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 |
|---|---|---|---|---|---|---|---|---|---|---|
"to * n" | := (n_act_action to n) : action_scope. | Notation | to * n | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"n_act_action",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dtuple_on | := [set t : n.-tuple sT | uniq t & t \subset S]. | Definition | dtuple_on | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"sT",
"tuple",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ntransitive | := [transitive A, on dtuple_on | to * n]. | Definition | ntransitive | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"dtuple_on",
"on",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dtuple_onP t :
reflect (injective (tnth t) /\ forall i, tnth t i \in S) (t \in dtuple_on). | Proof.
rewrite inE subset_all -forallb_tnth -[in uniq t]map_tnth_enum /=.
by apply: (iffP andP) => -[/injectiveP-f_inj /forallP].
Qed. | Lemma | dtuple_onP | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"apply",
"dtuple_on",
"f_inj",
"forallP",
"forallb_tnth",
"inE",
"injectiveP",
"map_tnth_enum",
"subset_all",
"tnth",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_act_dtuple t a :
a \in 'N(S | to) -> t \in dtuple_on -> n_act to t a \in dtuple_on. | Proof.
move/astabsP=> toSa /dtuple_onP[t_inj St]; apply/dtuple_onP.
split=> [i j | i]; rewrite !tnth_map ?[_ \in S]toSa //.
by move/act_inj; apply: t_inj.
Qed. | Lemma | n_act_dtuple | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"act_inj",
"apply",
"astabsP",
"dtuple_on",
"dtuple_onP",
"n_act",
"split",
"tnth_map",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n .-dtuple ( S )" | := (dtuple_on n S)
(format "n .-dtuple ( S )") : set_scope. | Notation | n .-dtuple ( S ) | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"dtuple_on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'transitive' ^ n A , 'on' S | to ]" | := (ntransitive n A S to)
(n at level 8,
format "[ 'transitive' ^ n A , 'on' S | to ]") : form_scope. | Notation | [ 'transitive' ^ n A , 'on' S | to ] | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"ntransitive",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_uniq_tuple n (t : n.-tuple sT) : uniq t -> #|t| = n. | Proof. by move/card_uniqP->; apply: size_tuple. Qed. | Lemma | card_uniq_tuple | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"apply",
"card_uniqP",
"sT",
"size_tuple",
"tuple",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_act0 (t : 0.-tuple sT) a : n_act to t a = [tuple]. | Proof. exact: tuple0. Qed. | Lemma | n_act0 | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"n_act",
"sT",
"to",
"tuple",
"tuple0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dtuple_on_add n x (t : n.-tuple sT) :
([tuple of x :: t] \in n.+1.-dtuple(S)) =
[&& x \in S, x \notin t & t \in n.-dtuple(S)]. | Proof. by rewrite !inE memtE !subset_all -!andbA; do !bool_congr. Qed. | Lemma | dtuple_on_add | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"inE",
"memtE",
"sT",
"subset_all",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dtuple_on_add_D1 n x (t : n.-tuple sT) :
([tuple of x :: t] \in n.+1.-dtuple(S))
= (x \in S) && (t \in n.-dtuple(S :\ x)). | Proof.
rewrite dtuple_on_add !inE (andbCA (~~ _)); do 2!congr (_ && _).
rewrite -!(eq_subset (in_set [in t])) setDE setIC subsetI; congr (_ && _).
by rewrite -setCS setCK sub1set !inE.
Qed. | Lemma | dtuple_on_add_D1 | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"dtuple_on_add",
"eq_subset",
"inE",
"in_set",
"sT",
"setCK",
"setCS",
"setDE",
"setIC",
"sub1set",
"subsetI",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dtuple_on_subset n (S1 S2 : {set sT}) t :
S1 \subset S2 -> t \in n.-dtuple(S1) -> t \in n.-dtuple(S2). | Proof. by move=> sS12 /[!inE] /andP[-> /subset_trans]; apply. Qed. | Lemma | dtuple_on_subset | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"S1",
"S2",
"apply",
"inE",
"sT",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_act_add n x (t : n.-tuple sT) a :
n_act to [tuple of x :: t] a = [tuple of to x a :: n_act to t a]. | Proof. exact: val_inj. Qed. | Lemma | n_act_add | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"n_act",
"sT",
"to",
"tuple",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ntransitive0 : [transitive^0 G, on S | to]. | Proof.
have dt0: [tuple] \in 0.-dtuple(S) by rewrite inE memtE subset_all.
apply/imsetP; exists [tuple of Nil sT] => //.
by apply/setP=> x; rewrite [x]tuple0 orbit_refl.
Qed. | Lemma | ntransitive0 | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"Nil",
"apply",
"imsetP",
"inE",
"memtE",
"on",
"orbit_refl",
"sT",
"setP",
"subset_all",
"to",
"tuple",
"tuple0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ntransitive_weak k m :
k <= m -> [transitive^m G, on S | to] -> [transitive^k G, on S | to]. | Proof.
move/subnKC <-; rewrite addnC; elim: {m}(m - k) => // m IHm.
rewrite addSn => tr_m1; apply: IHm; move: {m k}(m + k) tr_m1 => m tr_m1.
have ext_t t: t \in dtuple_on m S ->
exists x, [tuple of x :: t] \in m.+1.-dtuple(S).
- move=> dt.
have [sSt | /subsetPn[x Sx ntx]] := boolP (S \subset t); last first.
by ... | Lemma | ntransitive_weak | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"addSn",
"addnC",
"apply",
"card_uniq_tuple",
"dtuple_on",
"dtuple_on_add",
"imsetP",
"imset_f",
"inE",
"last",
"ltnn",
"n_act",
"n_act_add",
"on",
"setP",
"subnKC",
"subsetPn",
"subset_leq_card",
"subset_trans",
"to",
"tuple",
"tupleP",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ntransitive1 m :
0 < m -> [transitive^m G, on S | to] -> [transitive G, on S | to]. | Proof.
have trdom1 x: ([tuple x] \in 1.-dtuple(S)) = (x \in S).
by rewrite dtuple_on_add !inE memtE subset_all andbT.
move=> m_gt0 /(ntransitive_weak m_gt0) {m m_gt0}.
case/imsetP; case/tupleP=> x t0; rewrite {t0}(tuple0 t0) trdom1 => Sx trx.
apply/imsetP; exists x => //; apply/setP=> y; rewrite -trdom1 trx.
by apply... | Lemma | ntransitive1 | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"apply",
"dtuple_on_add",
"imsetP",
"inE",
"memtE",
"ntransitive_weak",
"on",
"setP",
"subset_all",
"to",
"tuple",
"tuple0",
"tupleP",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ntransitive_primitive m :
1 < m -> [transitive^m G, on S | to] -> [primitive G, on S | to]. | Proof.
