statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
cprodE G H : H \subset 'C(G) -> G \* H = G * H. | Proof. by move=> cGH; rewrite /cprod cGH pprodE ?cents_norm. Qed. | Lemma | cprodE | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cents_norm",
"cprod",
"pprodE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodEY G H : H \subset 'C(G) -> G \* H = G <*> H. | Proof. by move=> cGH; rewrite cprodE ?cent_joinEr. Qed. | Lemma | cprodEY | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cent_joinEr",
"cprodE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodWpp A B G : A \* B = G -> pprod A B = G. | Proof. by case/cprodP=> [[K H -> ->] <- /cents_norm/pprodE]. Qed. | Lemma | cprodWpp | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cents_norm",
"cprodP",
"pprod",
"pprodE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodW A B G : A \* B = G -> A * B = G. | Proof. by move/cprodWpp/pprodW. Qed. | Lemma | cprodW | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprodWpp",
"pprodW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodWC A B G : A \* B = G -> B * A = G. | Proof. by move/cprodWpp/pprodWC. Qed. | Lemma | cprodWC | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprodWpp",
"pprodWC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodWY A B G : A \* B = G -> A <*> B = G. | Proof. by move/cprodWpp/pprodWY. Qed. | Lemma | cprodWY | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprodWpp",
"pprodWY"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodJ A B x : (A \* B) :^ x = A :^ x \* B :^ x. | Proof.
by rewrite /cprod centJ conjSg -pprodJ; case: ifP => _ //; apply: imset0.
Qed. | Lemma | cprodJ | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"centJ",
"conjSg",
"cprod",
"imset0",
"pprodJ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprod_normal2 A B G : A \* B = G -> A <| G /\ B <| G. | Proof.
case/cprodP=> [[K H -> ->] <- cKH]; rewrite -cent_joinEr //.
by rewrite normalYl normalYr !cents_norm // centsC.
Qed. | Lemma | cprod_normal2 | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cKH",
"cent_joinEr",
"centsC",
"cents_norm",
"cprodP",
"normalYl",
"normalYr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcprodW I (r : seq I) P F G :
\big[cprod/1]_(i <- r | P i) F i = G -> \prod_(i <- r | P i) F i = G. | Proof.
elim/big_rec2: _ G => // i A B _ IH G /cprodP[[_ H _ defB] <- _].
by rewrite (IH H) defB.
Qed. | Lemma | bigcprodW | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"big_rec2",
"cprod",
"cprodP",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcprodWY I (r : seq I) P F G :
\big[cprod/1]_(i <- r | P i) F i = G -> << \bigcup_(i <- r | P i) F i >> = G. | Proof.
elim/big_rec2: _ G => [|i A B _ IH G]; first by rewrite gen0.
case/cprodP => [[K H -> defB] <- cKH].
by rewrite -[<<_>>]joing_idr (IH H) ?cent_joinEr -?defB.
Qed. | Lemma | bigcprodWY | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"big_rec2",
"cKH",
"cent_joinEr",
"cprod",
"cprodP",
"gen0",
"joing_idr",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
triv_cprod A B : (A \* B == 1) = (A == 1) && (B == 1). | Proof.
case A1: (A == 1); first by rewrite (eqP A1) cprod1g.
apply/eqP=> /cprodP[[G H defA ->]] /eqP.
by rewrite defA trivMg -defA A1.
Qed. | Lemma | triv_cprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"cprod1g",
"cprodP",
"trivMg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprod_ntriv A B : A != 1 -> B != 1 ->
A \* B =
if [&& group_set A, group_set B & B \subset 'C(A)] then A * B else set0. | Proof.
move=> A1 B1; rewrite /cprod; case: ifP => cAB; rewrite ?cAB ?andbF //=.
by rewrite /pprod -if_neg A1 -if_neg B1 cents_norm.
Qed. | Lemma | cprod_ntriv | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cents_norm",
"cprod",
"group_set",
"pprod",
"set0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trivg0 : (@set0 gT == 1) = false. | Proof. by rewrite eqEcard cards0 cards1 andbF. Qed. | Lemma | trivg0 | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cards0",
"cards1",
"eqEcard",
"gT",
"set0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group0 : group_set (@set0 gT) = false. | Proof. by rewrite /group_set inE. Qed. | Lemma | group0 | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"gT",
"group_set",
"inE",
"set0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprod0g A : set0 \* A = set0. | Proof. by rewrite /cprod centsC sub0set /pprod group0 trivg0 !if_same. Qed. | Lemma | cprod0g | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"centsC",
"cprod",
"group0",
"pprod",
"set0",
"sub0set",
"trivg0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodC : commutative cprod. | Proof.
rewrite /cprod => A B; case: ifP => cAB; rewrite centsC cAB // /pprod.
by rewrite andbCA normC !cents_norm // 1?centsC //; do 2!case: eqP => // ->.
