fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
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|---|---|---|---|---|---|---|
dtuple_on:= [set t : n.-tuple sT | uniq t & t \subset S].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
dtuple_on
| |
ntransitive:= [transitive A, on dtuple_on | to * n].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
ntransitive
| |
dtuple_onPt :
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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
dtuple_onP
| |
n_act_dtuplet 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
n_act_dtuple
| |
card_uniq_tuplen (t : n.-tuple sT) : uniq t -> #|t| = n.
Proof. by move/card_uniqP->; apply: size_tuple. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
card_uniq_tuple
| |
n_act0(t : 0.-tuple sT) a : n_act to t a = [tuple].
Proof. exact: tuple0. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
n_act0
| |
dtuple_on_addn 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
dtuple_on_add
| |
dtuple_on_add_D1n 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
dtuple_on_add_D1
| |
dtuple_on_subsetn (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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
dtuple_on_subset
| |
n_act_addn 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
n_act_add
| |
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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
ntransitive0
| |
ntransitive_weakk 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 exists x; rewrite dtuple_on_add andbA /= Sx ntx.
case/imsetP: tr_m1 dt => t1 /[!inE] /andP[Ut1 St1] _ /andP[Ut _].
have /subset_leq_card := subset_trans St1 sSt.
by rewrite !card_uniq_tuple // ltnn.
case/imsetP: (tr_m1); case/tupleP=> [x t]; rewrite dtuple_on_add.
case/and3P=> Sx ntx dt; set xt := [tuple of _] => tr_xt.
apply/imsetP; exists t => //.
apply/setP=> u; apply/idP/imsetP=> [du | [a Ga ->{u}]].
case: (ext_t u du) => y; rewrite tr_xt.
by case/imsetP=> a Ga [_ def_u]; exists a => //; apply: val_inj.
have: n_act to xt a \in dtuple_on _ S by rewrite tr_xt imset_f.
by rewrite n_act_add dtuple_on_add; case/and3P.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
ntransitive_weak
| |
ntransitive1m :
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/imsetP/imsetP=> [[a ? [->]]|[a ? ->]]; exists a => //; apply: val_inj.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
ntransitive1
| |
ntransitive_primitivem :
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 Ha nCa] sHG.
right; rewrite -(subgroup_transitiveP Sx sHG trG _) ?mulSGid //.
have actH := subset_trans sHG (atrans_acts trG).
pose y := to x a; have Sy: y \in S by rewrite (actsP actH).
have{nCa} yx: y != x by rewrite inE (sameP astab1P eqP) (subsetP sHG) in nCa.
apply/imsetP; exists y => //; apply/eqP.
rewrite eqEsubset acts_sub_orbit // Sy andbT; apply/subsetP=> z Sz.
have [-> | zx] := eqVneq z x; first by rewrite orbit_sym mem_orbit.
pose ty := [tuple y; x]; pose tz := [tuple z; x].
have [Sty Stz]: ty \in 2.-dtuple(S) /\ tz \in 2.-dtuple(S).
by rewrite !inE !memtE !subset_all /= !mem_seq1 !andbT; split; apply/and3P.
case: (atransP2 tr2G Sty Stz) => b Gb [->] /esym/astab1P cxb.
by rewrite mem_orbit // (subsetP sCH) // inE Gb.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
ntransitive_primitive
| |
stab_ntransitivem 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.+1.-dtuple(S).
by rewrite trt1 x1ax; apply/imsetP; exists a => //; apply: val_inj.
apply/imsetP; exists t; first by rewrite dtuple_on_add_D1 Sx in dxt.
apply/setP=> t2; apply/idP/imsetP => [dt2|[b]].
have: [tuple of x :: t2] \in dtuple_on _ S by rewrite dtuple_on_add_D1 Sx.
case/(atransP2 Gtr dxt)=> b Gb [xbx tbt2].
by exists b; [rewrite inE Gb; apply/astab1P | apply: val_inj].
case/setIP=> Gb /astab1P xbx ->{t2}.
rewrite n_act_dtuple //; last by rewrite dtuple_on_add_D1 Sx in dxt.
apply/astabsP=> y; rewrite !inE -{1}xbx (inj_eq (act_inj _ _)).
by rewrite (actsP (atrans_acts Gtr1)).
