fact stringlengths 8 1.54k | type stringclasses 19
values | library stringclasses 8
values | imports listlengths 1 10 | filename stringclasses 98
values | symbolic_name stringlengths 1 42 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
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... | 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 =... | 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 ex... | 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 := ntrans... | 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 = ... | 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 mov... | 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 nM... | 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 sGp... | 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 := or... | 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 //... | 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_pg... | 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... | 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 [->|]... | 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 ... | 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_Sylo... | 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_tran... | 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... | 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) //=.
pos... | 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_quotie... | 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 // pi... | 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 (nilpoten... | 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';... | 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_... | 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 -(ni... | 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.
... | 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 ... | 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:= for... | 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_... | 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... | 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 * 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 Definiti... | 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 <=... | 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 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.