Qed. | Lemma | cprodC | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"centsC",
"cents_norm",
"cprod",
"normC",
"pprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodA : associative cprod. | Proof.
move=> A B C; case A1: (A == 1); first by rewrite (eqP A1) !cprod1g.
case B1: (B == 1); first by rewrite (eqP B1) cprod1g cprodg1.
case C1: (C == 1); first by rewrite (eqP C1) !cprodg1.
rewrite !(triv_cprod, cprod_ntriv) ?{}A1 ?{}B1 ?{}C1 //.
case: isgroupP => [[G ->{A}] | _]; last by rewrite group0.
case: (isgr... | Lemma | cprodA | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"centM",
"cent_joinEr",
"cprod",
"cprod1g",
"cprod_ntriv",
"cprodg1",
"group0",
"groupP",
"isgroupP",
"last",
"mulG_subG",
"mulgA",
"subsetI",
"triv_cprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprod_modl A B G H :
A \* B = G -> A \subset H -> A \* (B :&: H) = G :&: H. | Proof.
case/cprodP=> [[U V -> -> {A B}]] defG cUV sUH.
by rewrite cprodE; [rewrite subIset ?cUV | rewrite group_modl ?defG].
Qed. | Lemma | cprod_modl | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprodE",
"cprodP",
"defG",
"group_modl",
"subIset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprod_modr A B G H :
A \* B = G -> B \subset H -> (H :&: A) \* B = H :&: G. | Proof. by rewrite -!(cprodC B) !(setIC H); apply: cprod_modl. Qed. | Lemma | cprod_modr | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"cprodC",
"cprod_modl",
"setIC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcprodYP (I : finType) (P : pred I) (H : I -> {group gT}) :
reflect (forall i j, P i -> P j -> i != j -> H i \subset 'C(H j))
(\big[cprod/1]_(i | P i) H i == (\prod_(i | P i) H i)%G). | Proof.
apply: (iffP eqP) => [defG i j Pi Pj neq_ij | cHH].
rewrite (bigD1 j) // (bigD1 i) /= ?cprodA in defG; first exact/andP.
by case/cprodP: defG => [[K _ /cprodP[//]]].
set Q := P; have sQP: subpred Q P by []; have [n leQn] := ubnP #|Q|.
elim: n => // n IHn in (Q) leQn sQP *.
have [i Qi | Q0] := pickP Q; last b... | Lemma | bigcprodYP | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"add1n",
"apply",
"bigD1",
"big_pred0",
"bigcupsP",
"bigprodGE",
"cardD1x",
"cprod",
"cprodA",
"cprodEY",
"cprodP",
"defG",
"gT",
"gen_subG",
"group",
"last",
"ltnS",
"pickP",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcprodEY I r (P : pred I) (H : I -> {group gT}) G :
abelian G -> (forall i, P i -> H i \subset G) ->
\big[cprod/1]_(i <- r | P i) H i = (\prod_(i <- r | P i) H i)%G. | Proof.
move=> cGG sHG; apply/eqP; rewrite !(big_tnth _ _ r).
by apply/bigcprodYP=> i j Pi Pj _; rewrite (sub_abelian_cent2 cGG) ?sHG.
Qed. | Lemma | bigcprodEY | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"abelian",
"apply",
"big_tnth",
"bigcprodYP",
"cGG",
"cprod",
"gT",
"group",
"sHG",
"sub_abelian_cent2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
perm_bigcprod (I : eqType) r1 r2 (A : I -> {set gT}) G x :
\big[cprod/1]_(i <- r1) A i = G -> {in r1, forall i, x i \in A i} ->
perm_eq r1 r2 ->
\prod_(i <- r1) x i = \prod_(i <- r2) x i. | Proof.
elim: r1 r2 G => [|i r1 IHr] r2 G defG Ax eq_r12.
by rewrite perm_sym in eq_r12; rewrite (perm_small_eq _ eq_r12) ?big_nil.
have /rot_to[n r3 Dr2]: i \in r2 by rewrite -(perm_mem eq_r12) mem_head.
transitivity (\prod_(j <- rot n r2) x j).
rewrite Dr2 !big_cons in defG Ax *; have [[_ G1 _ defG1] _ _] := cprod... | Lemma | perm_bigcprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"G1",
"allP",
"apply",
"big_cat",
"big_cons",
"big_mkord",
"big_nil",
"big_nth",
"bigcprodW",
"cat_take_drop",
"centsP",
"cprod",
"cprodP",
"defG",
"gT",
"mem_cat",
"mem_head",
"mem_nth",
"mem_prodg",
"perm_big",
"perm_cons",
"perm_eq",
"perm_mem",
"perm_rot",
"perm_s... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
reindex_bigcprod (I J : finType) (h : J -> I) P (A : I -> {set gT}) G x :
{on SimplPred P, bijective h} -> \big[cprod/1]_(i | P i) A i = G ->
{in SimplPred P, forall i, x i \in A i} ->
\prod_(i | P i) x i = \prod_(j | P (h j)) x (h j). | Proof.
case=> h1 hK h1K defG Ax; have [e big_e [Ue mem_e] _] := big_enumP P.
rewrite -!big_e in defG *; rewrite -(big_map h P x) -[RHS]big_filter filter_map.
apply: perm_bigcprod defG _ _ => [i|]; first by rewrite mem_e => /Ax.
have [r _ [Ur /= mem_r] _] := big_enumP; apply: uniq_perm Ue _ _ => [|i].
