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 |
|---|---|---|---|---|---|---|---|---|---|---|
cancel_index_extremal_groups :
cancel index_extremal_group_type (nth NotExtremal enum_extremal_groups). | Proof. by case. Qed. | Lemma | cancel_index_extremal_groups | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"enum_extremal_groups",
"index_extremal_group_type",
"nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extgK | := cancel_index_extremal_groups. | Notation | extgK | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"cancel_index_extremal_groups"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_extremal_groups (c : extremal_group_type) : pickle c < 6. | Proof. by case: c. Qed. | Lemma | bound_extremal_groups | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"extremal_group_type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extremal_class (A : {set gT}) | :=
let m := #|A| in let p := pdiv m in let n := logn p m in
if (n > 1) && (A \isog 'D_(2 ^ n)) then Dihedral else
if (n > 2) && (A \isog 'Q_(2 ^ n)) then Quaternion else
if (n > 3) && (A \isog 'SD_(2 ^ n)) then SemiDihedral else
if (n > 2) && (A \isog 'Mod_(p ^ n)) then ModularGroup else
NotExtremal. | Definition | extremal_class | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"gT",
"isog",
"logn",
"pdiv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extremal2 A | := extremal_class A \in behead enum_extremal_groups. | Definition | extremal2 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"behead",
"enum_extremal_groups",
"extremal_class"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dihedral_classP :
extremal_class G = Dihedral <-> (exists2 n, n > 1 & G \isog 'D_(2 ^ n)). | Proof.
rewrite /extremal_class; split=> [ | [n n_gt1 isoG]].
by move: (logn _ _) => n; do 4?case: ifP => //; case/andP; exists n.
rewrite (card_isog isoG) card_2dihedral // -(ltn_predK n_gt1) pdiv_pfactor //.
by rewrite pfactorK // (ltn_predK n_gt1) n_gt1 isoG.
Qed. | Lemma | dihedral_classP | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"card_2dihedral",
"card_isog",
"extremal_class",
"isoG",
"isog",
"logn",
"ltn_predK",
"pdiv_pfactor",
"pfactorK",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
quaternion_classP :
extremal_class G = Quaternion <-> (exists2 n, n > 2 & G \isog 'Q_(2 ^ n)). | Proof.
rewrite /extremal_class; split=> [ | [n n_gt2 isoG]].
by move: (logn _ _) => n; do 4?case: ifP => //; case/andP; exists n.
rewrite (card_isog isoG) card_quaternion // -(ltn_predK n_gt2) pdiv_pfactor //.
rewrite pfactorK // (ltn_predK n_gt2) n_gt2 isoG.
case: andP => // [[n_gt1 isoGD]].
have [[x y] genG [oy _ _... | Lemma | quaternion_classP | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"card_isog",
"card_quaternion",
"dihedral2_structure",
"extremal_class",
"generators_quaternion",
"isoG",
"isog",
"logn",
"ltn_predK",
"n_gt2",
"pdiv_pfactor",
"pfactorK",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
semidihedral_classP :
extremal_class G = SemiDihedral <-> (exists2 n, n > 3 & G \isog 'SD_(2 ^ n)). | Proof.
rewrite /extremal_class; split=> [ | [n n_gt3 isoG]].
by move: (logn _ _) => n; do 4?case: ifP => //; case/andP; exists n.
rewrite (card_isog isoG) card_semidihedral //.
rewrite -(ltn_predK n_gt3) pdiv_pfactor // pfactorK // (ltn_predK n_gt3) n_gt3.
have [[x y] genG [oy _]]:= generators_semidihedral n_gt3 isoG... | Lemma | semidihedral_classP | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"card_isog",
"card_semidihedral",
"cycle_id",
"dihedral2_structure",
"extremal_class",
"generators_semidihedral",
"groupMl",
"inE",
"isoG",
"isog",
"last",
"logn",
"ltn_predK",
"n_gt2",
"pdiv_pfactor",
"pfactorK",
"quaternion_structure",
"semidihedral_structure",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
odd_not_extremal2 : odd #|G| -> ~~ extremal2 G. | Proof.
rewrite /extremal2 /extremal_class; case: logn => // n'.
case: andP => [[n_gt1 isoG] | _].
by rewrite (card_isog isoG) card_2dihedral ?oddX.
case: andP => [[n_gt2 isoG] | _].
by rewrite (card_isog isoG) card_quaternion ?oddX.
case: andP => [[n_gt3 isoG] | _].