move=> lt1m /(ntransitive_weak lt1m) {m lt1m}tr2G.
have trG: [transitive G, on S | to] by apply: ntransitive1 tr2G.
have [x Sx _]:= imsetP trG; rewrite (trans_prim_astab Sx trG).
apply/maximal_eqP; split=> [|H]; first exact: subsetIl; rewrite subEproper.
case/predU1P; first by [left]; case/andP=> sCH /subsetPn[a... | Lemma | ntransitive_primitive | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"actsP",
"acts_sub_orbit",
"apply",
"astab1P",
"atransP2",
"atrans_acts",
"eqEsubset",
"eqVneq",
"imsetP",
"inE",
"maximal_eqP",
"mem_orbit",
"mem_seq1",
"memtE",
"mulSGid",
"ntransitive1",
"ntransitive_weak",
"on",
"orbit_sym",
"predU1P",
"primitive",
"sHG",
"split",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
stab_ntransitive m x :
0 < m -> x \in S -> [transitive^m.+1 G, on S | to] ->
[transitive^m 'C_G[x | to], on S :\ x | to]. | Proof.
move=> m_gt0 Sx Gtr; have sSxS: S :\ x \subset S by rewrite subsetDl.
case: (imsetP Gtr); case/tupleP=> x1 t1; rewrite dtuple_on_add.
case/and3P=> Sx1 nt1x1 dt1 trt1; have Gtr1 := ntransitive1 (ltn0Sn _) Gtr.
case: (atransP2 Gtr1 Sx1 Sx) => // a Ga x1ax.
pose t := n_act to t1 a.
have dxt: [tuple of x :: t] \in m... | Theorem | stab_ntransitive | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"act_inj",
"actsP",
"apply",
"astab1P",
"astabsP",
"atransP2",
"atrans_acts",
"dtuple_on",
"dtuple_on_add",
"dtuple_on_add_D1",
"imsetP",
"inE",
"inj_eq",
"last",
"ltn0Sn",
"n_act",
"n_act_dtuple",
"ntransitive1",
"on",
"setIP",
"setP",
"subsetDl",
"to",
"tuple",
"tup... | This is the forward implication of Aschbacher (15.12).1 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
stab_ntransitiveI m x :
x \in S -> [transitive G, on S | to] ->
[transitive^m 'C_G[x | to], on S :\ x | to] ->
[transitive^m.+1 G, on S | to]. | Proof.
move=> Sx Gtr Gntr.
have t_to_x t: t \in m.+1.-dtuple(S) ->
exists2 a, a \in G & exists2 t', t' \in m.-dtuple(S :\ x)
& t = n_act to [tuple of x :: t'] a.
- case/tupleP: t => y t St.
have Sy: y \in S by rewrite dtuple_on_add_D1 in St; case/andP: St.
rewrite -(atransP Gtr _ ... | Theorem | stab_ntransitiveI | solvable | solvable/primitive_action.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"div",
"seq",
"fintype",
"tuple",
"finset",
"fingroup",
"action",
"gseries"
] | [
"a2",
"actK",
"actM",
"apply",
"astab1P",
"atransP",
"atrans_acts",
"dtuple_on_add_D1",
"groupM",
"groupVr",
"imsetP",
"last",
"n_act",
"n_act_add",
"n_act_dtuple",
"on",
"setIP",
"setP",
"subsetP",
"to",
"tuple",
"tupleP"
] | This is the converse implication of Aschbacher (15.12).1 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
pgroup_fix_mod (p : nat) (G : {group aT}) (S : {set sT}) :
p.-group G -> [acts G, on S | to] -> #|S| = #|'Fix_(S | to)(G)| %[mod p]. | Proof.
move=> pG nSG; have sGD: G \subset D := acts_dom nSG.
apply/eqP; rewrite -(cardsID 'Fix_to(G)) eqn_mod_dvd (leq_addr, addKn) //.
have: [acts G, on S :\: 'Fix_to(G) | to]; last move/acts_sum_card_orbit <-.
rewrite actsD // -(setIidPr sGD); apply: subset_trans (acts_subnorm_fix _ _).
by rewrite setIS ?normG.
a... | Lemma | pgroup_fix_mod | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"aT",
"actsD",
"acts_dom",
"acts_subnorm_fix",
"acts_sum_card_orbit",
"addKn",
"afixP",
"apply",
"card_orbit1",
"card_orbit_in",
"cardsID",
"dvdn_indexg",
"dvdn_mulr",
"dvdn_sum",
"eqn_mod_dvd",
"expnS",
"group",
"imsetP",
"last",
"leq_addr",
"mem_orbit",
"nat",
"normG",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nontrivial_gacent_pgroup G M :
p.-group G -> p.-group M -> {acts G, on group M | to} ->
M :!=: 1 -> 'C_(M | to)(G) :!=: 1. | Proof.
move=> pG pM [nMG sMR] ntM; have [p_pr p_dv_M _] := pgroup_pdiv pM ntM.
rewrite -cardG_gt1 (leq_trans (prime_gt1 p_pr)) 1?dvdn_leq ?cardG_gt0 //= /dvdn.
by rewrite gacentE ?(acts_dom nMG) // setIA (setIidPl sMR) -pgroup_fix_mod.
Qed. | Lemma | nontrivial_gacent_pgroup | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"acts_dom",
"cardG_gt0",
"cardG_gt1",
"dvdn",
"dvdn_leq",
"gacentE",
"group",
"leq_trans",
"nMG",
"on",
"pG",
"p_pr",
"pgroup_fix_mod",
"pgroup_pdiv",
"prime_gt1",
"setIA",
"setIidPl",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcore_sub_astab_irr G M :
p.-group M -> M \subset R -> acts_irreducibly G M to ->
'O_p(G) \subset 'C_G(M | to). | Proof.
move=> pM sMR /mingroupP[/andP[ntM nMG] minM].
have /andP[sGpG nGpG]: 'O_p(G) <| G := gFnormal _ G.
have sGD := acts_dom nMG; have sGpD: 'O_p(G) \subset D := gFsub_trans _ sGD.
rewrite subsetI sGpG -gacentC //=; apply/setIidPl; apply: minM (subsetIl _ _).
rewrite nontrivial_gacent_pgroup ?pcore_pgroup //=.
by ... | Lemma | pcore_sub_astab_irr | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"acts_dom",
"acts_irreducibly",
"acts_subnorm_subgacent",
"apply",
"gFnormal",
"gFsub_trans",
"gacentC",
"group",
"mingroupP",
"nMG",
"nontrivial_gacent_pgroup",
"pcore_pgroup",
"sGD",
"setIidPl",
"split",
"subsetI",
"subsetIl",
"subset_trans",
"subxx",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcore_faithful_irr_act G M :
p.-group M -> M \subset R -> acts_irreducibly G M to ->
[faithful G, on M | to] ->
'O_p(G) = 1. | Proof.
move=> pM sMR irrG ffulG; apply/trivgP; apply: subset_trans ffulG.
exact: pcore_sub_astab_irr.