by rewrite map_i... | Lemma | reindex_bigcprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"big_enumP",
"big_filter",
"big_map",
"cprod",
"defG",
"filter_map",
"gT",
"last",
"mapP",
"map_inj_in_uniq",
"on",
"perm_bigcprod",
"uniq_perm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod1g : left_id 1 dprod. | Proof. by move=> A; rewrite /dprod subsetIl cprod1g. Qed. | Lemma | dprod1g | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprod1g",
"dprod",
"subsetIl"
] | Direct product | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
dprodg1 : right_id 1 dprod. | Proof. by move=> A; rewrite /dprod subsetIr cprodg1. Qed. | Lemma | dprodg1 | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprodg1",
"dprod",
"subsetIr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodP A B G :
A \x B = G -> [/\ are_groups A B, A * B = G, B \subset 'C(A) & A :&: B = 1]. | Proof.
rewrite /dprod; case: ifP => trAB; last by case/group_not0.
by case/cprodP=> gAB; split=> //; case: gAB trAB => ? ? -> -> /trivgP.
Qed. | Lemma | dprodP | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"are_groups",
"cprodP",
"dprod",
"group_not0",
"last",
"split",
"trivgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodE G H : H \subset 'C(G) -> G :&: H = 1 -> G \x H = G * H. | Proof. by move=> cGH trGH; rewrite /dprod trGH sub1G cprodE. Qed. | Lemma | dprodE | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprodE",
"dprod",
"sub1G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodEY G H : H \subset 'C(G) -> G :&: H = 1 -> G \x H = G <*> H. | Proof. by move=> cGH trGH; rewrite /dprod trGH subxx cprodEY. Qed. | Lemma | dprodEY | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprodEY",
"dprod",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodEcp A B : A :&: B = 1 -> A \x B = A \* B. | Proof. by move=> trAB; rewrite /dprod trAB subxx. Qed. | Lemma | dprodEcp | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"dprod",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodEsd A B : B \subset 'C(A) -> A \x B = A ><| B. | Proof. by rewrite /dprod /cprod => ->. Qed. | Lemma | dprodEsd | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprod",
"dprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodWcp A B G : A \x B = G -> A \* B = G. | Proof. by move=> defG; have [_ _ _ /dprodEcp <-] := dprodP defG. Qed. | Lemma | dprodWcp | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"defG",
"dprodEcp",
"dprodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodWsd A B G : A \x B = G -> A ><| B = G. | Proof. by move=> defG; have [_ _ /dprodEsd <-] := dprodP defG. Qed. | Lemma | dprodWsd | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"defG",
"dprodEsd",
"dprodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodW A B G : A \x B = G -> A * B = G. | Proof. by move/dprodWsd/sdprodW. Qed. | Lemma | dprodW | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"dprodWsd",
"sdprodW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodWC A B G : A \x B = G -> B * A = G. | Proof. by move/dprodWsd/sdprodWC. Qed. | Lemma | dprodWC | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"dprodWsd",
"sdprodWC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodWY A B G : A \x B = G -> A <*> B = G. | Proof. by move/dprodWsd/sdprodWY. Qed. | Lemma | dprodWY | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"dprodWsd",
"sdprodWY"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprod_card_dprod G A B :
A \* B = G -> #|A| * #|B| <= #|G| -> A \x B = G. | Proof. by case/cprodP=> [[K H -> ->] <- cKH] /cardMg_TI; apply: dprodE. Qed. | Lemma | cprod_card_dprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"cKH",
"cardMg_TI",
"cprodP",
"dprodE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodJ A B x : (A \x B) :^ x = A :^ x \x B :^ x. | Proof.
rewrite /dprod -conjIg sub_conjg conjs1g -cprodJ.
by case: ifP => _ //; apply: imset0.
Qed. | Lemma | dprodJ | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"conjIg",
"conjs1g",
"cprodJ",
"dprod",
"imset0",
"sub_conjg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_normal2 A B G : A \x B = G -> A <| G /\ B <| G. | Proof. by move/dprodWcp/cprod_normal2. Qed. | Lemma | dprod_normal2 | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprod_normal2",
"dprodWcp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodYP K H : reflect (K \x H = K <*> H) (H \subset 'C(K) :\: K^#). | Proof.
rewrite subsetD -setI_eq0 setIDA setD_eq0 setIC subG1 /=.
by apply: (iffP andP) => [[cKH /eqP/dprodEY->] | /dprodP[_ _ -> ->]].
Qed. | Lemma | dprodYP | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"cKH",
"dprodEY",
"dprodP",
"setD_eq0",
"setIC",
"setIDA",
"setI_eq0",
"subG1",
"subsetD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodC : commutative dprod. | Proof. by move=> A B; rewrite /dprod setIC cprodC. Qed. | Lemma | dprodC | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprodC",
"dprod",
"setIC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodWsdC A B G : A \x B = G -> B ><| A = G. | Proof. by rewrite dprodC => /dprodWsd. Qed. | Lemma | dprodWsdC | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"dprodC",
"dprodWsd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodA : associative dprod. | Proof.
move=> A B C; case A1: (A == 1); first by rewrite (eqP A1) !dprod1g.
case B1: (B == 1); first by rewrite (eqP B1) dprod1g dprodg1.
case C1: (C == 1); first by rewrite (eqP C1) !dprodg1.
rewrite /dprod (fun_if (cprod A)) (fun_if (cprod^~ C)) -cprodA.