Qed.
|
Theorem
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
stab_ntransitive
| |
stab_ntransitiveIm 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 _ Sy) in Sx; case/imsetP: Sx => a Ga toya.
exists a^-1; first exact: groupVr.
exists (n_act to t a); last by rewrite n_act_add toya !actK.
move/(n_act_dtuple (subsetP (atrans_acts Gtr) a Ga)): St.
by rewrite n_act_add -toya dtuple_on_add_D1 => /andP[].
case: (imsetP Gntr) => t dt S_tG; pose xt := [tuple of x :: t].
have dxt: xt \in m.+1.-dtuple(S) by rewrite dtuple_on_add_D1 Sx.
apply/imsetP; exists xt => //; apply/setP=> t2.
apply/esym; apply/imsetP/idP=> [[a Ga ->] | ].
by apply: n_act_dtuple; rewrite // (subsetP (atrans_acts Gtr)).
case/t_to_x=> a2 Ga2 [t2']; rewrite S_tG.
case/imsetP=> a /setIP[Ga /astab1P toxa] -> -> {t2 t2'}.
by exists (a * a2); rewrite (groupM, actM) //= !n_act_add toxa.
Qed.
|
Theorem
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat",
"From mathcomp Require Import div seq fintype tuple finset",
"From mathcomp Require Import fingroup action gseries"
] |
solvable/primitive_action.v
|
stab_ntransitiveI
| |
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.
apply: dvdn_sum => _ /imsetP[x /setDP[_ nfx] ->].
have [k oGx]: {k | #|orbit to G x| = (p ^ k)%N}.
by apply: p_natP; apply: pnat_dvd pG; rewrite card_orbit_in ?dvdn_indexg.
case: k oGx => [/card_orbit1 fix_x | k ->]; last by rewrite expnS dvdn_mulr.
by case/afixP: nfx => a Ga; apply/set1P; rewrite -fix_x mem_orbit.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
pgroup_fix_mod
| |
nontrivial_gacent_pgroupG 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
nontrivial_gacent_pgroup
| |
pcore_sub_astab_irrG 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 //=; last first.
by split; rewrite ?gFsub_trans.
by apply: subset_trans (acts_subnorm_subgacent sGpD nMG); rewrite subsetI subxx.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
pcore_sub_astab_irr
| |
pcore_faithful_irr_actG 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
pcore_faithful_irr_act
| |
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/andP[].
have SmaxN P Q: Q \in S -> Q \subset 'N(P) -> maxp 'N_G(P) Q.
rewrite inE => /maxgroupP[/andP[sQG pQ] maxQ] nPQ.
apply/maxgroupP; rewrite /psubgroup subsetI sQG nPQ.
by split=> // R; rewrite subsetI -andbA andbCA => /andP[_]; apply: maxQ.
have nrmG P: P \subset G -> P <| 'N_G(P).
by move=> sPG; rewrite /normal subsetIr subsetI sPG normG.
have sylS P: P \in S -> p.-Sylow('N_G(P)) P.
move=> S_P; have [sPG pP] := S_pG P S_P.
by rewrite normal_max_pgroup_Hall ?nrmG //; apply: SmaxN; rewrite ?normG.
have{SmaxN} defCS P: P \in S -> 'Fix_(S |'JG)(P) = [set P].
move=> S_P; apply/setP=> Q; rewrite {1}in_setI {1}afixJG.
apply/andP/set1P=> [[S_Q nQP]|->{Q}]; last by rewrite normG.
apply/esym/val_inj; case: (S_pG Q) => //= sQG _.
by apply: uniq_normal_Hall (SmaxN Q _ _ _) => //=; rewrite ?sylS ?nrmG.
have{defCS} oG_mod: {in S &, forall P Q, #|oG P| = (Q \in oG P) %[mod p]}.
move=> P Q S_P S_Q; have [sQG pQ] := S_pG _ S_Q.
have soP_S: oG
...
|
Theorem
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Sylow's_theorem
| |
max_pgroup_SylowP : [max P | p.-subgroup(G) P] = p.-Sylow(G) P.
Proof. by case Sylow's_theorem. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
max_pgroup_Sylow
| |
Sylow_supersetQ :
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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Sylow_superset
| |
Sylow_exists: {P : {group gT} | p.-Sylow(G) P}.
Proof. by case: (Sylow_superset (sub1G G) (pgroup1 _ p)) => P; exists P. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Sylow_exists
| |
Syl_trans: [transitive G, on 'Syl_p(G) | 'JG].
