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 |
|---|---|---|---|---|---|---|---|---|---|---|
Sv (sc : square) : square | := tnth [tuple c3; c2; c1; c0] sc. | Definition | Sv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c0",
"c1",
"c2",
"c3",
"square",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sv_inj : injective Sv. | Proof.
by apply: (can_inj (g:= Sv)); case; do 4?case=> //=; move=> H; apply/eqP.
Qed. | Lemma | Sv_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Sv",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sv | := (perm Sv_inj). | Definition | sv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Sv_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sv_inv : sv^-1 = sv. | Proof.
apply: (mulIg sv); rewrite mulVg; apply/permP.
by case; do 4?case=> //=; move=> H; rewrite !permE /= !permE; apply/eqP.
Qed. | Lemma | sv_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"mulIg",
"mulVg",
"permE",
"permP",
"sv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sd1 (sc : square) : square | := tnth [tuple c0; c3; c2; c1] sc. | Definition | Sd1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c0",
"c1",
"c2",
"c3",
"square",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sd1_inj : injective Sd1. | Proof.
by apply: can_inj Sd1 _; case; do 4?case=> //=; move=> H; apply/eqP.
Qed. | Lemma | Sd1_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Sd1",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sd1 | := (perm Sd1_inj). | Definition | sd1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Sd1_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sd1_inv : sd1^-1 = sd1. | Proof.
apply: (mulIg sd1); rewrite mulVg; apply/permP.
by case; do 4?case=> //=; move=> H; rewrite !permE /= !permE; apply/eqP.
Qed. | Lemma | sd1_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"mulIg",
"mulVg",
"permE",
"permP",
"sd1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sd2 (sc : square) : square | := tnth [tuple c2; c1; c0; c3] sc. | Definition | Sd2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c0",
"c1",
"c2",
"c3",
"square",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sd2_inj : injective Sd2. | Proof.
by apply: can_inj Sd2 _; case; do 4?case=> //=; move=> H; apply/eqP.
Qed. | Lemma | Sd2_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Sd2",
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sd2 | := (perm Sd2_inj). | Definition | sd2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Sd2_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sd2_inv : sd2^-1 = sd2. | Proof.
apply: (mulIg sd2); rewrite mulVg; apply/permP.
by case; do 4?case=> //=; move=> H; rewrite !permE /= !permE; apply/eqP.
Qed. | Lemma | sd2_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"mulIg",
"mulVg",
"permE",
"permP",
"sd2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ord_enum4 : enum 'I_4 = [:: c0; c1; c2; c3]. | Proof. by apply: (inj_map val_inj); rewrite val_enum_ord. Qed. | Lemma | ord_enum4 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"c0",
"c1",
"c2",
"c3",
"enum",
"inj_map",
"val_enum_ord",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
diff_id_sh : 1 != sh. | Proof.
by apply/eqP; move/(congr1 (fun p : {perm square} => p c0)); rewrite !permE.
Qed. | Lemma | diff_id_sh | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"c0",
"permE",
"sh",
"square"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isometries2 | := [set 1; sh]. | Definition | isometries2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"sh"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_iso2 : #|isometries2| = 2. | Proof. by rewrite cards2 diff_id_sh. Qed. | Lemma | card_iso2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"cards2",
"diff_id_sh",
"isometries2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_set_iso2 : group_set isometries2. | Proof.
apply/group_setP; split => [|x y]; rewrite !inE ?eqxx //.
do 2![case/orP; move/eqP->]; rewrite ?(mul1g, mulg1) ?eqxx ?orbT//.
by rewrite -/sh -{1}sh_inv mulVg eqxx.
Qed. | Lemma | group_set_iso2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"eqxx",
"group_set",
"group_setP",
"inE",
"isometries2",
"mul1g",
"mulVg",
"mulg1",
"sh",
"sh_inv",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iso2_group | := Group group_set_iso2. | Canonical | iso2_group | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"group_set_iso2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isometries | :=
[set p | [|| p == 1, p == r1, p == r2, p == r3,
p == sh, p == sv, p == sd1 | p == sd2 ]]. | Definition | isometries | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"r1",
"r2",
"r3",
"sd1",
"sd2",
"sh",
"sv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp (sc : square) | := tnth [tuple c2; c3; c0; c1] sc. | Definition | opp | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c0",
"c1",
"c2",
"c3",
"square",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_iso (p : {perm square}) | := forall ci, p (opp ci) = opp (p ci). | Definition | is_iso | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"opp",
"square"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isometries_iso : forall p, p \in isometries -> is_iso p. | Proof.
move=> p; rewrite inE.
by do ?case/orP; move/eqP=> -> a; rewrite !permE; case: a; do 4?case.