by rewrite (card_isog isoG) card_semidihedral ?... | Lemma | odd_not_extremal2 | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"card_2dihedral",
"card_isog",
"card_quaternion",
"card_semidihedral",
"extremal2",
"extremal_class",
"isoG",
"logn",
"n'",
"n_gt2",
"odd",
"oddX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
modular_group_classP :
extremal_class G = ModularGroup
<-> (exists2 p, prime p &
exists2 n, n >= (p == 2) + 3 & G \isog 'Mod_(p ^ n)). | Proof.
rewrite /extremal_class; split=> [ | [p p_pr [n n_gt23 isoG]]].
move: (pdiv _) => p; set n := logn p _; do 4?case: ifP => //.
case/andP=> n_gt2 isoG _ _; rewrite ltnW //= => not_isoG _.
exists p; first by move: n_gt2; rewrite /n lognE; case (prime p).
exists n => //; case: eqP => // p2; rewrite ltn_neqAl... | Lemma | modular_group_classP | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"apply",
"card_2dihedral",
"card_quaternion",
"card_semidihedral",
"def_n",
"dihedral2_structure",
"extremal_class",
"generators_modular_group",
"isoG",
"isog",
"isogEcard",
"isog_sym",
"leqNgt",
"leq_addl",
"leq_exp2r",
"leq_trans",
"logn",
"lognE",
"ltnNge",
"ltnS",
"ltnW",... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
extremal2_structure (gT : finGroupType) (G : {group gT}) n x y :
let cG := extremal_class G in
let m := (2 ^ n)%N in let q := (2 ^ n.-1)%N in let r := (2 ^ n.-2)%N in
let X := <[x]> in let yG := y ^: G in let xyG := (x * y) ^: G in
let My := <<yG>> in let Mxy := <<xyG>> in
extremal_generators G 2 n (x, y) ... | Proof.
move=> cG m q r X yG xyG My Mxy genG; have [oG _ _ _] := genG.
have logG: logn (pdiv #|G|) #|G| = n by rewrite oG pfactorKpdiv.
rewrite /extremal2 -/cG; do [rewrite {1}/extremal_class /= {}logG] in cG *.
case: ifP => [isoG | _] in cG * => [_ _ /=|].
case/andP: isoG => n_gt1 isoG.
have:= dihedral2_structure n... | Theorem | extremal2_structure | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Mho",
"Mxy",
"My",
"Ohm_leq",
"Ohm_sub",
"True",
"abelem",
"add0n",
"add1n",
"apply",
"cyclic",
"def2",
"defG",
"dihedral2_structure",
"dihedral_classP",
"disjoint",
"eqEsubset",
"eqnP",
"eqn_leq",
"extremal2",
"extremal_class",
"extremal_generators",
"gT",
"group",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maximal_cycle_extremal gT p (G X : {group gT}) :
p.-group G -> ~~ abelian G -> cyclic X -> X \subset G -> #|G : X| = p ->
(extremal_class G == ModularGroup) || (p == 2) && extremal2 G. | Proof.
move=> pG not_cGG cycX sXG iXG; rewrite /extremal2; set cG := extremal_class G.
have [|p_pr _ _] := pgroup_pdiv pG.
by case: eqP not_cGG => // ->; rewrite abelian1.
have p_gt1 := prime_gt1 p_pr; have p_gt0 := ltnW p_gt1.
have [n oG] := p_natP pG; have n_gt2: n > 2.
apply: contraR not_cGG; rewrite -leqNgt => ... | Lemma | maximal_cycle_extremal | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Aut",
"Aut_aut",
"Aut_cyclic_abelian",
"Grp_2dihedral",
"Grp_modular_group",
"Grp_quaternion",
"Grp_semidihedral",
"OhmE",
"OhmS",
"Ohm_dprod",
"Ohm_p_cycle",
"Sylow",
"abelian",
"abelian1",
"abelianE",
"abelianM",
"abelian_nil",
"addSnnS",
"addnC",
"addnn",
"addrK",
"appl... | This is Aschbacher (23.4). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cyclic_SCN gT p (G U : {group gT}) :
p.-group G -> U \in 'SCN(G) -> ~~ abelian G -> cyclic U ->
[/\ p = 2, #|G : U| = 2 & extremal2 G]
\/ exists M : {group gT},
[/\ M :=: 'C_G('Mho^1(U)), #|M : U| = p, extremal_class M = ModularGroup,
'Ohm_1(M)%G \in 'E_p^2(G) & 'Ohm_1(M) \char G]. | Proof.
move=> pG /SCN_P[nsUG scUG] not_cGG cycU; have [sUG nUG] := andP nsUG.