Proof. by case Sylow's_theorem. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Syl_trans
| |
Sylow_transP 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Sylow_trans
| |
Sylow_subJP 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Sylow_subJ
| |
Sylow_JsubP 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Sylow_Jsub
| |
card_SylP : p.-Sylow(G) P -> #|'Syl_p(G)| = #|G : 'N_G(P)|.
Proof. by case: Sylow's_theorem P. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
card_Syl
| |
card_Syl_dvd: #|'Syl_p(G)| %| #|G|.
Proof. by case Sylow_exists => P /card_Syl->; apply: dvdn_indexg. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
card_Syl_dvd
| |
card_Syl_mod: prime p -> #|'Syl_p(G)| %% p = 1%N.
Proof. by case Sylow's_theorem. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
card_Syl_mod
| |
Frattini_argH 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 nGH x Hx) pHallJ2.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Frattini_arg
| |
Sylow_setI_normalG 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 :&: R by rewrite subsetI sQG.
by rewrite /psubgroup subsetIl (pgroupS _ pR) ?subsetIr.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Sylow_setI_normal
| |
normal_sylowPG :
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_Syl sylP) (eqP syl1) muln1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
normal_sylowP
| |
trivg_center_pgroupP : 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
trivg_center_pgroup
| |
p2group_abelianP : 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_pgroup 'Z(P) pP; case/(pgroup_pdiv pPq) => _ _ [j oPq].
rewrite prime_cyclic // oPq; case: j oPq lePp2 => //= j.
rewrite card_quotient ?gFnorm //.
by rewrite -(Lagrange sZP) lognM // => ->; rewrite oZ !pfactorK ?addnS.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
p2group_abelian
| |
card_p2group_abelianP : 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
card_p2group_abelian
| |
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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Sylow_transversal_gen
| |
Sylow_genG : <<\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 congr <<_>>; apply: eq_bigl => P; rewrite inE.
- by rewrite inE => /and3P[].
by case: (Sylow_exists q G) => P sylP; exists P; rewrite // inE (p_Sylow sylP).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Sylow_gen
| |
Hall_pJsubp 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Hall_pJsub
| |
Hall_psubJp 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Hall_psubJ
| |
Hall_setI_normalpi 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_normal nsKG (subHall_Sylow hallH pi_p sylP).
rewrite -!p_part -(card_Hall sylPK); symmetry; apply: card_Hall.
by rewrite (pHall_subl _ sHK_K) //= setIC setSI ?(pHall_sub sylP).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Hall_setI_normal
| |
coprime_mulG_setI_normH 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 [hallK hallR] := coprime_mulpG_Hall defG piK pi'R.
have nsHG: H :&: G <| G by rewrite /normal subsetIr normsI ?normG.
rewrite -!(setIC H) defG -(partnC pi (cardG_gt0 _)).
rewrite -(card_Hall (Hall_setI_normal nsHG hallR)) /= setICA.
rewrite -(card_Hall (Hall_setI_normal nsHG hallK)) /= setICA.
by rewrite -defG (setIidPl (mulG_subl _ _)) (setIidPl (mulG_subr _ _)).
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
coprime_mulG_setI_norm
| |
pgroup_nilp 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 ?subsetIl //.
by rewrite ltn_Pdiv // ltnNge -trivg_card_le1.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
pgroup_nil
| |
pgroup_solp P : p.-group P -> solvable P.
Proof. by move/pgroup_nil; apply: nilpotent_sol. Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
pgroup_sol
| |
small_nil_classG : 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
small_nil_class
| |
nil_class2G : (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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
nil_class2
| |
nil_class3G : (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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
nil_class3
| |
nilpotent_maxp_normalpi 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 sHG; apply/setIidPl/esym.
apply: nilpotent_sub_norm; rewrite ?subsetIl ?setIS //= char_norms //.
by congr (_ \char _): (pcore_char pi 'N_G(H)); apply: normal_Hall_pcore.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
nilpotent_maxp_normal
| |
nilpotent_Hall_pcorepi 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) //.
by rewrite (normal_sub_max_pgroup maxH) ?pcore_pgroup ?pcore_normal.
exact: nilpotent_maxp_normal maxH.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
nilpotent_Hall_pcore
| |
nilpotent_pcore_Hallpi 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
nilpotent_pcore_Hall
| |
nilpotent_pcoreCpi 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 //.
apply/eqP; rewrite eqEcard mul_subG ?pcore_sub // (TI_cardMg trO).