Qed. | Lemma | pcore_faithful_irr_act | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"acts_irreducibly",
"apply",
"faithful",
"group",
"irrG",
"on",
"pcore_sub_astab_irr",
"subset_trans",
"to",
"trivgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylow's_theorem :
[/\ forall P, [max P | p.-subgroup(G) P] = p.-Sylow(G) P,
[transitive G, on 'Syl_p(G) | 'JG],
forall P, p.-Sylow(G) P -> #|'Syl_p(G)| = #|G : 'N_G(P)|
& prime p -> #|'Syl_p(G)| %% p = 1%N]. | Proof.
pose maxp A P := [max P | p.-subgroup(A) P]; pose S := [set P | maxp G P].
pose oG := orbit 'JG%act G.
have actS: [acts G, on S | 'JG].
apply/subsetP=> x Gx; rewrite 3!inE; apply/subsetP=> P; rewrite 3!inE.
exact: max_pgroupJ.
have S_pG P: P \in S -> P \subset G /\ p.-group P.
by rewrite inE => /maxgroupp/... | Theorem | Sylow's_theorem | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Hall_max",
"LagrangeI",
"Sylow",
"act",
"actsP",
"acts_sub_orbit",
"addn0",
"afixJG",
"apply",
"astab1JG",
"atransP",
"atransP2",
"cardG_gt0",
"cardJg",
"card_Hall",
"card_orbit",
"cards0",
"cardsD1",
"dvdn",
"eqEsubset",
"eqVneq",
"group",
"group_inj",
"imsetP",
"in... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max_pgroup_Sylow P : [max P | p.-subgroup(G) P] = p.-Sylow(G) P. | Proof. by case Sylow's_theorem. Qed. | Lemma | max_pgroup_Sylow | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow",
"Sylow's_theorem",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylow_superset Q :
Q \subset G -> p.-group Q -> {P : {group gT} | p.-Sylow(G) P & Q \subset P}. | Proof.
move=> sQG pQ.
have [|P] := @maxgroup_exists _ (p.-subgroup(G)) Q; first exact/andP.
by rewrite max_pgroup_Sylow; exists P.
Qed. | Lemma | Sylow_superset | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow",
"gT",
"group",
"max_pgroup_Sylow",
"maxgroup_exists"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylow_exists : {P : {group gT} | p.-Sylow(G) P}. | Proof. by case: (Sylow_superset (sub1G G) (pgroup1 _ p)) => P; exists P. Qed. | Lemma | Sylow_exists | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow",
"Sylow_superset",
"gT",
"group",
"pgroup1",
"sub1G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Syl_trans : [transitive G, on 'Syl_p(G) | 'JG]. | Proof. by case Sylow's_theorem. Qed. | Lemma | Syl_trans | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow's_theorem",
"on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylow_trans P Q :
p.-Sylow(G) P -> p.-Sylow(G) Q -> exists2 x, x \in G & Q :=: P :^ x. | Proof.
move=> sylP sylQ; have /[!inE] := (atransP2 Syl_trans) P Q.
by case=> // x Gx ->; exists x.
Qed. | Lemma | Sylow_trans | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Syl_trans",
"Sylow",
"atransP2",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylow_subJ P Q :
p.-Sylow(G) P -> Q \subset G -> p.-group Q ->
exists2 x, x \in G & Q \subset P :^ x. | Proof.
move=> sylP sQG pQ; have [Px sylPx] := Sylow_superset sQG pQ.
by have [x Gx ->] := Sylow_trans sylP sylPx; exists x.
Qed. | Lemma | Sylow_subJ | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Px",
"Sylow",
"Sylow_superset",
"Sylow_trans",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylow_Jsub P Q :
p.-Sylow(G) P -> Q \subset G -> p.-group Q ->
exists2 x, x \in G & Q :^ x \subset P. | Proof.
move=> sylP sQG pQ; have [x Gx] := Sylow_subJ sylP sQG pQ.
by exists x^-1; rewrite (groupV, sub_conjgV).
Qed. | Lemma | Sylow_Jsub | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow",
"Sylow_subJ",
"group",
"groupV",
"sub_conjgV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_Syl P : p.-Sylow(G) P -> #|'Syl_p(G)| = #|G : 'N_G(P)|. | Proof. by case: Sylow's_theorem P. Qed. | Lemma | card_Syl | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow",
"Sylow's_theorem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_Syl_dvd : #|'Syl_p(G)| %| #|G|. | Proof. by case Sylow_exists => P /card_Syl->; apply: dvdn_indexg. Qed. | Lemma | card_Syl_dvd | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow_exists",
"apply",
"card_Syl",
"dvdn_indexg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_Syl_mod : prime p -> #|'Syl_p(G)| %% p = 1%N. | Proof. by case Sylow's_theorem. Qed. | Lemma | card_Syl_mod | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow's_theorem",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Frattini_arg H P : G <| H -> p.-Sylow(G) P -> G * 'N_H(P) = H. | Proof.
case/andP=> sGH nGH sylP; rewrite -normC ?subIset ?nGH ?orbT // -astab1JG.
move/subgroup_transitiveP: Syl_trans => ->; rewrite ?inE //.
apply/imsetP; exists P; rewrite ?inE //.
apply/eqP; rewrite eqEsubset -{1}((atransP Syl_trans) P) ?inE // imsetS //=.
by apply/subsetP=> _ /imsetP[x Hx ->]; rewrite inE -(normsP... | Lemma | Frattini_arg | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Syl_trans",
"Sylow",
"apply",
"astab1JG",
"atransP",
"eqEsubset",
"imsetP",
"imsetS",
"inE",
"normC",
"normsP",
"pHallJ2",
"sGH",
"subIset",
"subgroup_transitiveP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylow_setI_normal G H P :
G <| H -> p.-Sylow(H) P -> p.-Sylow(G) (G :&: P). | Proof.
case/normalP=> sGH nGH sylP; have [Q sylQ] := Sylow_exists p G.
have /maxgroupP[/andP[sQG pQ] maxQ] := Hall_max sylQ.
have [R sylR sQR] := Sylow_superset (subset_trans sQG sGH) pQ.
have [[x Hx ->] pR] := (Sylow_trans sylR sylP, pHall_pgroup sylR).