have [cUU pU] := (cyclic_abelian cycU, pgroupS sUG pG).
have ltUG: ~~ (G \subset U).
by apply: contra not_cGG => sGU; apply: abelianS cUU.
have ntU: U :!=: 1.
by apply: contraNneq ltUG => U1; rewrite -scUG subsetIidl U1 cents1.
have [p_pr... | Lemma | cyclic_SCN | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Aut",
"Aut_conj_aut",
"Aut_cyclic_abelian",
"Mho",
"MhoS",
"Mho_p_cycle",
"Mho_sub",
"OhmE",
"OhmS",
"Ohm_char",
"Ohm_dprod",
"Ohm_id",
"Ohm_p_cycle",
"Ohm_sub",
"SCN",
"SCN_P",
"Sylow",
"TI_Ohm1",
"TI_cardMg",
"Uu",
"abelem_Ohm1",
"abelem_abelian",
"abelian",
"abelian... | This is Aschbacher (23.5) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
normal_rank1_structure gT p (G : {group gT}) :
p.-group G -> (forall X : {group gT}, X <| G -> abelian X -> cyclic X) ->
cyclic G \/ [&& p == 2, extremal2 G & (#|G| >= 16) || (G \isog 'Q_8)]. | Proof.
move=> pG dn_G_1.
have [cGG | not_cGG] := boolP (abelian G); first by left; rewrite dn_G_1.
have [X maxX]: {X | [max X | X <| G & abelian X]}.
by apply: ex_maxgroup; exists 1%G; rewrite normal1 abelian1.
have cycX: cyclic X by rewrite dn_G_1; case/andP: (maxgroupp maxX).
have scX: X \in 'SCN(G) := max_SCN pG m... | Lemma | normal_rank1_structure | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"SCN",
"abelE",
"abelem_abelian",
"abelem_cyclic",
"abelian",
"abelian1",
"apply",
"cGG",
"char_normal",
"cyclic",
"cyclic_SCN",
"dihedral2_structure",
"eq_sym",
"eqxx",
"ex_maxgroup",
"extremal2",
"extremal_class",
"gT",
"generators_2dihedral",
"group",
"inE",
"isoG",
"i... | This is Aschbacher, exercise (8.4) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
odd_pgroup_rank1_cyclic gT p (G : {group gT}) :
p.-group G -> odd #|G| -> cyclic G = ('r_p(G) <= 1). | Proof.
move=> pG oddG; rewrite -rank_pgroup //; apply/idP/idP=> [cycG | dimG1].
by rewrite -abelian_rank1_cyclic ?cyclic_abelian.
have [X nsXG cXX|//|] := normal_rank1_structure pG; last first.
by rewrite (negPf (odd_not_extremal2 oddG)) andbF.
by rewrite abelian_rank1_cyclic // (leq_trans (rankS (normal_sub nsXG))... | Lemma | odd_pgroup_rank1_cyclic | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"abelian_rank1_cyclic",
"apply",
"cyclic",
"cyclic_abelian",
"gT",
"group",
"last",
"leq_trans",
"normal_rank1_structure",
"normal_sub",
"odd",
"odd_not_extremal2",
"pG",
"rankS",
"rank_pgroup"
] | Replacement for Section 4 proof. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
prime_Ohm1P gT p (G : {group gT}) :
p.-group G -> G :!=: 1 ->
reflect (#|'Ohm_1(G)| = p)
(cyclic G || (p == 2) && (extremal_class G == Quaternion)). | Proof.
move=> pG ntG; have [p_pr p_dvd_G _] := pgroup_pdiv pG ntG.
apply: (iffP idP) => [|oG1p].
case/orP=> [cycG|]; first exact: Ohm1_cyclic_pgroup_prime.
case/andP=> /eqP p2 /eqP/quaternion_classP[n n_gt2 isoG].
rewrite p2; have [[x y]] := generators_quaternion n_gt2 isoG.
by case/quaternion_structure=> // _ ... | Lemma | prime_Ohm1P | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Ohm1_cyclic_pgroup_prime",
"OhmS",
"abelian_rank1_cyclic",
"apply",
"cardG_gt0",
"cardSg",
"card_2dihedral",
"card_isog",
"cyclic",
"dihedral2_structure",
"dihedral_classP",
"dvdn_leq_log",
"extremal2",
"extremal_class",
"gT",
"generators_2dihedral",
"generators_quaternion",
"gene... | This is the second part of Aschbacher, exercise (8.4). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
symplectic_type_group_structure gT p (G : {group gT}) :
p.-group G -> (forall X : {group gT}, X \char G -> abelian X -> cyclic X) ->
exists2 E : {group gT}, E :=: 1 \/ extraspecial E
& exists R : {group gT},
[/\ cyclic R \/ [/\ p = 2, extremal2 R & #|R| >= 16],
E \* R = G
& E :&: R = 'Z(E)]. | Proof.