by rewrite !(card_Hall (nilpotent_pcore_Hall _ _)) // partnC ?leqnn.
rewrite (sameP commG1P trivgP) -trO subsetI commg_subl commg_subr.
by rewrite !gFsub_trans ?gFnorm.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
nilpotent_pcoreC
| |
sub_nilpotent_cent2H 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
sub_nilpotent_cent2
| |
pi_center_nilpotentG : 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_Hall p nilG)) p_part.
by rewrite (ltn_exp2l 0 _ (prime_gt1 p_pr)) logn_gt0.
have pZ: p.-group Z := pgroupS (subsetIl _ _) (pcore_pgroup _ _).
have{ntZ pZ} [_ pZ _] := pgroup_pdiv pZ ntZ.
by rewrite p_pr (dvdn_trans pZ) // cardSg ?subsetIr.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
pi_center_nilpotent
| |
Sylow_subnormp 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 subEproper eq_sym eqEcard subsetI sPQ sPN dvdn_leq //.
rewrite -(part_pnat_id (pgroupS (subsetIl _ _) pQ)) (card_Hall sylP).
by rewrite partn_dvd // cardSg ?setSI.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Sylow_subnorm
| |
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.
rewrite -(subnKC e_gt2) ltnS (leq_trans (IHc _ _ _ pPq)) //.
by rewrite nil_class_quotient_center ?def_c.
rewrite geq_max /= -add1n -leq_subLR -subn1 -subnDA -subSS leq_sub2r //.
rewrite ltn_log_quotient //= -(setIidPr sZP) meet_center_nil //.
by rewrite -nil_class0 def_c.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
nil_class_pgroup
| |
Zgroup(gT : finGroupType) (A : {set gT}) :=
[forall (V : {group gT} | Sylow A V), cyclic V].
|
Definition
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Zgroup
| |
ZgroupSG 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
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
ZgroupS
| |
morphim_ZgroupG : 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_trans (pHall_sub sylP) sGD.
by rewrite cyclicJ morphim_cyclic ?(forall_inP zgG) //; apply/SylowP; exists p.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
morphim_Zgroup
| |
nil_Zgroup_cyclicG : 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_Hall p nilG)).
have /cyclicP[x' def_p']: cyclic 'O_p^'(G).
have sp'G := pcore_sub p^' G.
apply: IHn (leq_trans _ leGn) (ZgroupS sp'G _) (nilpotentS sp'G _) => //.
rewrite proper_card // properEneq sp'G andbT; case: eqP => //= def_p'.
by have:= pcore_pgroup p^' G; rewrite def_p' /pgroup p'natE ?pG.
apply/cyclicP; exists (x * x'); rewrite -{}defG def_p def_p' cycleM //.
by red; rewrite -(centsP Cpp') // (def_p, def_p') cycle_id.
by rewrite /order -def_p -def_p' (@pnat_coprime p) //; apply: pcore_pgroup.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
nil_Zgroup_cyclic
| |
normal_pgroupr 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/pgroup_pdiv=> // p_pr /Cauchy[// | z].
rewrite -cycle_subG !subsetI => /and3P[szN szP cPz] ozp _.
have{cPz} nzP: P \subset 'N(<[z]>) by rewrite cents_norm // centsC.
have: N / <[z]> <| P / <[z]> by rewrite morphim_normal.
case/IHr=> [||Qb [sQNb nQPb]]; first exact: morphim_pgroup.
rewrite card_quotient ?(subset_trans (normal_sub nNP)) // -ltnS.
apply: (leq_trans le_r); rewrite -(Lagrange szN) [#|_|]ozp.
by rewrite lognM // ?prime_gt0 // logn_prime ?eqxx.
case/(inv_quotientN _): nQPb sQNb => [|Q -> szQ nQP]; first exact/andP.
have nzQ := subset_trans (normal_sub nQP) nzP.
rewrite quotientSGK // card_quotient // => sQN izQ.
by exists Q; split=> //; rewrite expnS -izQ -ozp Lagrange.