rewrite -(nGH x Hx) -conjIg pHallJ2.
have /maxQ-> //: Q \subset G... | Lemma | Sylow_setI_normal | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Hall_max",
"Sylow",
"Sylow_exists",
"Sylow_superset",
"Sylow_trans",
"conjIg",
"maxgroupP",
"normalP",
"pHallJ2",
"pHall_pgroup",
"pgroupS",
"psubgroup",
"sGH",
"subsetI",
"subsetIl",
"subsetIr",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normal_sylowP G :
reflect (exists2 P : {group gT}, p.-Sylow(G) P & P <| G)
(#|'Syl_p(G)| == 1%N). | Proof.
apply: (iffP idP) => [syl1 | [P sylP nPG]]; last first.
by rewrite (card_Syl sylP) (setIidPl _) (indexgg, normal_norm).
have [P sylP] := Sylow_exists p G; exists P => //.
rewrite /normal (pHall_sub sylP); apply/setIidPl; apply/eqP.
rewrite eqEcard subsetIl -(LagrangeI G 'N(P)) -indexgI /=.
by rewrite -(card_Sy... | Lemma | normal_sylowP | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"LagrangeI",
"Sylow",
"Sylow_exists",
"apply",
"card_Syl",
"eqEcard",
"gT",
"group",
"indexgI",
"indexgg",
"last",
"muln1",
"normal",
"normal_norm",
"pHall_sub",
"setIidPl",
"subsetIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trivg_center_pgroup P : p.-group P -> 'Z(P) = 1 -> P :=: 1. | Proof.
move=> pP Z1; apply/eqP/idPn=> ntP.
have{ntP} [p_pr p_dv_P _] := pgroup_pdiv pP ntP.
suff: p %| #|'Z(P)| by rewrite Z1 cards1 gtnNdvd ?prime_gt1.
by rewrite /center /dvdn -afixJ -pgroup_fix_mod // astabsJ normG.
Qed. | Lemma | trivg_center_pgroup | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"afixJ",
"apply",
"astabsJ",
"cards1",
"center",
"dvdn",
"group",
"gtnNdvd",
"normG",
"pP",
"p_pr",
"pgroup_fix_mod",
"pgroup_pdiv",
"prime_gt1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p2group_abelian P : p.-group P -> logn p #|P| <= 2 -> abelian P. | Proof.
move=> pP lePp2; pose Z := 'Z(P); have sZP: Z \subset P := center_sub P.
have [/(trivg_center_pgroup pP) ->|] := eqVneq Z 1; first exact: abelian1.
case/(pgroup_pdiv (pgroupS sZP pP)) => p_pr _ [k oZ].
apply: cyclic_center_factor_abelian.
have [->|] := eqVneq (P / Z) 1; first exact: cyclic1.
have pPq := quotient... | Lemma | p2group_abelian | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Lagrange",
"abelian",
"abelian1",
"addnS",
"apply",
"card_quotient",
"center_sub",
"cyclic1",
"cyclic_center_factor_abelian",
"eqVneq",
"gFnorm",
"group",
"logn",
"lognM",
"oZ",
"pP",
"p_pr",
"pfactorK",
"pgroupS",
"pgroup_pdiv",
"prime_cyclic",
"quotient_pgroup",
"trivg... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_p2group_abelian P : prime p -> #|P| = (p ^ 2)%N -> abelian P. | Proof.
move=> primep oP; have pP: p.-group P by rewrite /pgroup oP pnatX pnat_id.
by rewrite (p2group_abelian pP) // oP pfactorK.
Qed. | Lemma | card_p2group_abelian | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"abelian",
"group",
"p2group_abelian",
"pP",
"pfactorK",
"pgroup",
"pnatX",
"pnat_id",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylow_transversal_gen (T : {set {group gT}}) G :
(forall P, P \in T -> P \subset G) ->
(forall p, p \in \pi(G) -> exists2 P, P \in T & p.-Sylow(G) P) ->
<< \bigcup_(P in T) P >> = G. | Proof.
move=> G_T T_G; apply/eqP; rewrite eqEcard gen_subG.
apply/andP; split; first exact/bigcupsP.
apply: dvdn_leq (cardG_gt0 _) _; apply/dvdn_partP=> // q /T_G[P T_P sylP].
by rewrite -(card_Hall sylP); apply: cardSg; rewrite sub_gen // bigcup_sup.
Qed. | Lemma | Sylow_transversal_gen | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow",
"apply",
"bigcup_sup",
"bigcupsP",
"cardG_gt0",
"cardSg",
"card_Hall",
"dvdn_leq",
"dvdn_partP",
"eqEcard",
"gT",
"gen_subG",
"group",
"pi",
"split",
"sub_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylow_gen G : <<\bigcup_(P : {group gT} | Sylow G P) P>> = G. | Proof.
set T := [set P : {group gT} | Sylow G P].
rewrite -{2}(@Sylow_transversal_gen T G) => [P | q _|].
- by rewrite inE => /and3P[].
- by case: (Sylow_exists q G) => P sylP; exists P; rewrite // inE (p_Sylow sylP).
- by congr <<_>>; apply: eq_bigl => P; rewrite inE.
Qed. | Lemma | Sylow_gen | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow",
"Sylow_exists",
"Sylow_transversal_gen",
"apply",
"eq_bigl",
"gT",
"group",
"inE",
"p_Sylow"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hall_pJsub p pi G H P :
pi.-Hall(G) H -> p \in pi -> P \subset G -> p.-group P ->
exists2 x, x \in G & P :^ x \subset H. | Proof.
move=> hallH pi_p sPG pP.
have [S sylS] := Sylow_exists p H; have sylS_G := subHall_Sylow hallH pi_p sylS.
have [x Gx sPxS] := Sylow_Jsub sylS_G sPG pP; exists x => //.
exact: subset_trans sPxS (pHall_sub sylS).
Qed. | Lemma | Hall_pJsub | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Hall",
"Sylow_Jsub",
"Sylow_exists",
"group",
"pHall_sub",
"pP",
"pi",
"pi_p",
"subHall_Sylow",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hall_psubJ p pi G H P :
pi.-Hall(G) H -> p \in pi -> P \subset G -> p.-group P ->
exists2 x, x \in G & P \subset H :^ x. | Proof.
move=> hallH pi_p sPG pP; have [x Gx sPxH] := Hall_pJsub hallH pi_p sPG pP.
by exists x^-1; rewrite ?groupV -?sub_conjg.