move=> pG sympG; have [H [charH]] := Thompson_critical pG.
have sHG := char_sub charH; have pH := pgroupS sHG pG.
set U := 'Z(H) => sPhiH_U sHG_U defU; set Z := 'Ohm_1(U).
have sZU: Z \subset U by rewrite Ohm_sub.
have charU: U \char G := gFchar_trans _ charH.
have cUU: abelian U := center_abelian H.
have cycU: ... | Theorem | symplectic_type_group_structure | solvable | solvable/extremal.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"bigop",
"finset",
"prime",
"binomial",
"fingroup",
"morphism",
"perm",
"automorphism",
"presentation",
"quotient",
"action",
"commutator",
"gprodu... | [
"Euclid_dvdX",
"Gg",
"Hh",
"Mho",
"Mho_p_cycle",
"Mho_p_elt",
"My",
"Ohm1_abelem",
"Ohm1_cyclic_pgroup_prime",
"Ohm1_id",
"OhmE",
"OhmS",
"Ohm_char",
"Ohm_dprod",
"Ohm_sub",
"Phi_joing",
"SCN",
"SCN_P",
"TI_cardMg",
"Thompson_critical",
"abelE",
"abelem",
"abelemS",
"ab... | This is Aschbacher (23.9) | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
fmod_of (gT : finGroupType) (A : {group gT}) (abelA : abelian A) | :=
Fmod x & x \in A. | Inductive | fmod_of | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"abelian",
"gT",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
f2sub_magma (gT : finGroupType) (A : {group gT}) (abA : abelian A) | :=
fun u : fmod_of abA => let : Fmod x Ax := u in Subg Ax : Magma.sort _. | Let | f2sub_magma | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"abelian",
"fmod_of",
"gT",
"group",
"sort"
] | TODO: understand why FinGroup has to be changed to Magma here. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
f2sub_magma : fmod_of >-> Magma.sort. | Coercion | f2sub_magma | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmod_of",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
f2sub_baseGroup (gT : finGroupType) (A : {group gT}) (abA : abelian A) | :=
fun u : fmod_of abA => let : Fmod x Ax := u in Subg Ax : BaseGroup.sort _. | Let | f2sub_baseGroup | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"abelian",
"fmod_of",
"gT",
"group",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
f2sub_baseGroup : fmod_of >-> BaseGroup.sort. | Coercion | f2sub_baseGroup | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmod_of",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
fmodA | := (fmod_of abelA). | Notation | fmodA | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmod_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub2f (s : [subg A]) | := Fmod abelA (valP s). | Let | sub2f | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"subg",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmval u | := val (f2sub_magma u). | Definition | fmval | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"f2sub_magma",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
valA | := (val: fmodA -> gT) (only parsing). | Notation | valA | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmodA",
"gT",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmod x | := sub2f (subg A x). | Definition | fmod | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"sub2f",
"subg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actr u x | := if x \in 'N(A) then fmod (fmval u ^ x) else u. | Definition | actr | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmod",
"fmval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmod_opp u | := sub2f u^-1. | Definition | fmod_opp | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"sub2f"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmod_add u v | := sub2f (u * v). | Definition | fmod_add | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"sub2f"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmod_add0r : left_id (sub2f 1) fmod_add. | Proof. by move=> u; apply: val_inj; apply: mul1g. Qed. | Fact | fmod_add0r | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"apply",
"fmod_add",
"mul1g",
"sub2f",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmod_addrA : associative fmod_add. | Proof. by move=> u v w; apply: val_inj; apply: mulgA. Qed. | Fact | fmod_addrA | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"apply",
"fmod_add",
"mulgA",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmod_addNr : left_inverse (sub2f 1) fmod_opp fmod_add. | Proof. by move=> u; apply: val_inj; apply: mulVg. Qed. | Fact | fmod_addNr | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"apply",
"fmod_add",
"fmod_opp",
"mulVg",
"sub2f",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmod_addrC : commutative fmod_add. | Proof. by case=> x Ax [y Ay]; apply: val_inj; apply: (centsP abelA). Qed. | Fact | fmod_addrC | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"apply",
"centsP",
"fmod_add",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmodP u : val u \in A. | Proof. exact: valP. Qed. | Lemma | fmodP | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"val",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmod_inj : injective fmval. | Proof. exact: val_inj. Qed. | Lemma | fmod_inj | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmval",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