Qed.
|
Lemma
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
normal_pgroup
| |
Baer_Suzukix 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 sEG: <<E>> \subset G by rewrite gen_subG class_subG.
have nEG: G \subset 'N(E) by apply: class_norm.
have Ex: x \in E by apply: class_refl.
have [P Px sylP]: exists2 P : {group gT}, x \in P & p.-Sylow(<<E>>) P.
have sxxE: <<[set x; x]>> \subset <<E>> by rewrite genS // setUid sub1set.
have{sxxE} [P sylP sxxP] := Sylow_superset sxxE (pE _ _ Ex Ex).
by exists P => //; rewrite (subsetP sxxP) ?mem_gen ?setU11.
case sEP: (E \subset P).
apply: subsetP Ex; rewrite -gen_subG; apply: pcore_max.
by apply: pgroupS (pHall_pgroup sylP); rewrite gen_subG.
by rewrite /normal gen_subG class_subG // norms_gen.
pose P_yD D := [pred y in E :\: P | p.-group <<y |: D>>].
pose P_D := [pred D : {set gT} | D \subset P :&: E & [exists y, P_yD D y]].
have{Ex Px}: P_D [set x].
rewrite /= sub1set inE Px Ex; apply/existsP=> /=.
by case/subsetPn: sEP => y Ey Py; exists y; rewrite inE Ey Py pE.
case/(@maxset_exists _ P_D)=> D /maxsetP[]; rewrite {P_yD P_D}/=.
rewrite subsetI sub1set -andbA => /and3P[sDP sDE /existsP[y0]].
set B := _ |: D; rewrite inE -andbA => /and3P[Py0 Ey0 pB] maxD Dx
...
|
Theorem
|
solvable
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype prime bigop finset fingroup morphism",
"From mathcomp Require Import automorphism quotient action cyclic gproduct ",
"From mathcomp Require Import gfunctor commutator pgroup center nilpotent"
] |
solvable/sylow.v
|
Baer_Suzuki
| |
tree:= Node { children : seq tree }.
|
Inductive
|
test_suite
|
[
"From mathcomp Require Import all_boot"
] |
test_suite/test_guard.v
|
tree
| |
ptree(T : Type) := singleton of T | branch of list (ptree T).
|
Inductive
|
test_suite
|
[
"From mathcomp Require Import all_boot"
] |
test_suite/test_guard.v
|
ptree
| |
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
|
test_suite
|
[
"From mathcomp Require Import all_boot"
] |
test_suite/test_guard.v
|
tree_has
| |
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
|
test_suite
|
[
"From mathcomp Require Import all_boot"
] |
test_suite/test_guard.v
|
tree_all
| |
traverse_id(t : tree) : tree :=
Node (map traverse_id (children t)).
|
Fixpoint
|
test_suite
|
[
"From mathcomp Require Import all_boot"
] |
test_suite/test_guard.v
|
traverse_id
| |
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
|
test_suite
|
[
"From mathcomp Require Import all_boot"
] |
test_suite/test_guard.v
|
tree_foldr
| |
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
|
test_suite
|
[
"From mathcomp Require Import all_boot"
] |
test_suite/test_guard.v
|
tree_foldl
| |
eq_tree(x y : tree) {struct x} : bool :=
all2 eq_tree (children x) (children y).
|
Fixpoint
|
test_suite
|
[
"From mathcomp Require Import all_boot"
] |
test_suite/test_guard.v
|
eq_tree
| |
test_dup1: forall n : nat, odd n.
Proof. move=> /[dup] m n; suff: odd n by []. Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat"
] |
test_suite/test_intro_rw.v
|
test_dup1
| |
test_dup2: let n := 1 in False.
Proof. move=> /[dup] m n; have : m = n := erefl. Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat"
] |
test_suite/test_intro_rw.v
|
test_dup2
| |
test_swap1: forall (n : nat) (b : bool), odd n = b.
Proof. move=> /[swap] b n; suff: odd n = b by []. Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat"
] |
test_suite/test_intro_rw.v
|
test_swap1
| |
test_swap1: let n := 1 in let b := true in False.
Proof. move=> /[swap] b n; have : odd n = b := erefl. Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat"
] |
test_suite/test_intro_rw.v
|
test_swap1
| |
test_applyA B : forall (f : A -> B) (a : A), False.
Proof.
move=> /[apply] b.
Check (b : B).
Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat"
] |
test_suite/test_intro_rw.v
|
test_apply
| |
test_swap_plusP Q : P -> Q -> False.
Proof.
move=> + /[dup] q.
suff: P -> Q -> False by [].
Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat"
] |
test_suite/test_intro_rw.v
|
test_swap_plus
| |
test_dup_plus2P : P -> let x := 0 in False.
Proof.
move=> + /[dup] y.
suff: P -> let x := 0 in False by [].
Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat"
] |
test_suite/test_intro_rw.v
|
test_dup_plus2
| |
test_swap_plusP Q R : P -> Q -> R -> False.
Proof.
move=> + /[swap].
suff: P -> R -> Q -> False by [].
Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat"
] |
test_suite/test_intro_rw.v
|
test_swap_plus
| |
test_swap_plus2P : 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_suite
|
[
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat"
] |
test_suite/test_intro_rw.v
|
test_swap_plus2
| |
test_orb(a b c d : bool) : (a || b) || (c || d) = (a || c) || (b || d).
Proof. time by rewrite orbACA. Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import all_boot ssralg"
] |
test_suite/test_ssrAC.v
|
test_orb
| |
test_orb(a b c d : bool) : (a || b) || (c || d) = (a || c) || (b || d).
Proof. time by rewrite (AC (2*2) ((1*3)*(2*4))). Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import all_boot ssralg"
] |
test_suite/test_ssrAC.v
|
test_orb
| |
test_orb(a b c d : bool) : (a || b) || (c || d) = (a || c) || (b || d).
Proof. time by rewrite orb.[AC (2*2) ((1*3)*(2*4))]. Qed.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import all_boot ssralg"
] |
test_suite/test_ssrAC.v
|
test_orb
| |
test_addn(a b c d : nat) : a + b + c + d = a + c + b + d.
Proof. time by rewrite -addnA addnAC addnA addnAC. Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import all_boot ssralg"
] |
test_suite/test_ssrAC.v
|
test_addn
| |
test_addn(a b c d : nat) : a + b + c + d = a + c + b + d.
Proof. time by rewrite (ACl (1*3*2*4)). Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import all_boot ssralg"
] |
test_suite/test_ssrAC.v
|
test_addn
| |
test_addn(a b c d : nat) : a + b + c + d = a + c + b + d.
Proof. time by rewrite addn.[ACl 1*3*2*4]. Qed.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import all_boot ssralg"
] |
test_suite/test_ssrAC.v
|
test_addn
| |
test_addr(R : comRingType) (a b c d : R) : (a + b + c + d = a + c + b + d)%R.
Proof. time by rewrite -GRing.addrA GRing.addrAC GRing.addrA GRing.addrAC. Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import all_boot ssralg"
] |
test_suite/test_ssrAC.v
|
test_addr
| |
test_addr(R : comRingType) (a b c d : R) : (a + b + c + d = a + c + b + d)%R.
Proof. time by rewrite (ACl (1*3*2*4)). Abort.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import all_boot ssralg"
] |
test_suite/test_ssrAC.v
|
test_addr
| |
test_addr(R : comRingType) (a b c d : R) : (a + b + c + d = a + c + b + d)%R.
Proof. time by rewrite (@GRing.add R).[ACl 1*3*2*4]. Qed.
Local Open Scope ring_scope.
Import GRing.Theory.
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import all_boot ssralg"
] |
test_suite/test_ssrAC.v
|
test_addr
| |
test_mulr(R : comRingType) (x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 : R)
(x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 : R) :
(x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x7 * x8 * x9) *
(x10 * x11) * (x12 * x13) * (x14 * x15) * (x16 * x17 * x18 * x19) *
(x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x7 * x8 * x9) *
(x10 * x11) * (x12 * x13) * (x14 * x15) * (x16 * x17 * x18 * x19) =
(x0 * x2 * x4 * x9) * (x1 * x3 * x5 * x7) * x6 * x8 *
(x10 * x12 * x14 * x19) * (x11 * x13 * x15 * x17) * x16 * x18 * (x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x7 * x8 * x9) *
(x10 * x11) * (x12 * x13) * (x14 * x15) * (x16 * x17 * x18 * x19)
*(x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x7 * x8 * x9) *
(x10 * x11) * (x12 * x13) * (x14 * x15) * (x16 * x17 * x18 * x19) *
(x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x7 * x8 * x9) *(x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x7 * x8 * x9) *
(x10 * x11) * (x12 * x13) * (x14 * x15) * (x16 * x17 * x18 * x19) *
(x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x7 * x8 * x9) * (x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x7 * x8 * x9) *
(x10 * x11) * (x12 * x13) * (x14 * x15) * (x16 * x17 * x18 * x19)
*(x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x7 * x8 * x9) *
(x10 * x11) * (x12 * x13) * (x14 * x15) * (x16 * x17 * x18 * x19) *
(x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x7 * x8 * x9) *(x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x7 * x8 * x9) *
(x10 * x11) * (x12 * x13) * (x14 * x15) * (x16 * x17 * x18 * x19) *
(x0 * x1) * (x2 * x3) * (x4 * x5) * (x6 * x
...
