ord",
"coprime",
"eqEsubset",
"eq_expg_ord",
"eq_sym",
"eqxx",
"expgD_Zp",
"fK",
"gK",
"gen_tpermn_circular_shift",
"groupX",
"gtnNdvd",
"imsetU1",
"imset_set1",
"inE",
"in_setT",
"inj_card_bij",
"injm... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
solvable_AltF : 4 < #|T| -> solvable 'Alt_T = false. | Proof.
move=> card_T; apply/negP => Alt_solvable.
have/simple_Alt5 Alt_simple := card_T.
have := simple_sol_prime Alt_solvable Alt_simple.
have lt_T n : n <= 4 -> n < #|T| by move/leq_ltn_trans; apply.
have -> : #|('Alt_T)%G| = #|T|`! %/ 2 by rewrite -card_Alt ?mulKn ?lt_T.
move/even_prime => [/eqP|]; apply/negP.
rew... | Lemma | solvable_AltF | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"apply",
"card_Alt",
"card_T",
"dvdn2",
"dvdn_divRL",
"dvdn_fact",
"even_prime",
"leq_divRL",
"leq_ltn_trans",
"ltnW",
"ltn_fact",
"mulKn",
"mulnC",
"neq_ltn",
"simple_Alt5",
"simple_sol_prime",
"solvable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
solvable_SymF : 4 < #|T| -> solvable 'Sym_T = false. | Proof. by rewrite (series_sol (Alt_normal T)) => /solvable_AltF->. Qed. | Lemma | solvable_SymF | solvable | solvable/alt.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"fintype",
"tuple",
"bigop",
"prime",
"finset",
"ssralg",
"zmodp",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"cyclic",
"pgroup",... | [
"Alt_normal",
"series_sol",
"solvable",
"solvable_AltF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
burnside_formula : forall (gT : finGroupType) s (G : {group gT}),
uniq s -> s =i G ->
forall (sT : finType) (to : {action gT &-> sT}),
(#|orbit to G @: setT| * size s)%N = \sum_(p <- s) #|'Fix_to[p]|. | Proof.
move=> gT s G Us sG sT to.
rewrite big_uniq // -(card_uniqP Us) (eq_card sG) -Frobenius_Cauchy.
by apply/actsP=> ? _ ?; rewrite !inE.
by apply: eq_big => // p _; rewrite setTI.
Qed. | Lemma | burnside_formula | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Frobenius_Cauchy",
"action",
"actsP",
"apply",
"big_uniq",
"card_uniqP",
"eq_big",
"eq_card",
"gT",
"group",
"inE",
"orbit",
"sG",
"sT",
"setT",
"setTI",
"size",
"to",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
colors | := 'I_n. | Definition | colors | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
square | := 'I_4. | Definition | square | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mksquare i : square | := Sub (i %% _) (ltn_mod i 4). | Definition | mksquare | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Sub",
"ltn_mod",
"square"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
c0 | := mksquare 0. | Definition | c0 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"mksquare"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
c1 | := mksquare 1. | Definition | c1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"mksquare"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
c2 | := mksquare 2. | Definition | c2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"mksquare"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
c3 | := mksquare 3. | Definition | c3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"mksquare"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R1 (sc : square) : square | := tnth [tuple c1; c2; c3; c0] sc. | Definition | R1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c0",
"c1",
"c2",
"c3",
"square",
"tnth",
"tuple"
] | rotations | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
R2 (sc : square) : square | := tnth [tuple c2; c3; c0; c1] sc. | Definition | R2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c0",
"c1",
"c2",
"c3",
"square",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R3 (sc : square) : square | := tnth [tuple c3; c0; c1; c2] sc. | Definition | R3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c0",
"c1",
"c2",
"c3",
"square",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
get_inv elt l | :=
match l with
| (_, (elt, ?x)) => x
| (elt, ?x) => x
| (?x, _) => get_inv elt x
end. | Ltac | get_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_inv | := ((R1, R3), (R2, R2), (R3, R1)). | Definition | rot_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R1",
"R2",
"R3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_tac | :=
move: (erefl rot_inv); unfold rot_inv;
match goal with |- ?X = _ -> injective ?Y =>
move=> _; let x := get_inv Y X in
apply: (can_inj (g:=x)); move=> [val H1]
end. | Ltac | inj_tac | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"get_inv",
"rot_inv",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R1_inj : injective R1. | Proof. by inj_tac; repeat (destruct val => //=; first by apply/eqP). Qed. | Lemma | R1_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R1",
"apply",
"inj_tac",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R2_inj : injective R2. | Proof. by inj_tac; repeat (destruct val => //=; first by apply/eqP). Qed. | Lemma | R2_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R2",
"apply",
"inj_tac",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R3_inj : injective R3. | Proof. by inj_tac; repeat (destruct val => //=; first by apply/eqP). Qed. | Lemma | R3_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R3",
"apply",
"inj_tac",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r1 | := (perm R1_inj). | Definition | r1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R1_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r2 | := (perm R2_inj). | Definition | r2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R2_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r3 | := (perm R3_inj). | Definition | r3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R3_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
id1 | := (1 : {perm square}). | Definition | id1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"square"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_rot (r : {perm _}) | := (r * r1 == r1 * r). | Definition | is_rot | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"r1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot | := [set r | is_rot r]. | Definition | rot | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"is_rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_set_rot : group_set rot. | Proof.