congr_fmod u v : u = v -> fmval u = fmval v. | Proof. exact: congr1. Qed. | Lemma | congr_fmod | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmvalA : {morph valA : x y / x + y >-> (x * y)%g}. | Proof. by []. Qed. | Lemma | fmvalA | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"valA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmvalN : {morph valA : x / - x >-> x^-1%g}. | Proof. by []. Qed. | Lemma | fmvalN | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"valA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmval0 : valA 0 = 1%g. | Proof. by []. Qed. | Lemma | fmval0 | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"valA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmval_morphism | := @Morphism _ _ setT fmval (in2W fmvalA). | Canonical | fmval_morphism | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmval",
"fmvalA",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmval_sum | := big_morph fmval fmvalA fmval0. | Definition | fmval_sum | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"big_morph",
"fmval",
"fmval0",
"fmvalA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmvalZ n : {morph valA : x / x *+ n >-> (x ^+ n)%g}. | Proof. by move=> u; rewrite /= morphX ?inE. Qed. | Lemma | fmvalZ | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"inE",
"morphX",
"valA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmodKcond x : val (fmod x) = if x \in A then x else 1%g. | Proof. by rewrite /= /fmval /= val_insubd. Qed. | Lemma | fmodKcond | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmod",
"fmval",
"val",
"val_insubd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmodK : {in A, cancel fmod val}. | Proof. exact: subgK. Qed. | Lemma | fmodK | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmod",
"subgK",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmvalK : cancel val fmod. | Proof. by case=> x Ax; apply: val_inj; rewrite /fmod /= sgvalK. Qed. | Lemma | fmvalK | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"apply",
"fmod",
"sgvalK",
"val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmod1 : fmod 1 = 0. | Proof. by rewrite -fmval0 fmvalK. Qed. | Lemma | fmod1 | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmod",
"fmval0",
"fmvalK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmodM : {in A &, {morph fmod : x y / (x * y)%g >-> x + y}}. | Proof. by move=> x y Ax Ay /=; apply: val_inj; rewrite /fmod morphM. Qed. | Lemma | fmodM | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"apply",
"fmod",
"morphM",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmod_morphism | := Morphism fmodM. | Canonical | fmod_morphism | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmodM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmodX n : {in A, {morph fmod : x / (x ^+ n)%g >-> x *+ n}}. | Proof. exact: morphX. Qed. | Lemma | fmodX | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmod",
"morphX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmodV : {morph fmod : x / x^-1%g >-> - x}. | Proof.
move=> x; apply: val_inj; rewrite fmvalN !fmodKcond groupV.
by case: (x \in A); rewrite ?invg1.
Qed. | Lemma | fmodV | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"apply",
"fmod",
"fmodKcond",
"fmvalN",
"groupV",
"invg1",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_fmod : 'injm fmod. | Proof.
by apply/injmP=> x y Ax Ay []; move/val_inj; apply: (injmP (injm_subg A)).
Qed. | Lemma | injm_fmod | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"apply",
"fmod",
"injmP",
"injm_subg",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"u ^@ x" | := (actr u x) : ring_scope. | Notation | u ^@ x | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmvalJcond u x :
val (u ^@ x) = if x \in 'N(A) then val u ^ x else val u. | Proof. by case: ifP => Nx; rewrite /actr Nx ?fmodK // memJ_norm ?fmodP. Qed. | Lemma | fmvalJcond | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actr",
"fmodK",
"fmodP",
"memJ_norm",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmvalJ u x : x \in 'N(A) -> val (u ^@ x) = val u ^ x. | Proof. by move=> Nx; rewrite fmvalJcond Nx. Qed. | Lemma | fmvalJ | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmvalJcond",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmodJ x y : y \in 'N(A) -> fmod (x ^ y) = fmod x ^@ y. | Proof.
move=> Ny; apply: val_inj; rewrite fmvalJ ?fmodKcond ?memJ_norm //.
by case: ifP => // _; rewrite conj1g.
Qed. | Lemma | fmodJ | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"apply",
"conj1g",
"fmod",
"fmodKcond",
"fmvalJ",
"memJ_norm",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actr_is_action : is_action 'N(A) actr. | Proof.
split=> [a u v eq_uv_a | u a b Na Nb].
case Na: (a \in 'N(A)); last by rewrite /actr Na in eq_uv_a.
by apply: val_inj; apply: (conjg_inj a); rewrite -!fmvalJ ?eq_uv_a.
by apply: val_inj; rewrite !fmvalJ ?groupM ?conjgM.