rewrite -(cprodC set0) !cprod0g cprod_ntriv ?B1 ?{}C1 //.
case:... | Lemma | dprodA | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"cent_joinEr",
"cprod",
"cprod0g",
"cprodA",
"cprodC",
"cprodEY",
"cprod_ntriv",
"dprod",
"dprod1g",
"dprodg1",
"group_modl",
"group_modr",
"isgroupP",
"joing_subl",
"joing_subr",
"last",
"mul1g",
"mulGS",
"mulG_subG",
"mulSG",
"mulSg",
"mulg1",
"mulgS",
"nor... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigdprodWcp I (r : seq I) P F G :
\big[dprod/1]_(i <- r | P i) F i = G -> \big[cprod/1]_(i <- r | P i) F i = G. | Proof.
elim/big_rec2: _ G => // i A B _ IH G /dprodP[[K H -> defB] <- cKH _].
by rewrite (IH H) // cprodE -defB.
Qed. | Lemma | bigdprodWcp | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"big_rec2",
"cKH",
"cprod",
"cprodE",
"dprod",
"dprodP",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigdprodW I (r : seq I) P F G :
\big[dprod/1]_(i <- r | P i) F i = G -> \prod_(i <- r | P i) F i = G. | Proof. by move/bigdprodWcp; apply: bigcprodW. Qed. | Lemma | bigdprodW | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"bigcprodW",
"bigdprodWcp",
"dprod",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigdprodWY I (r : seq I) P F G :
\big[dprod/1]_(i <- r | P i) F i = G -> << \bigcup_(i <- r | P i) F i >> = G. | Proof. by move/bigdprodWcp; apply: bigcprodWY. Qed. | Lemma | bigdprodWY | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"bigcprodWY",
"bigdprodWcp",
"dprod",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigdprodYP (I : finType) (P : pred I) (F : I -> {group gT}) :
reflect (forall i, P i ->
(\prod_(j | P j && (j != i)) F j)%G \subset 'C(F i) :\: (F i)^#)
(\big[dprod/1]_(i | P i) F i == (\prod_(i | P i) F i)%G). | Proof.
apply: (iffP eqP) => [defG i Pi | dxG].
rewrite !(bigD1 i Pi) /= in defG; have [[_ G' _ defG'] _ _ _] := dprodP defG.
by apply/dprodYP; rewrite -defG defG' bigprodGE (bigdprodWY defG').
set Q := P; have sQP: subpred Q P by []; have [n leQn] := ubnP #|Q|.
elim: n => // n IHn in (Q) leQn sQP *.
have [i Qi | Q0... | Lemma | bigdprodYP | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"G'",
"add1n",
"apply",
"bigD1",
"big_pred0",
"bigcup_max",
"bigcupsP",
"bigdprodWY",
"bigprodGE",
"cardD1x",
"defG",
"dprod",
"dprodP",
"dprodYP",
"gT",
"genS",
"group",
"last",
"ltnS",
"pickP",
"subset_trans",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_modl A B G H :
A \x B = G -> A \subset H -> A \x (B :&: H) = G :&: H. | Proof.
case/dprodP=> [[U V -> -> {A B}]] defG cUV trUV sUH.
rewrite dprodEcp; last by apply: cprod_modl; rewrite ?cprodE.
by rewrite setIA trUV (setIidPl _) ?sub1G.
Qed. | Lemma | dprod_modl | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"cprodE",
"cprod_modl",
"defG",
"dprodEcp",
"dprodP",
"last",
"setIA",
"setIidPl",
"sub1G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_modr A B G H :
A \x B = G -> B \subset H -> (H :&: A) \x B = H :&: G. | Proof. by rewrite -!(dprodC B) !(setIC H); apply: dprod_modl. Qed. | Lemma | dprod_modr | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"dprodC",
"dprod_modl",
"setIC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subcent_dprod B C G A :
B \x C = G -> A \subset 'N(B) :&: 'N(C) -> 'C_B(A) \x 'C_C(A) = 'C_G(A). | Proof.
move=> defG; have [_ _ cBC _] := dprodP defG; move: defG.
by rewrite !dprodEsd 1?(centSS _ _ cBC) ?subsetIl //; apply: subcent_sdprod.
Qed. | Lemma | subcent_dprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"centSS",
"defG",
"dprodEsd",
"dprodP",
"subcent_sdprod",
"subsetIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprod_card A B G : A \x B = G -> (#|A| * #|B|)%N = #|G|. | Proof. by case/dprodP=> [[H K -> ->] <- _]; move/TI_cardMg. Qed. | Lemma | dprod_card | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"TI_cardMg",
"dprodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigdprod_card I r (P : pred I) E G :
\big[dprod/1]_(i <- r | P i) E i = G ->
(\prod_(i <- r | P i) #|E i|)%N = #|G|. | Proof.
elim/big_rec2: _ G => [G <- | i A B _ IH G defG]; first by rewrite cards1.
have [[_ H _ defH] _ _ _] := dprodP defG.
by rewrite -(dprod_card defG) (IH H) defH.