Qed. | Lemma | Hall_psubJ | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Hall",
"Hall_pJsub",
"group",
"groupV",
"pP",
"pi",
"pi_p",
"sub_conjg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hall_setI_normal pi G K H :
K <| G -> pi.-Hall(G) H -> pi.-Hall(K) (H :&: K). | Proof.
move=> nsKG hallH; have [sHG piH _] := and3P hallH.
have [sHK_H sHK_K] := (subsetIl H K, subsetIr H K).
rewrite pHallE sHK_K /= -(part_pnat_id (pgroupS sHK_H piH)); apply/eqP.
rewrite (widen_partn _ (subset_leq_card sHK_K)); apply: eq_bigr => p pi_p.
have [P sylP] := Sylow_exists p H.
have sylPK := Sylow_setI_no... | Lemma | Hall_setI_normal | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Hall",
"Sylow_exists",
"Sylow_setI_normal",
"apply",
"card_Hall",
"eq_bigr",
"nsKG",
"pHallE",
"pHall_sub",
"pHall_subl",
"p_part",
"part_pnat_id",
"pgroupS",
"pi",
"pi_p",
"sHG",
"setIC",
"setSI",
"subHall_Sylow",
"subsetIl",
"subsetIr",
"subset_leq_card",
"widen_partn"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprime_mulG_setI_norm H G K R :
K * R = G -> G \subset 'N(H) -> coprime #|K| #|R| ->
(K :&: H) * (R :&: H) = G :&: H. | Proof.
move=> defG nHG coKR; apply/eqP; rewrite eqEcard mulG_subG /= -defG.
rewrite !setSI ?mulG_subl ?mulG_subr //=.
rewrite coprime_cardMg ?(coKR, coprimeSg (subsetIl _ _), coprime_sym) //=.
pose pi := \pi(K); have piK: pi.-group K by apply: pgroup_pi.
have pi'R: pi^'.-group R by rewrite /pgroup -coprime_pi' /=.
have... | Lemma | coprime_mulG_setI_norm | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Hall_setI_normal",
"apply",
"cardG_gt0",
"card_Hall",
"coprime",
"coprimeSg",
"coprime_cardMg",
"coprime_mulpG_Hall",
"coprime_pi'",
"coprime_sym",
"defG",
"eqEcard",
"group",
"mulG_subG",
"mulG_subl",
"mulG_subr",
"nHG",
"normG",
"normal",
"normsI",
"nsHG",
"partnC",
"p... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pgroup_nil p P : p.-group P -> nilpotent P. | Proof.
move: {2}_.+1 (ltnSn #|P|) => n.
elim: n gT P => // n IHn pT P; rewrite ltnS=> lePn pP.
have [Z1 | ntZ] := eqVneq 'Z(P) 1.
by rewrite (trivg_center_pgroup pP Z1) nilpotent1.
rewrite -quotient_center_nil IHn ?morphim_pgroup // (leq_trans _ lePn) //.
rewrite card_quotient ?normal_norm ?center_normal // -divgS ?s... | Lemma | pgroup_nil | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"card_quotient",
"center_normal",
"divgS",
"eqVneq",
"gT",
"group",
"leq_trans",
"ltnNge",
"ltnS",
"ltnSn",
"ltn_Pdiv",
"morphim_pgroup",
"nilpotent",
"nilpotent1",
"normal_norm",
"pP",
"quotient_center_nil",
"subsetIl",
"trivg_card_le1",
"trivg_center_pgroup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pgroup_sol p P : p.-group P -> solvable P. | Proof. by move/pgroup_nil; apply: nilpotent_sol. Qed. | Lemma | pgroup_sol | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"apply",
"group",
"nilpotent_sol",
"pgroup_nil",
"solvable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
small_nil_class G : nil_class G <= 5 -> nilpotent G. | Proof.
move=> leK5; case: (ltnP 5 #|G|) => [lt5G | leG5 {leK5}].
by rewrite nilpotent_class (leq_ltn_trans leK5).
apply: pgroup_nil (pdiv #|G|) _ _; apply/andP; split=> //.
by case: #|G| leG5 => //; do 5!case=> //.
Qed. | Lemma | small_nil_class | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"apply",
"leq_ltn_trans",
"ltnP",
"nil_class",
"nilpotent",
"nilpotent_class",
"pdiv",
"pgroup_nil",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nil_class2 G : (nil_class G <= 2) = (G^`(1) \subset 'Z(G)). | Proof.
rewrite subsetI der_sub; apply/idP/commG1P=> [clG2 | L3G1].
by apply/(lcn_nil_classP 2); rewrite ?small_nil_class ?(leq_trans clG2).
by apply/(lcn_nil_classP 2) => //; apply/lcnP; exists 2.
Qed. | Lemma | nil_class2 | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"apply",
"commG1P",
"der_sub",
"lcnP",
"lcn_nil_classP",
"leq_trans",
"nil_class",
"small_nil_class",
"subsetI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nil_class3 G : (nil_class G <= 3) = ('L_3(G) \subset 'Z(G)). | Proof.
rewrite subsetI lcn_sub; apply/idP/commG1P=> [clG3 | L4G1].
by apply/(lcn_nil_classP 3); rewrite ?small_nil_class ?(leq_trans clG3).
by apply/(lcn_nil_classP 3) => //; apply/lcnP; exists 3.
Qed. | Lemma | nil_class3 | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"apply",
"commG1P",
"lcnP",
"lcn_nil_classP",
"lcn_sub",
"leq_trans",
"nil_class",
"small_nil_class",
"subsetI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nilpotent_maxp_normal pi G H :
nilpotent G -> [max H | pi.-subgroup(G) H] -> H <| G. | Proof.
move=> nilG /maxgroupP[/andP[sHG piH] maxH].
have nHN: H <| 'N_G(H) by rewrite normal_subnorm.
have{maxH} hallH: pi.-Hall('N_G(H)) H.
apply: normal_max_pgroup_Hall => //; apply/maxgroupP.
rewrite /psubgroup normal_sub // piH; split=> // K.
by rewrite subsetI -andbA andbCA => /andP[_ /maxH].
rewrite /normal... | Lemma | nilpotent_maxp_normal | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Hall",
"apply",
"char",
"char_norms",
"max",
"maxgroupP",
"nilpotent",
"nilpotent_sub_norm",
"normal",
"normal_Hall_pcore",
"normal_max_pgroup_Hall",
"normal_sub",
"normal_subnorm",
"pcore_char",
"pi",
"psubgroup",
"sHG",
"setIS",
"setIidPl",
"split",
"subsetI",
"subsetIl"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nilpotent_Hall_pcore pi G H :
nilpotent G -> pi.-Hall(G) H -> H :=: 'O_pi(G). | Proof.
move=> nilG hallH; have maxH := Hall_max hallH; apply/eqP.
rewrite eqEsubset pcore_max ?(pHall_pgroup hallH) //.
exact: nilpotent_maxp_normal maxH.
by rewrite (normal_sub_max_pgroup maxH) ?pcore_pgroup ?pcore_normal.