Qed. | Fact | actr_is_action | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actr",
"apply",
"conjgM",
"conjg_inj",
"fmvalJ",
"groupM",
"is_action",
"last",
"split",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actr_action | := Action actr_is_action. | Canonical | actr_action | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actr_is_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M'" | := actr_action : action_scope. | Notation | ''M' | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actr_action"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
act0r x : 0 ^@ x = 0. | Proof. by rewrite /actr conj1g morph1 if_same. Qed. | Lemma | act0r | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actr",
"conj1g",
"morph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actAr x : {morph actr^~ x : u v / u + v}. | Proof.
by move=> u v; apply: val_inj; rewrite !(fmvalA, fmvalJcond) conjMg; case: ifP.
Qed. | Lemma | actAr | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actr",
"apply",
"conjMg",
"fmvalA",
"fmvalJcond",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actr_sum x | := big_morph _ (actAr x) (act0r x). | Definition | actr_sum | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"act0r",
"actAr",
"big_morph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actNr x : {morph actr^~ x : u / - u}. | Proof. by move=> u; apply: (addrI (u ^@ x)); rewrite -actAr !subrr act0r. Qed. | Lemma | actNr | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"act0r",
"actAr",
"actr",
"addrI",
"apply",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actZr x n : {morph actr^~ x : u / u *+ n}. | Proof.
by move=> u; elim: n => [|n IHn]; rewrite ?act0r // !mulrS actAr IHn.
Qed. | Lemma | actZr | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"act0r",
"actAr",
"actr",
"mulrS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actr_is_groupAction : is_groupAction setT 'M. | Proof.
move=> a Na /[1!inE]; apply/andP; split; first by apply/subsetP=> u _ /[1!inE].
by apply/morphicP=> u v _ _; rewrite !permE /= actAr.
Qed. | Fact | actr_is_groupAction | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actAr",
"apply",
"inE",
"is_groupAction",
"morphicP",
"permE",
"setT",
"split",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actr_groupAction | := GroupAction actr_is_groupAction. | Canonical | actr_groupAction | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actr_is_groupAction"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M'" | := actr_groupAction : groupAction_scope. | Notation | ''M' | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actr_groupAction"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actr1 u : u ^@ 1 = u. | Proof. exact: act1. Qed. | Lemma | actr1 | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"act1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actrM : {in 'N(A) &, forall x y u, u ^@ (x * y) = u ^@ x ^@ y}. | Proof.
by move=> x y Nx Ny /= u; apply: val_inj; rewrite !fmvalJ ?conjgM ?groupM.
Qed. | Lemma | actrM | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"apply",
"conjgM",
"fmvalJ",
"groupM",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actrK x : cancel (actr^~ x) (actr^~ x^-1%g). | Proof.
move=> u; apply: val_inj; rewrite !fmvalJcond groupV.
by case: ifP => -> //; rewrite conjgK.
Qed. | Lemma | actrK | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actr",
"apply",
"conjgK",
"fmvalJcond",
"groupV",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actrKV x : cancel (actr^~ x^-1%g) (actr^~ x). | Proof. by move=> u; rewrite /= -{2}(invgK x) actrK. Qed. | Lemma | actrKV | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actr",
"actrK",
"invgK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(nsHG : H <| G) (sHP : H \subset P) (sPG : P \subset G). | Hypotheses | nsHG | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
(abelH : abelian H) (coHiPG : coprime #|H| #|G : P|). | Hypotheses | abelH | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"abelian",
"coprime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
nHG | := subsetP (normal_norm nsHG). | Let | nHG | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"normal_norm",
"nsHG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
m | := (expg_invn H #|G : P|). | Let | m | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"expg_invn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmod | := (fmod abelH). | Notation | fmod | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"abelH"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gaschutz_split : [splits G, over H] = [splits P, over H]. | Proof.
apply/splitsP/splitsP=> [[K /complP[tiHK eqHK]] | [Q /complP[tiHQ eqHQ]]].
exists (K :&: P)%G; rewrite inE setICA (setIidPl sHP) setIC tiHK eqxx.
by rewrite group_modl // eqHK (sameP eqP setIidPr).
have sQP: Q \subset P by rewrite -eqHQ mulG_subr.
pose rP x := repr (P :* x); pose pP x := x * (rP x)^-1.