Qed. | Lemma | bigdprod_card | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"big_rec2",
"cards1",
"defG",
"dprod",
"dprodP",
"dprod_card"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcprod_card_dprod I r (P : pred I) (A : I -> {set gT}) G :
\big[cprod/1]_(i <- r | P i) A i = G ->
\prod_(i <- r | P i) #|A i| <= #|G| ->
\big[dprod/1]_(i <- r | P i) A i = G. | Proof.
elim: r G => [|i r IHr]; rewrite !(big_nil, big_cons) //; case: ifP => _ // G.
case/cprodP=> [[K H -> defH]]; rewrite defH => <- cKH leKH_G.
have /implyP := leq_trans leKH_G (dvdn_leq _ (dvdn_cardMg K H)).
rewrite muln_gt0 leq_pmul2l !cardG_gt0 //= => /(IHr H defH){}defH.
by rewrite defH dprodE // cardMg_TI // -... | Lemma | bigcprod_card_dprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"big_cons",
"big_nil",
"bigdprod_card",
"cKH",
"cardG_gt0",
"cardMg_TI",
"cprod",
"cprodP",
"dprod",
"dprodE",
"dvdn_cardMg",
"dvdn_leq",
"gT",
"leq_pmul2l",
"leq_trans",
"muln_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bigcprod_coprime_dprod (I : finType) (P : pred I) (A : I -> {set gT}) G :
\big[cprod/1]_(i | P i) A i = G ->
(forall i j, P i -> P j -> i != j -> coprime #|A i| #|A j|) ->
\big[dprod/1]_(i | P i) A i = G. | Proof.
move=> defG coA; set Q := P in defG *; have sQP: subpred Q P by [].
have [m leQm] := ubnP #|Q|; elim: m => // m IHm in (Q) leQm G defG sQP *.
have [i Qi | Q0] := pickP Q; last by rewrite !big_pred0 in defG *.
move: defG; rewrite !(bigD1 i Qi) /= => /cprodP[[Hi Gi defAi defGi] <-].
rewrite defAi defGi => cHGi.
ha... | Lemma | bigcprod_coprime_dprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"bigD1",
"big_pred0",
"big_rec",
"bigdprod_card",
"cardD1x",
"coprime",
"coprimeMr",
"coprime_TIg",
"coprime_sym",
"coprimen1",
"cprod",
"cprodP",
"defG",
"dprod",
"dprodE",
"gT",
"last",
"pickP",
"ubnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_dprod G A B x : A \x B = G -> x \in G ->
exists y, exists z,
[/\ y \in A, z \in B, x = y * z &
{in A & B, forall u t, x = u * t -> u = y /\ t = z}]. | Proof.
move=> defG; have [_ _ cBA _] := dprodP defG.
by apply: mem_sdprod; rewrite -dprodEsd.
Qed. | Lemma | mem_dprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"defG",
"dprodEsd",
"dprodP",
"mem_sdprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_bigdprod (I : finType) (P : pred I) F G x :
\big[dprod/1]_(i | P i) F i = G -> x \in G ->
exists c, [/\ forall i, P i -> c i \in F i, x = \prod_(i | P i) c i
& forall e, (forall i, P i -> e i \in F i) ->
x = \prod_(i | P i) e i ->
forall i, P i -> e i = ... | Proof.
move=> defG; rewrite -(bigdprodW defG) => /prodsgP[c Fc ->].
have [r big_r [_ mem_r] _] := big_enumP P.
exists c; split=> // e Fe eq_ce i Pi; rewrite -!{}big_r in defG eq_ce.
have{Pi}: i \in r by rewrite mem_r.
have{mem_r}: all P r by apply/allP=> j; rewrite mem_r.
elim: r G defG eq_ce => // j r IHr G.
rewrite !... | Lemma | mem_bigdprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"all",
"allP",
"all_nthP",
"apply",
"big_cons",
"big_enumP",
"big_mkord",
"big_nth",
"bigdprodW",
"defG",
"divgr",
"divgrMid",
"dprod",
"dprodP",
"inE",
"mem_prodg",
"mulgI",
"predU1P",
"prodsgP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comm_prodG I r (G : I -> {group gT}) (P : {pred I}) :
{in P &, forall i j, commute (G i) (G j)} ->
(\prod_(i <- r | P i) G i)%G = \prod_(i <- r | P i) G i :> {set gT}. | Proof.
elim: r => /= [|i {}r IHr]; rewrite !(big_nil, big_cons)//=.
case: ifP => //= Pi Gcomm; rewrite comm_joingE {}IHr// /commute.
elim: r => [|j r IHr]; first by rewrite big_nil mulg1 mul1g.
by rewrite big_cons; case: ifP => //= Pj; rewrite mulgA Gcomm// -!mulgA IHr.
Qed. | Lemma | comm_prodG | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"big_cons",
"big_nil",
"comm_joingE",
"commute",
"gT",
"group",
"mul1g",
"mulg1",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_pprod : pprod K H = G -> pprod (f @* K) (f @* H) = f @* G. | Proof.
case/pprodP=> _ defG mKH; rewrite pprodE ?morphim_norms //.
by rewrite -morphimMl ?(subset_trans _ sGD) -?defG // mulG_subl.