apply/group_setP; split; first by rewrite /rot inE /is_rot mulg1 mul1g.
move=> x1 y; rewrite /rot !inE /= /is_rot; move/eqP => hx1; move/eqP => hy.
by rewrite -mulgA hy !mulgA hx1.
Qed. | Lemma | group_set_rot | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"group_set",
"group_setP",
"inE",
"is_rot",
"mul1g",
"mulg1",
"mulgA",
"rot",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_group | := Group group_set_rot. | Canonical | rot_group | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"group_set_rot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotations | := [set id1; r1; r2; r3]. | Definition | rotations | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"id1",
"r1",
"r2",
"r3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_eq_c0 : forall r s : {perm square},
is_rot r -> is_rot s -> r c0 = s c0 -> r = s. | Proof.
rewrite /is_rot => r s; move/eqP => hr; move/eqP=> hs hrs; apply/permP => a.
have ->: a = (r1 ^+ a) c0
by apply/eqP; case: a; do 4?case=> //=; rewrite ?permM !permE.
by rewrite -!permM -!commuteX // !permM hrs.
Qed. | Lemma | rot_eq_c0 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"c0",
"commuteX",
"is_rot",
"permE",
"permM",
"permP",
"r1",
"square"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_r1 : forall r, is_rot r -> r = r1 ^+ (r c0). | Proof.
move=> r hr; apply: rot_eq_c0 => //; apply/eqP.
by symmetry; apply: commuteX.
by case: (r c0); do 4?case=> //=; rewrite ?permM !permE /=.
Qed. | Lemma | rot_r1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"c0",
"commuteX",
"is_rot",
"permE",
"permM",
"r1",
"rot_eq_c0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotations_is_rot : forall r, r \in rotations -> is_rot r. | Proof.
move=> r Dr; apply/eqP; apply/permP => a; rewrite !inE -!orbA !permM in Dr *.
by case/or4P: Dr; move/eqP->; rewrite !permE //; case: a; do 4?case.
Qed. | Lemma | rotations_is_rot | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"inE",
"is_rot",
"permE",
"permM",
"permP",
"rotations"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rot_is_rot : rot = rotations. | Proof.
apply/setP=> r; apply/idP/idP => [|/rotations_is_rot] /[!inE]// h.
have -> : r = r1 ^+ (r c0) by apply: rot_eq_c0; rewrite // -rot_r1.
have e2: 2 = r2 c0 by rewrite permE /=.
have e3: 3 = r3 c0 by rewrite permE /=.
case (r c0); do 4?[case] => // ?; rewrite ?(expg1, eqxx, orbT) //.
by rewrite [nat_of_ord _]/= e... | Lemma | rot_is_rot | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"c0",
"eqxx",
"expg1",
"inE",
"nat_of_ord",
"permE",
"r1",
"r2",
"r3",
"rot",
"rot_eq_c0",
"rot_r1",
"rotations",
"rotations_is_rot",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sh (sc : square) : square | := tnth [tuple c1; c0; c3; c2] sc. | Definition | Sh | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c0",
"c1",
"c2",
"c3",
"square",
"tnth",
"tuple"
] | symmetries | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Sh_inj : injective Sh. | Proof.
by apply: (can_inj (g:= Sh)); case; do 4?case=> //=; move=> H; apply/eqP.
Qed. | Lemma | Sh_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Sh",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sh | := (perm Sh_inj). | Definition | sh | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Sh_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sh_inv : sh^-1 = sh. | Proof.
apply: (mulIg sh); rewrite mulVg; apply/permP.
by case; do 4?case=> //=; move=> H; rewrite !permE /= !permE; apply/eqP.
Qed. | Lemma | sh_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"mulIg",
"mulVg",
"permE",
"permP",
"sh"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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