have ... | Theorem | Gaschutz_split | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"Hh",
"Px",
"abelH",
"actZr",
"actr_sum",
"actsRs_rcosets",
"addKr",
"addrC",
"apply",
"big_split",
"complP",
"conjgC",
"conjgCV",
"conjgE",
"conjgM",
"eq_bigr",
"eqxx",
"expgK",
"fM",
"fmod",
"fmodK",
"fmodP",
"fmvalA",
"fmvalJ",
"fmvalN",
"fmvalZ",
"groupM",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Gaschutz_transitive : {in [complements to H in G] &,
forall K L, K :&: P = L :&: P -> exists2 x, x \in H & L :=: K :^ x}. | Proof.
move=> K L /=; set Q := K :&: P => /complP[tiHK eqHK] cpHL QeqLP.
have [trHL eqHL] := complP cpHL.
pose nu x := fmod (divgr H L x^-1).
have sKG: {subset K <= G} by apply/subsetP; rewrite -eqHK mulG_subr.
have sLG: {subset L <= G} by apply/subsetP; rewrite -eqHL mulG_subr.
have val_nu x: x \in G -> val (nu x) = d... | Theorem | Gaschutz_transitive | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"TI_cardMg",
"act0r",
"actZr",
"actr_sum",
"actsRs_rcosets",
"add0r",
"addKr",
"addrC",
"apply",
"cardG_gt0",
"cardJg",
"complP",
"conjVg",
"conjgC",
"conjgE",
"divgS",
"divgr",
"divnMl",
"eqEcard",
"eq_bigr",
"eq_sym",
"expgK",
"fmod",
"fmodK",
"fmodP",
"fmval",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coprime_abel_cent_TI (gT : finGroupType) (A G : {group gT}) :
A \subset 'N(G) -> coprime #|G| #|A| -> abelian G -> 'C_[~: G, A](A) = 1. | Proof.
move=> nGA coGA abG; pose f x := val (\sum_(a in A) fmod abG x ^@ a)%R.
have fM: {in G &, {morph f : x y / x * y}}.
move=> x y Gx Gy /=; rewrite -fmvalA -big_split /=; congr (fmval _).
by apply: eq_bigr => a Aa; rewrite fmodM // actAr.
have nfA x a: a \in A -> f (x ^ a) = f x.
move=> Aa; rewrite {2}/f (rei... | Lemma | coprime_abel_cent_TI | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"abelian",
"actAr",
"apply",
"big_split",
"centP",
"coGA",
"commgEl",
"commg_subl",
"conjg",
"conjgM",
"coprime",
"eq_big",
"eq_bigr",
"expg1n",
"expgK",
"fM",
"fmod",
"fmodJ",
"fmodK",
"fmodM",
"fmval",
"fmvalA",
"fmvalZ",
"gT",
"gen_subG",
"group",
"groupM",
"... | This Lemma is used in maximal.v for the proof of Aschbacher 24.7. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
(sHG : H \subset G) (abelA : abelian (alpha @* H)). | Hypotheses | sHG | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"abelian",
"alpha"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
HG | := (rcosets (gval H) (gval G)). | Notation | HG | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"rcosets"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
transfer_morph_subproof : H \subset alpha @*^-1 (alpha @* H). | Proof. by rewrite -sub_morphim_pre. Qed. | Fact | transfer_morph_subproof | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"alpha",
"sub_morphim_pre"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmalpha | := restrm transfer_morph_subproof (fmod abelA \o alpha). | Let | fmalpha | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"alpha",
"fmod",
"restrm",
"transfer_morph_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
V (rX : {set gT} -> gT) g | :=
\sum_(Hx in rcosets H G) fmalpha (rX Hx * g * (rX (Hx :* g))^-1). | Let | V | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"fmalpha",
"gT",
"rcosets"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
transfer g | := V repr g. | Definition | transfer | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
transferM : {in G &, {morph transfer : x y / (x * y)%g >-> x + y}}. | Proof.
move=> s t Gs Gt /=.
rewrite [transfer t](reindex_acts 'Rs _ Gs) ?actsRs_rcosets //= -big_split /=.
apply: eq_bigr => _ /rcosetsP[x Gx ->]; rewrite !rcosetE -!rcosetM.
rewrite -zmodMgE -morphM -?mem_rcoset; last by rewrite !mulgA mulgKV rcosetM.
by rewrite rcoset_repr rcosetM mem_rcoset mulgK mem_repr_rcoset.