Qed. | Lemma | morphim_pprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"defG",
"morphimMl",
"morphim_norms",
"mulG_subl",
"pprod",
"pprodE",
"pprodP",
"sGD",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_coprime_sdprod :
K ><| H = G -> coprime #|K| #|H| -> f @* K ><| f @* H = f @* G. | Proof.
rewrite /sdprod => defG coHK; move: defG.
by rewrite !coprime_TIg ?coprime_morph // !subxx; apply: morphim_pprod.
Qed. | Lemma | morphim_coprime_sdprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"coprime",
"coprime_TIg",
"coprime_morph",
"defG",
"morphim_pprod",
"sdprod",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_sdprod : 'injm f -> K ><| H = G -> f @* K ><| f @* H = f @* G. | Proof.
move=> inj_f; case/sdprodP=> _ defG nKH tiKH.
by rewrite /sdprod -injmI // tiKH morphim1 subxx morphim_pprod // pprodE.
Qed. | Lemma | injm_sdprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"defG",
"inj_f",
"injmI",
"morphim1",
"morphim_pprod",
"nKH",
"pprodE",
"sdprod",
"sdprodP",
"subxx",
"tiKH"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_cprod : K \* H = G -> f @* K \* f @* H = f @* G. | Proof.
case/cprodP=> _ defG cKH; rewrite /cprod morphim_cents // morphim_pprod //.
by rewrite pprodE // cents_norm // centsC.
Qed. | Lemma | morphim_cprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cKH",
"centsC",
"cents_norm",
"cprod",
"cprodP",
"defG",
"morphim_cents",
"morphim_pprod",
"pprodE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_dprod : 'injm f -> K \x H = G -> f @* K \x f @* H = f @* G. | Proof.
move=> inj_f; case/dprodP=> _ defG cHK tiKH.
by rewrite /dprod -injmI // tiKH morphim1 subxx morphim_cprod // cprodE.
Qed. | Lemma | injm_dprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"cprodE",
"defG",
"dprod",
"dprodP",
"inj_f",
"injmI",
"morphim1",
"morphim_cprod",
"subxx",
"tiKH"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_coprime_dprod :
K \x H = G -> coprime #|K| #|H| -> f @* K \x f @* H = f @* G. | Proof.
rewrite /dprod => defG coHK; move: defG.
by rewrite !coprime_TIg ?coprime_morph // !subxx; apply: morphim_cprod.
Qed. | Lemma | morphim_coprime_dprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"coprime",
"coprime_TIg",
"coprime_morph",
"defG",
"dprod",
"morphim_cprod",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_bigcprod I r (P : pred I) (H : I -> {group gT}) G :
G \subset D -> \big[cprod/1]_(i <- r | P i) H i = G ->
\big[cprod/1]_(i <- r | P i) f @* H i = f @* G. | Proof.
elim/big_rec2: _ G => [|i fB B Pi def_fB] G sGD defG.
by rewrite -defG morphim1.
case/cprodP: defG (defG) => [[Hi Gi -> defB] _ _]; rewrite defB => defG.
rewrite (def_fB Gi) //; last exact: morphim_cprod.
by apply: subset_trans sGD; case/cprod_normal2: defG => _ /andP[].
Qed. | Lemma | morphim_bigcprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"big_rec2",
"cprod",
"cprodP",
"cprod_normal2",
"defG",
"gT",
"group",
"last",
"morphim1",
"morphim_cprod",
"sGD",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_bigdprod I r (P : pred I) (H : I -> {group gT}) G :
G \subset D -> 'injm f -> \big[dprod/1]_(i <- r | P i) H i = G ->
\big[dprod/1]_(i <- r | P i) f @* H i = f @* G. | Proof.
move=> sGD injf; elim/big_rec2: _ G sGD => [|i fB B Pi def_fB] G sGD defG.
by rewrite -defG morphim1.
case/dprodP: defG (defG) => [[Hi Gi -> defB] _ _ _]; rewrite defB => defG.
rewrite (def_fB Gi) //; last exact: injm_dprod.
by apply: subset_trans sGD; case/dprod_normal2: defG => _ /andP[].
Qed. | Lemma | injm_bigdprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"big_rec2",
"defG",
"dprod",
"dprodP",
"dprod_normal2",
"gT",
"group",
"injf",
"injm_dprod",
"last",
"morphim1",
"sGD",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_coprime_bigdprod (I : finType) P (H : I -> {group gT}) G :
G \subset D -> \big[dprod/1]_(i | P i) H i = G ->
(forall i j, P i -> P j -> i != j -> coprime #|H i| #|H j|) ->
\big[dprod/1]_(i | P i) f @* H i = f @* G. | Proof.
move=> sGD /bigdprodWcp defG coH; have def_fG := morphim_bigcprod sGD defG.
by apply: bigcprod_coprime_dprod => // i j *; rewrite coprime_morph ?coH.