Qed. | Lemma | nilpotent_Hall_pcore | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Hall",
"Hall_max",
"apply",
"eqEsubset",
"nilpotent",
"nilpotent_maxp_normal",
"normal_sub_max_pgroup",
"pHall_pgroup",
"pcore_max",
"pcore_normal",
"pcore_pgroup",
"pi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nilpotent_pcore_Hall pi G : nilpotent G -> pi.-Hall(G) 'O_pi(G). | Proof.
move=> nilG; case: (@maxgroup_exists _ (psubgroup pi G) 1) => [|H maxH _].
by rewrite /psubgroup sub1G pgroup1.
have hallH := normal_max_pgroup_Hall maxH (nilpotent_maxp_normal nilG maxH).
by rewrite -(nilpotent_Hall_pcore nilG hallH).
Qed. | Lemma | nilpotent_pcore_Hall | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Hall",
"maxgroup_exists",
"nilpotent",
"nilpotent_Hall_pcore",
"nilpotent_maxp_normal",
"normal_max_pgroup_Hall",
"pgroup1",
"pi",
"psubgroup",
"sub1G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nilpotent_pcoreC pi G : nilpotent G -> 'O_pi(G) \x 'O_pi^'(G) = G. | Proof.
move=> nilG; have trO: 'O_pi(G) :&: 'O_pi^'(G) = 1.
by apply: coprime_TIg; apply: (@pnat_coprime pi); apply: pcore_pgroup.
rewrite dprodE //.
rewrite (sameP commG1P trivgP) -trO subsetI commg_subl commg_subr.
by rewrite !gFsub_trans ?gFnorm.
apply/eqP; rewrite eqEcard mul_subG ?pcore_sub // (TI_cardMg trO)... | Lemma | nilpotent_pcoreC | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"TI_cardMg",
"apply",
"card_Hall",
"commG1P",
"commg_subl",
"commg_subr",
"coprime_TIg",
"dprodE",
"eqEcard",
"gFnorm",
"gFsub_trans",
"leqnn",
"mul_subG",
"nilpotent",
"nilpotent_pcore_Hall",
"partnC",
"pcore_pgroup",
"pcore_sub",
"pi",
"pnat_coprime",
"subsetI",
"trivgP"
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_nilpotent_cent2 H K G :
nilpotent G -> K \subset G -> H \subset G -> coprime #|K| #|H| ->
H \subset 'C(K). | Proof.
move=> nilG sKG sHG; rewrite coprime_pi' // => p'H.
have sub_Gp := sub_Hall_pcore (nilpotent_pcore_Hall _ nilG).
have [_ _ cGpp' _] := dprodP (nilpotent_pcoreC \pi(K) nilG).
by apply: centSS cGpp'; rewrite sub_Gp ?pgroup_pi.
Qed. | Lemma | sub_nilpotent_cent2 | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"apply",
"centSS",
"coprime",
"coprime_pi'",
"dprodP",
"nilpotent",
"nilpotent_pcoreC",
"nilpotent_pcore_Hall",
"pgroup_pi",
"pi",
"sHG",
"sKG",
"sub_Hall_pcore"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pi_center_nilpotent G : nilpotent G -> \pi('Z(G)) = \pi(G). | Proof.
move=> nilG; apply/eq_piP => /= p.
apply/idP/idP=> [|pG]; first exact: (piSg (center_sub _)).
move: (pG); rewrite !mem_primes !cardG_gt0; case/andP=> p_pr _.
pose Z := 'O_p(G) :&: 'Z(G); have ntZ: Z != 1.
rewrite meet_center_nil ?pcore_normal // trivg_card_le1 -ltnNge.
rewrite (card_Hall (nilpotent_pcore_Hal... | Lemma | pi_center_nilpotent | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"apply",
"cardG_gt0",
"cardSg",
"card_Hall",
"center_sub",
"dvdn_trans",
"eq_piP",
"group",
"logn_gt0",
"ltnNge",
"ltn_exp2l",
"meet_center_nil",
"mem_primes",
"nilpotent",
"nilpotent_pcore_Hall",
"pG",
"pZ",
"p_part",
"p_pr",
"pcore_normal",
"pcore_pgroup",
"pgroupS",
"p... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sylow_subnorm p G P : p.-Sylow('N_G(P)) P = p.-Sylow(G) P. | Proof.
apply/idP/idP=> sylP; last first.
apply: pHall_subl (subsetIl _ _) (sylP).
by rewrite subsetI normG (pHall_sub sylP).
have [/subsetIP[sPG sPN] pP _] := and3P sylP.
have [Q sylQ sPQ] := Sylow_superset sPG pP; have [sQG pQ _] := and3P sylQ.
rewrite -(nilpotent_sub_norm (pgroup_nil pQ) sPQ) {sylQ}//.
rewrite su... | Lemma | Sylow_subnorm | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow",
"Sylow_superset",
"apply",
"cardSg",
"card_Hall",
"dvdn_leq",
"eqEcard",
"eq_sym",
"last",
"nilpotent_sub_norm",
"normG",
"pHall_sub",
"pHall_subl",
"pP",
"part_pnat_id",
"partn_dvd",
"pgroupS",
"pgroup_nil",
"setSI",
"subEproper",
"subsetI",
"subsetIP",
"subsetI... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nil_class_pgroup (gT : finGroupType) (p : nat) (P : {group gT}) :
p.-group P -> nil_class P <= maxn 1 (logn p #|P|).-1. | Proof.
move=> pP; move def_c: (nil_class P) => c.
elim: c => // c IHc in gT P def_c pP *; set e := logn p _.
have nilP := pgroup_nil pP; have sZP := center_sub P.
have [e_le2 | e_gt2] := leqP e 2.
by rewrite -def_c leq_max nil_class1 (p2group_abelian pP).
have pPq: p.-group (P / 'Z(P)) by apply: quotient_pgroup.
rewr... | Lemma | nil_class_pgroup | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"add1n",
"apply",
"center_sub",
"gT",
"geq_max",
"group",
"leqP",
"leq_max",
"leq_sub2r",
"leq_subLR",
"leq_trans",
"logn",
"ltnS",
"ltn_log_quotient",
"maxn",
"meet_center_nil",
"nat",
"nilP",
"nil_class",
"nil_class0",
"nil_class1",
"nil_class_quotient_center",
"p2group... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Zgroup (gT : finGroupType) (A : {set gT}) | :=
[forall (V : {group gT} | Sylow A V), cyclic V]. | Definition | Zgroup | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Sylow",
"cyclic",
"gT",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ZgroupS G H : H \subset G -> Zgroup G -> Zgroup H. | Proof.
move=> sHG /forallP zgG; apply/forall_inP=> V /SylowP[p p_pr /and3P[sVH]].
case/(Sylow_superset (subset_trans sVH sHG))=> P sylP sVP _.
by have:= zgG P; rewrite (p_Sylow sylP); apply: cyclicS.