... | Lemma | transferM | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actsRs_rcosets",
"apply",
"big_split",
"eq_bigr",
"last",
"mem_rcoset",
"mem_repr_rcoset",
"morphM",
"mulgA",
"mulgK",
"mulgKV",
"rcosetE",
"rcosetM",
"rcoset_repr",
"rcosetsP",
"reindex_acts",
"transfer",
"zmodMgE"
] | This is Aschbacher (37.2). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
transfer_morphism | := Morphism transferM. | Canonical | transfer_morphism | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"transferM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
transfer_indep X (rX := transversal_repr 1 X) :
is_transversal X HG G -> {in G, transfer =1 V rX}. | Proof.
move=> trX g Gg; have mem_rX := repr_mem_pblock trX 1; rewrite -/rX in mem_rX.
apply: (addrI (\sum_(Hx in HG) fmalpha (repr Hx * (rX Hx)^-1))).
rewrite {1}(reindex_acts 'Rs _ Gg) ?actsRs_rcosets // -!big_split /=.
apply: eq_bigr => _ /rcosetsP[x Gx ->]; rewrite !rcosetE -!rcosetM.
case: repr_rcosetP => h1 Hh1; c... | Lemma | transfer_indep | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"Gg",
"HG",
"actsRs_rcosets",
"addrC",
"addrI",
"apply",
"big_split",
"eq_bigr",
"fmalpha",
"groupM",
"groupV",
"imset_f",
"invMg",
"is_transversal",
"morphM",
"mulKVg",
"mulKg",
"mulgA",
"rcosetE",
"rcosetM",
"rcosetP",
"rcosets",
"rcosetsP",
"reindex_acts",
"repr",
... | This is Aschbacher (37.1). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
Gg : g \in G. | Hypothesis | Gg | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
sgG : <[g]> \subset G. | Proof. by rewrite cycle_subG. Qed. | Let | sgG | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"cycle_subG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
H_g_rcosets x : {set {set gT}} | := rcosets (H :* x) <[g]>. | Let | H_g_rcosets | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"gT",
"rcosets"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_ x | := #|<[g]> : H :* x|. | Let | n_ | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulg_exp_card_rcosets x : x * (g ^+ n_ x) \in H :* x. | Proof.
rewrite /n_ /indexg -orbitRs -porbit_actperm ?inE //.
rewrite -{2}(iter_porbit (actperm 'Rs g) (H :* x)) -permX -morphX ?inE //.
by rewrite actpermE //= rcosetE -rcosetM rcoset_refl.
Qed. | Lemma | mulg_exp_card_rcosets | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"actperm",
"actpermE",
"inE",
"indexg",
"iter_porbit",
"morphX",
"n_",
"orbitRs",
"permX",
"porbit_actperm",
"rcosetE",
"rcosetM",
"rcoset_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
HGg : {set {set {set gT}}} | := orbit 'Rs <[g]> @: HG. | Let | HGg | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"HG",
"gT",
"orbit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
partHG : partition HG G | := rcosets_partition sHG. | Let | partHG | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"HG",
"partition",
"rcosets_partition",
"sHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actsgHG : [acts <[g]>, on HG | 'Rs]. | Proof. exact: subset_trans sgG (actsRs_rcosets H G). Qed. | Let | actsgHG | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"HG",
"actsRs_rcosets",
"on",
"sgG",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
partHGg : partition HGg HG | := orbit_partition actsgHG. | Let | partHGg | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"HG",
"HGg",
"actsgHG",
"orbit_partition",
"partition"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injHGg : {in HGg &, injective cover}. | Proof. by have [] := partition_partition partHG partHGg. Qed. | Let | injHGg | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"HGg",
"cover",
"partHG",
"partHGg",
"partition_partition"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
defHGg : HG :* <[g]> = cover @: HGg. | Proof.
rewrite -imset_comp [_ :* _]imset2_set1r; apply: eq_imset => Hx /=.
by rewrite cover_imset -curry_imset2r.
Qed. | Let | defHGg | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"HG",
"HGg",
"apply",
"cover",
"cover_imset",
"curry_imset2r",
"eq_imset",
"imset2_set1r",
"imset_comp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rcosets_cycle_partition : partition (HG :* <[g]>) G. | Proof. by rewrite defHGg; have [] := partition_partition partHG partHGg. Qed. | Lemma | rcosets_cycle_partition | solvable | solvable/finmodule.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"bigop",
"ssralg",
"finset",
"fingroup",
"morphism",
"perm",
"finalg",
"action",
"gproduct",
"commutator",
"cyclic",
"GRing.Theory",
"Fin... | [
"HG",
"defHGg",
"partHG",
"partHGg",
"partition",
"partition_partition"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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