Qed. | Lemma | morphim_coprime_bigdprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"bigcprod_coprime_dprod",
"bigdprodWcp",
"coprime",
"coprime_morph",
"defG",
"dprod",
"gT",
"group",
"morphim_bigcprod",
"sGD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nMG: G \subset 'N(M). | Hypothesis | nMG | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
quotient_pprod : pprod K H = G -> pprod (K / M) (H / M) = G / M. | Proof. exact: morphim_pprod. Qed. | Lemma | quotient_pprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"morphim_pprod",
"pprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_coprime_sdprod :
K ><| H = G -> coprime #|K| #|H| -> (K / M) ><| (H / M) = G / M. | Proof. exact: morphim_coprime_sdprod. Qed. | Lemma | quotient_coprime_sdprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"coprime",
"morphim_coprime_sdprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_cprod : K \* H = G -> (K / M) \* (H / M) = G / M. | Proof. exact: morphim_cprod. Qed. | Lemma | quotient_cprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"morphim_cprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quotient_coprime_dprod :
K \x H = G -> coprime #|K| #|H| -> (K / M) \x (H / M) = G / M. | Proof. exact: morphim_coprime_dprod. Qed. | Lemma | quotient_coprime_dprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"coprime",
"morphim_coprime_dprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extprod_mulg (x y : gT1 * gT2) | := (x.1 * y.1, x.2 * y.2). | Definition | extprod_mulg | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extprod_invg (x : gT1 * gT2) | := (x.1^-1, x.2^-1). | Definition | extprod_invg | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extprod_mul1g : left_id (1, 1) extprod_mulg. | Proof. by case=> x1 x2; congr (_, _); apply: mul1g. Qed. | Lemma | extprod_mul1g | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"extprod_mulg",
"mul1g"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extprod_mulVg : left_inverse (1, 1) extprod_invg extprod_mulg. | Proof. by move=> x; congr (_, _); apply: mulVg. Qed. | Lemma | extprod_mulVg | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"extprod_invg",
"extprod_mulg",
"mulVg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extprod_mulgA : associative extprod_mulg. | Proof. by move=> x y z; congr (_, _); apply: mulgA. Qed. | Lemma | extprod_mulgA | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"extprod_mulg",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_setX (H1 : {group gT1}) (H2 : {group gT2}) : group_set (setX H1 H2). | Proof.
apply/group_setP; split; first by rewrite !inE !group1.
by case=> [x1 x2] [y1 y2] /[!inE] /andP[Hx1 Hx2] /andP[Hy1 Hy2] /[!groupM].
Qed. | Lemma | group_setX | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"group",
"group1",
"groupM",
"group_set",
"group_setP",
"inE",
"setX",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setX_group H1 H2 | := Group (group_setX H1 H2). | Canonical | setX_group | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"group_setX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pairg1 x : gT1 * gT2 | := (x, 1). | Definition | pairg1 | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair1g x : gT1 * gT2 | := (1, x). | Definition | pair1g | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pairg1_morphM : {morph pairg1 : x y / x * y}. | Proof. by move=> x y /=; rewrite {2}/mul /= /mul_pair/= mul1g. Qed. | Lemma | pairg1_morphM | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"mul",
"mul1g",
"mul_pair",
"pairg1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pairg1_morphism | := @Morphism _ _ setT _ (in2W pairg1_morphM). | Canonical | pairg1_morphism | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"pairg1_morphM",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair1g_morphM : {morph pair1g : x y / x * y}. | Proof. by move=> x y /=; rewrite {2}/mul /= /mul_pair/= mul1g. Qed. | Lemma | pair1g_morphM | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"mul",
"mul1g",
"mul_pair",
"pair1g"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair1g_morphism | := @Morphism _ _ setT _ (in2W pair1g_morphM). | Canonical | pair1g_morphism | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"pair1g_morphM",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fst_morphM : {morph (@fst gT1 gT2) : x y / x * y}. | Proof. by []. Qed. | Lemma | fst_morphM | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
snd_morphM : {morph (@snd gT1 gT2) : x y / x * y}. | Proof. by []. Qed. | Lemma | snd_morphM | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fst_morphism | := @Morphism _ _ setT _ (in2W fst_morphM). | Canonical | fst_morphism | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"fst_morphM",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
snd_morphism | := @Morphism _ _ setT _ (in2W snd_morphM). | Canonical | snd_morphism | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"setT",
"snd_morphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_pair1g : 'injm pair1g. | Proof. by apply/subsetP=> x /morphpreP[_ /set1P[->]]; apply: set11. Qed. | Lemma | injm_pair1g | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"morphpreP",
"pair1g",
"set11",
"set1P",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_pairg1 : 'injm pairg1. | Proof. by apply/subsetP=> x /morphpreP[_ /set1P[->]]; apply: set11. Qed. | Lemma | injm_pairg1 | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"morphpreP",
"pairg1",
"set11",
"set1P",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_pairg1 (H1 : {set gT1}) : pairg1 @* H1 = setX H1 1. | Proof. by rewrite -imset2_pair imset2_set1r morphimEsub ?subsetT. Qed. | Lemma | morphim_pairg1 | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"imset2_pair",
"imset2_set1r",
"morphimEsub",
"pairg1",
"setX",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_pair1g (H2 : {set gT2}) : pair1g @* H2 = setX 1 H2. | Proof. by rewrite -imset2_pair imset2_set1l morphimEsub ?subsetT. Qed. | Lemma | morphim_pair1g | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"imset2_pair",
"imset2_set1l",
"morphimEsub",
"pair1g",
"setX",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_fstX (H1: {set gT1}) (H2 : {group gT2}) :
[morphism of fun x => x.1] @* setX H1 H2 = H1. | Proof.
apply/eqP; rewrite eqEsubset morphimE setTI /=.
apply/andP; split; apply/subsetP=> x.
by case/imsetP=> x0 /[1!inE] /andP[Hx1 _] ->.
move=> Hx1; apply/imsetP; exists (x, 1); last by trivial.
by rewrite in_setX Hx1 /=.