Qed. | Lemma | ZgroupS | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"SylowP",
"Sylow_superset",
"Zgroup",
"apply",
"cyclicS",
"forallP",
"forall_inP",
"p_Sylow",
"p_pr",
"sHG",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_Zgroup G : Zgroup G -> Zgroup (f @* G). | Proof.
move=> zgG; wlog sGD: G zgG / G \subset D.
by rewrite -morphimIdom; apply; rewrite (ZgroupS _ zgG, subsetIl) ?subsetIr.
apply/forall_inP=> fV /SylowP[p pr_p sylfV].
have [P sylP] := Sylow_exists p G.
have [|z _ ->] := @Sylow_trans p _ _ (f @* P)%G _ _ sylfV.
by apply: morphim_pHall (sylP); apply: subset_tran... | Lemma | morphim_Zgroup | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"SylowP",
"Sylow_exists",
"Sylow_trans",
"Zgroup",
"ZgroupS",
"apply",
"cyclicJ",
"forall_inP",
"morphimIdom",
"morphim_cyclic",
"morphim_pHall",
"pHall_sub",
"pr_p",
"sGD",
"subsetIl",
"subsetIr",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nil_Zgroup_cyclic G : Zgroup G -> nilpotent G -> cyclic G. | Proof.
have [n] := ubnP #|G|; elim: n G => // n IHn G /ltnSE-leGn ZgG nilG.
have [->|[p pr_p pG]] := trivgVpdiv G; first by rewrite -cycle1 cycle_cyclic.
have /dprodP[_ defG Cpp' _] := nilpotent_pcoreC p nilG.
have /cyclicP[x def_p]: cyclic 'O_p(G).
have:= forallP ZgG 'O_p(G)%G.
by rewrite (p_Sylow (nilpotent_pcore... | Lemma | nil_Zgroup_cyclic | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Zgroup",
"ZgroupS",
"apply",
"centsP",
"cycle1",
"cycleM",
"cycle_cyclic",
"cycle_id",
"cyclic",
"cyclicP",
"defG",
"def_p",
"dprodP",
"forallP",
"leq_trans",
"ltnSE",
"nilpotent",
"nilpotentS",
"nilpotent_pcoreC",
"nilpotent_pcore_Hall",
"order",
"p'natE",
"pG",
"p_Sy... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normal_pgroup r P N :
p.-group P -> N <| P -> r <= logn p #|N| ->
exists Q : {group gT}, [/\ Q \subset N, Q <| P & #|Q| = (p ^ r)%N]. | Proof.
elim: r gT P N => [|r IHr] gTr P N pP nNP le_r.
by exists (1%G : {group gTr}); rewrite sub1G normal1 cards1.
have [NZ_1 | ntNZ] := eqVneq (N :&: 'Z(P)) 1.
by rewrite (TI_center_nil (pgroup_nil pP)) // cards1 logn1 in le_r.
have: p.-group (N :&: 'Z(P)) by apply: pgroupS pP; rewrite /= setICA subsetIl.
case/pg... | Lemma | normal_pgroup | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Cauchy",
"Lagrange",
"TI_center_nil",
"apply",
"card_quotient",
"cards1",
"centsC",
"cents_norm",
"cycle_subG",
"eqVneq",
"eqxx",
"expnS",
"gT",
"group",
"inv_quotientN",
"leq_trans",
"logn",
"logn1",
"lognM",
"logn_prime",
"ltnS",
"morphim_normal",
"morphim_pgroup",
"... | B & G 1.22 p.9 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Baer_Suzuki x G :
x \in G -> (forall y, y \in G -> p.-group <<[set x; x ^ y]>>) ->
x \in 'O_p(G). | Proof.
have [n] := ubnP #|G|; elim: n G x => // n IHn G x /ltnSE-leGn Gx pE.
set E := x ^: G; have{} pE: {in E &, forall x1 x2, p.-group <<[set x1; x2]>>}.
move=> _ _ /imsetP[y1 Gy1 ->] /imsetP[y2 Gy2 ->].
rewrite -(mulgKV y1 y2) conjgM -2!conjg_set1 -conjUg genJ pgroupJ.
by rewrite pE // groupMl ?groupV.
have sE... | Theorem | Baer_Suzuki | solvable | solvable/sylow.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"prime",
"bigop",
"finset",
"fingroup",
"morphism",
"automorphism",
"quotient",
"action",
"cyclic",
"gproduct",
"gfunctor",
"commutator",
"pgroup",
"center",
"nilpotent"
] | [
"Dx",
"Px",
"Sylow",
"Sylow_Jsub",
"Sylow_superset",
"apply",
"cardJg",
"cardsU1",
"class_norm",
"class_refl",
"class_subG",
"conjSg",
"conjUg",
"conjgM",
"conjg_set1",
"eqEsubset",
"existsP",
"gT",
"genGid",
"genJ",
"genS",
"gen_subG",
"group",
"groupJ",
"groupMl",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tree | := Node { children : seq tree }. | Inductive | tree | test_suite | test_suite/test_guard.v | [
"mathcomp",
"boot"
] | [
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ptree (T : Type) | := singleton of T | branch of list (ptree T). | Inductive | ptree | test_suite | test_suite/test_guard.v | [
"mathcomp",
"boot"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tree_has (T : Type) (p : pred T) (t : ptree T) : bool | :=
match t with
| singleton x => p x
| branch ts => has (tree_has p) ts
end. | Fixpoint | tree_has | test_suite | test_suite/test_guard.v | [
"mathcomp",
"boot"
] | [
"has",
"ptree"
] | has | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
tree_all (T : Type) (p : pred T) (t : ptree T) : bool | :=
match t with
| singleton x => p x
| branch ts => all (tree_all p) ts
end. | Fixpoint | tree_all | test_suite | test_suite/test_guard.v | [
"mathcomp",
"boot"
] | [
"all",
"ptree"
] | all | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
traverse_id (t : tree) : tree | :=
Node (map traverse_id (children t)). | Fixpoint | traverse_id | test_suite | test_suite/test_guard.v | [
"mathcomp",
"boot"
] | [
"map",
"tree"
] | map | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
tree_foldr (T R : Type) (f : T -> R -> R) (z : R) (t : ptree T) : R | :=
match t with
| singleton x => f x z
| branch ts => foldr (fun t z' => tree_foldr f z' t) z ts
end. | Fixpoint | tree_foldr | test_suite | test_suite/test_guard.v | [
"mathcomp",
"boot"
] | [
"foldr",
"ptree"
] | foldr | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
tree_foldl (T R : Type) (f : R -> T -> R) (z : R) (t : ptree T) : R | :=
match t with
| singleton x => f z x
| branch ts => foldl (tree_foldl f) z ts
end. | Fixpoint | tree_foldl | test_suite | test_suite/test_guard.v | [
"mathcomp",
"boot"
] | [
"foldl",
"ptree"
] | foldl | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eq_tree (x y : tree) {struct x} : bool | :=
all2 eq_tree (children x) (children y). | Fixpoint | eq_tree | test_suite | test_suite/test_guard.v | [
"mathcomp",
"boot"
] | [
"all2",
"tree"
] | all2 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
s_of_pq (p q : {i01 R}) : {i01 R} | :=
(1 - ((1 - p%:num)%:i01%:num * (1 - q%:num)%:i01%:num))%:i01. | Definition | s_of_pq | test_suite | test_suite/test_interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"interval_inference",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory"