|
Lemma
|
test_suite
|
[
"From mathcomp Require Import all_boot ssralg"
] |
test_suite/test_ssrAC.v
|
test_mulr
| |
RecordZmodule_isSemiNormed (R : POrderedZmodule.type) M
of GRing.Zmodule M := {
norm : M -> R;
ler_normD : forall x y, norm (x + y) <= norm x + norm y;
normrMn : forall x n, norm (x *+ n) = norm x *+ n;
normrN : forall x, norm (- x) = norm x;
}.
#[short(type="semiNormedZmodType")]
HB.structure Definition SemiNormedZmodule (R : porderZmodType) :=
{ M of Zmodule_isSemiNormed R M & GRing.Zmodule M }.
HB.mixin Record SemiNormedZmodule_isPositiveDefinite
(R : POrderedZmodule.type) M of @SemiNormedZmodule R M := {
normr0_eq0 : forall x : M, norm x = 0 -> x = 0;
}.
#[short(type="normedZmodType")]
HB.structure Definition NormedZmodule (R : porderZmodType) :=
{ M of SemiNormedZmodule_isPositiveDefinite R M & SemiNormedZmodule R M }.
Arguments norm {R M} x : rename.
HB.factory Record Zmodule_isNormed (R : POrderedZmodule.type) M
of GRing.Zmodule M := {
norm : M -> R;
ler_normD : forall x y, norm (x + y) <= norm x + norm y;
normr0_eq0 : forall x, norm x = 0 -> x = 0;
normrMn : forall x n, norm (x *+ n) = norm x *+ n;
normrN : forall x, norm (- x) = norm x;
}.
HB.builders Context (R : POrderedZmodule.type) M of Zmodule_isNormed R M.
HB.instance Definition _ :=
Zmodule_isSemiNormed.Build R M ler_normD normrMn normrN.
HB.instance Definition _ :=
SemiNormedZmodule_isPositiveDefinite.Build R M normr0_eq0.
HB.end.
|
HB.mixin
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
Record
| |
RecordisNumRing R of GRing.NzRing R & POrderedZmodule R
& NormedZmodule (POrderedZmodule.clone R _) R := {
addr_gt0 : forall x y : R, 0 < x -> 0 < y -> 0 < (x + y);
ger_leVge : forall x y : R, 0 <= x -> 0 <= y -> (x <= y) || (y <= x);
normrM : {morph (norm : R -> R) : x y / x * y};
ler_def : forall x y : R, (x <= y) = (norm (y - x) == (y - x));
}.
#[short(type="numDomainType")]
HB.structure Definition NumDomain := { R of
GRing.IntegralDomain R &
POrderedZmodule R &
NormedZmodule (POrderedZmodule.clone R _) R &
isNumRing R
}.
Arguments addr_gt0 {_} [x y] : rename.
Arguments ger_leVge {_} [x y] : rename.
|
HB.mixin
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
Record
| |
normr:= norm.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
normr
| |
sgr(x : R) : R := if x == 0 then 0 else if x < 0 then -1 else 1.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
sgr
| |
sg:= sgr.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
sg
| |
real_axiom: Prop := forall x : R, x \is real.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_axiom
| |
archimedean_axiom: Prop := forall x : R, exists ub, `|x| < ub%:R.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
archimedean_axiom
| |
real_closed_axiom: Prop :=
forall (p : {poly R}) (a b : R),
a <= b -> p.[a] <= 0 <= p.[b] -> exists2 x, a <= x <= b & root p x.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
real_closed_axiom
| |
DefinitionRealDomain :=
{ R of Order.Total ring_display R & NumDomain R }.
|
HB.structure
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
Definition
| |
ger0_defx : (0 <= x) = (`|x| == x).
Proof. by rewrite ler_def subr0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
ger0_def
| |
subr_ge0x y : (0 <= x - y) = (y <= x).
Proof. by rewrite ger0_def -ler_def. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import ssrAC div fintype path bigop order finset fingroup",
"From mathcomp Require Import ssralg poly orderedzmod"
] |
algebra/num_theory/numdomain.v
|
subr_ge0
|
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