Qed. | Lemma | morphim_fstX | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"eqEsubset",
"group",
"imsetP",
"inE",
"in_setX",
"last",
"morphimE",
"morphism",
"setTI",
"setX",
"split",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_sndX (H1: {group gT1}) (H2 : {set gT2}) :
[morphism of fun x => x.2] @* setX H1 H2 = H2. | Proof.
apply/eqP; rewrite eqEsubset morphimE setTI /=.
apply/andP; split; apply/subsetP=> x.
by case/imsetP=> x0 /[1!inE] /andP[_ Hx2] ->.
move=> Hx2; apply/imsetP; exists (1, x); last by [].
by rewrite in_setX Hx2 andbT.
Qed. | Lemma | morphim_sndX | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"eqEsubset",
"group",
"imsetP",
"inE",
"in_setX",
"last",
"morphimE",
"morphism",
"setTI",
"setX",
"split",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setX_prod (H1 : {set gT1}) (H2 : {set gT2}) :
setX H1 1 * setX 1 H2 = setX H1 H2. | Proof.
apply/setP=> [[x y]]; rewrite !inE /=.
apply/imset2P/andP=> [[[x1 u1] [v1 y1]] | [Hx Hy]].
rewrite !inE /= => /andP[Hx1 /eqP->] /andP[/eqP-> Hx] [-> ->].
by rewrite mulg1 mul1g.
exists (x, 1 : gT2) (1 : gT1, y); rewrite ?inE ?Hx ?eqxx //.
by rewrite /mul /= /mul_pair /= mulg1 mul1g.
Qed. | Lemma | setX_prod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"eqxx",
"imset2P",
"inE",
"mul",
"mul1g",
"mul_pair",
"mulg1",
"setP",
"setX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setX_dprod (H1 : {group gT1}) (H2 : {group gT2}) :
setX H1 1 \x setX 1 H2 = setX H1 H2. | Proof.
rewrite dprodE ?setX_prod //.
apply/centsP=> [[x u]] /[!inE]/= /andP[/eqP-> _] [v y].
by rewrite !inE /= => /andP[_ /eqP->]; congr (_, _); rewrite ?mul1g ?mulg1.
apply/trivgP; apply/subsetP=> [[x y]]; rewrite !inE /= -!andbA.
by case/and4P=> _ /eqP-> /eqP->; rewrite eqxx.
Qed. | Lemma | setX_dprod | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"centsP",
"dprodE",
"eqxx",
"group",
"inE",
"mul1g",
"mulg1",
"setX",
"setX_prod",
"subsetP",
"trivgP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isog_setX1 (H1 : {group gT1}) : isog H1 (setX H1 1). | Proof.
apply/isogP; exists [morphism of restrm (subsetT H1) pairg1].
by rewrite injm_restrm ?injm_pairg1.
by rewrite morphim_restrm morphim_pairg1 setIid.
Qed. | Lemma | isog_setX1 | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"group",
"injm_pairg1",
"injm_restrm",
"isog",
"isogP",
"morphim_pairg1",
"morphim_restrm",
"morphism",
"pairg1",
"restrm",
"setIid",
"setX",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isog_set1X (H2 : {group gT2}) : isog H2 (setX 1 H2). | Proof.
apply/isogP; exists [morphism of restrm (subsetT H2) pair1g].
by rewrite injm_restrm ?injm_pair1g.
by rewrite morphim_restrm morphim_pair1g setIid.
Qed. | Lemma | isog_set1X | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"group",
"injm_pair1g",
"injm_restrm",
"isog",
"isogP",
"morphim_pair1g",
"morphim_restrm",
"morphism",
"pair1g",
"restrm",
"setIid",
"setX",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setX_gen (H1 : {set gT1}) (H2 : {set gT2}) :
1 \in H1 -> 1 \in H2 -> <<setX H1 H2>> = setX <<H1>> <<H2>>. | Proof.
move=> H1_1 H2_1; apply/eqP.
rewrite eqEsubset gen_subG setXS ?subset_gen //.
(* TODO: investigate why the occurrence selection changed *)
rewrite -[in X in X \subset _]setX_prod.
rewrite -morphim_pair1g -morphim_pairg1 !morphim_gen ?subsetT //.
by rewrite morphim_pair1g morphim_pairg1 mul_subG // genS // setXS ... | Lemma | setX_gen | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"apply",
"eqEsubset",
"genS",
"gen_subG",
"morphim_gen",
"morphim_pair1g",
"morphim_pairg1",
"mul_subG",
"setX",
"setXS",
"setX_prod",
"sub1set",
"subsetT",
"subset_gen"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gTn | := {dffun forall i, gT i}. | Notation | gTn | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"gT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extnprod_mulg (x y : gTn) : gTn | := [ffun i => (x i * y i)%g]. | Definition | extnprod_mulg | finite_group | finite_group/gproduct.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"quotient",
"action",
"finfun"
] | [
"gTn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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