] | [
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
s_of_p0 (p : {i01 R}) : s_of_pq p 0%:i01 = p. | Proof. by apply/val_inj; rewrite /= subr0 mulr1 subKr. Qed. | Lemma | s_of_p0 | test_suite | test_suite/test_interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"interval_inference",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory"
] | [
"apply",
"mulr1",
"s_of_pq",
"subKr",
"subr0",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_dup1 : forall n : nat, odd n. | Proof. move=> /[dup] m n; suff: odd n by []. Abort. | Lemma | test_dup1 | test_suite | test_suite/test_intro_rw.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat"
] | [
"nat",
"odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_dup2 : let n := 1 in False. | Proof. move=> /[dup] m n; have : m = n := erefl. Abort. | Lemma | test_dup2 | test_suite | test_suite/test_intro_rw.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat"
] | [
"False"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_swap1 : forall (n : nat) (b : bool), odd n = b. | Proof. move=> /[swap] b n; suff: odd n = b by []. Abort. | Lemma | test_swap1 | test_suite | test_suite/test_intro_rw.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat"
] | [
"nat",
"odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_swap1 : let n := 1 in let b := true in False. | Proof. move=> /[swap] b n; have : odd n = b := erefl. Abort. | Lemma | test_swap1 | test_suite | test_suite/test_intro_rw.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat"
] | [
"False",
"odd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_apply A B : forall (f : A -> B) (a : A), False. | Proof.
move=> /[apply] b.
Check (b : B).
Abort. | Lemma | test_apply | test_suite | test_suite/test_intro_rw.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat"
] | [
"False",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_swap_plus P Q : P -> Q -> False. | Proof.
move=> + /[dup] q.
suff: P -> Q -> False by [].
Abort. | Lemma | test_swap_plus | test_suite | test_suite/test_intro_rw.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat"
] | [
"False"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_dup_plus2 P : P -> let x := 0 in False. | Proof.
move=> + /[dup] y.
suff: P -> let x := 0 in False by [].
Abort. | Lemma | test_dup_plus2 | test_suite | test_suite/test_intro_rw.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat"
] | [
"False"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_swap_plus P Q R : P -> Q -> R -> False. | Proof.
move=> + /[swap].
suff: P -> R -> Q -> False by [].
Abort. | Lemma | test_swap_plus | test_suite | test_suite/test_intro_rw.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat"
] | [
"False"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_swap_plus2 P : P -> let x := 0 in let y := 1 in False. | Proof.
move=> + /[swap].
suff: P -> let y := 1 in let x := 0 in False by [].
Abort. | Lemma | test_swap_plus2 | test_suite | test_suite/test_intro_rw.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat"
] | [
"False"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test (F : realFieldType) (x y : F) :
x + 2 * y <= 3 -> 2 * x + y <= 3 -> x + y <= 2. | Proof. lra. Qed. | Lemma | test | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_rat (x y : rat) : x + 2 * y <= 3 -> 2 * x + y <= 3 -> x + y <= 2. | Proof. lra. Qed. | Lemma | test_rat | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [
"rat"
] | Print Assumptions test. (* Closed under the global context *) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
test_realDomain (R : realDomainType) (x y : R) :
x + 2 * y <= 3 -> 2 * x + y <= 3 -> x + y <= 2. | Proof. lra. Qed. | Lemma | test_realDomain | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_realDomain' (R : realDomainType) (x : int) (y : R) :
x%:~R + 2 * y <= 3 -> (2 * x)%:~R + y <= 3 -> x%:~R + y <= 2. | Proof. lra. Qed. | Lemma | test_realDomain' | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_cast : 0 <= 2 :> F. | Proof. lra. Qed. | Lemma | test_cast | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_div x y : x / 2 + y <= 3 -> x + y / 2 <= 3 -> x + y <= 4. | Proof. lra. Qed. | Example | test_div | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_div_mul x : 1 / (2 * x) <= 1 / 2 / x + 1. | Proof. lra. Qed. | Example | test_div_mul | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_div_inv x : 1 / x^-1 <= x + 1. | Proof. lra. Qed. | Example | test_div_inv | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_div_opp x : (- x)^-1 <= - x^-1 + 1. | Proof. lra. Qed. | Example | test_div_opp | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_div_exp x : (x ^+ 2) ^-1 <= x ^-1 ^+ 2 + 1. | Proof. lra. Qed. | Example | test_div_exp | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_lt x y : x + 2 * y < 3 -> 2 * x + y <= 3 -> x + y < 2. | Proof. lra. Qed. | Lemma | test_lt | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_eq x y : x + 2 * y = 3 -> 2 * x + y <= 3 -> x + y <= 2. | Proof. lra. Qed. | Lemma | test_eq | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_eqop x y : x + 2 * y == 3 -> 2 * x + y <= 3 -> x + y <= 2. | Proof. lra. Qed. | Lemma | test_eqop | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_prop_in_middle (C : Prop) x : x <= 2 -> C -> x <= 3. | Proof. lra. Qed. | Lemma | test_prop_in_middle | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_opp x : x <= 2 -> -x >= -2. | Proof. lra. Qed. | Lemma | test_opp | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
test_iff x : x <= 2 <-> -x >= -2. | Proof. lra. Qed. | Lemma | test_iff | test_suite | test_suite/test_lra.v | [
"mathcomp",
"all_boot",
"ssralg",
"ssrnum",
"ssrint",
"rat",
"arithmetic_tactic"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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