Qed. | Lemma | isometries_iso | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"inE",
"is_iso",
"isometries",
"permE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
non_inj p a1 a2 heq1 heq2 | :=
let h1:= fresh "h1" in
(absurd (p a1 = p a2); first (by red => - h1; move: (perm_inj h1));
by rewrite heq1 heq2; apply/eqP). | Ltac | non_inj | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"a1",
"a2",
"apply",
"perm_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_isoPtac p f e0 e1 e2 e3 | :=
suff ->: p = f by [rewrite inE eqxx ?orbT];
let e := fresh "e" in apply/permP;
(do 5?[case] => // ?; [move: e0 | move: e1 | move: e2 | move: e3]) => e;
apply: etrans (congr1 p _) (etrans e _); apply/eqP; rewrite // permE. | Ltac | is_isoPtac | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"e0",
"eqxx",
"inE",
"permE",
"permP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
is_isoP : forall p, reflect (is_iso p) (p \in isometries). | Proof.
move=> p; apply: (iffP idP) => [|iso_p]; first exact: isometries_iso.
move e1: (p c1) (iso_p c1) => k1; move e0: (p c0) (iso_p c0) k1 e1 => k0.
case: k0 e0; do 4?[case] => //= ? e0 e2; do 5?[case] => //= ? e1 e3;
try by [non_inj p c0 c1 e0 e1 | non_inj p c0 c3 e0 e3].
by is_isoPtac p id1 e0 e1 e2 e3.
by is_isoP... | Lemma | is_isoP | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"c0",
"c1",
"c3",
"e0",
"id1",
"is_iso",
"is_isoPtac",
"isometries",
"isometries_iso",
"non_inj",
"r1",
"r2",
"r3",
"sd1",
"sd2",
"sh",
"sv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_set_iso : group_set isometries. | Proof.
apply/group_setP; split; first by rewrite inE eqxx /=.
by move=> x y hx hy; apply/is_isoP => ci; rewrite !permM !isometries_iso.
Qed. | Lemma | group_set_iso | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"eqxx",
"group_set",
"group_setP",
"inE",
"is_isoP",
"isometries",
"isometries_iso",
"permM",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
iso_group | := Group group_set_iso. | Canonical | iso_group | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"group_set_iso"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_rot : #|rot| = 4. | Proof.
rewrite -[4]/(size [:: id1; r1; r2; r3]) -(card_uniqP _); last first.
by apply: eq_card => x; rewrite rot_is_rot !inE -!orbA.
by apply: map_uniq (fun p : {perm square} => p c0) _ _; rewrite /= !permE.
Qed. | Lemma | card_rot | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"c0",
"card_uniqP",
"eq_card",
"id1",
"inE",
"last",
"map_uniq",
"permE",
"r1",
"r2",
"r3",
"rot",
"rot_is_rot",
"size",
"square"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
group_set_rotations : group_set rotations. | Proof. by rewrite -rot_is_rot group_set_rot. Qed. | Lemma | group_set_rotations | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"group_set",
"group_set_rot",
"rot_is_rot",
"rotations"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rotations_group | := Group group_set_rotations. | Canonical | rotations_group | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"group_set_rotations"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
col_squares | := {ffun square -> colors}. | Notation | col_squares | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"colors",
"square"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
act_f (sc : col_squares) (p : {perm square}) : col_squares | :=
[ffun z => sc (p^-1 z)]. | Definition | act_f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"col_squares",
"square"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
act_f_1 : forall k, act_f k 1 = k. | Proof. by move=> k; apply/ffunP=> a; rewrite ffunE invg1 permE. Qed. | Lemma | act_f_1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"act_f",
"apply",
"ffunE",
"ffunP",
"invg1",
"permE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
act_f_morph : forall k x y, act_f k (x * y) = act_f (act_f k x) y. | Proof. by move=> k x y; apply/ffunP=> a; rewrite !ffunE invMg permE. Qed. | Lemma | act_f_morph | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"act_f",
"apply",
"ffunE",
"ffunP",
"invMg",
"permE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
to | := TotalAction act_f_1 act_f_morph. | Definition | to | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"TotalAction",
"act_f_1",
"act_f_morph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
square_coloring_number2 | := #|orbit to isometries2 @: setT|. | Definition | square_coloring_number2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"isometries2",
"orbit",
"setT",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
square_coloring_number4 | := #|orbit to rotations @: setT|. | Definition | square_coloring_number4 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"orbit",
"rotations",
"setT",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
square_coloring_number8 | := #|orbit to isometries @: setT|. | Definition | square_coloring_number8 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"isometries",
"orbit",
"setT",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Fid : 'Fix_to(1) = setT. | Proof. by apply/setP=> x /=; rewrite in_setT; apply/afix1P; apply: act1. Qed. | Lemma | Fid | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"act1",
"afix1P",
"apply",
"in_setT",
"setP",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_Fid : #|'Fix_to(1)| = (n ^ 4)%N. | Proof.
rewrite -[4]card_ord -[n]card_ord -card_ffun_on Fid cardsE.
by symmetry; apply: eq_card => f; apply/ffun_onP.
Qed. | Lemma | card_Fid | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Fid",
"apply",
"card_ffun_on",
"card_ord",
"cardsE",
"eq_card",
"ffun_onP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coin0 (sc : col_squares) : colors | := sc c0. | Definition | coin0 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c0",
"col_squares",
"colors"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coin1 (sc : col_squares) : colors | := sc c1. | Definition | coin1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c1",
"col_squares",
"colors"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coin2 (sc : col_squares) : colors | := sc c2. | Definition | coin2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c2",
"col_squares",
"colors"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coin3 (sc : col_squares) : colors | := sc c3. | Definition | coin3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"c3",
"col_squares",
"colors"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqperm_map : forall p1 p2 : col_squares,
(p1 == p2) = all (fun s => p1 s == p2 s) [:: c0; c1; c2; c3]. | Proof.
move=> p1 p2; apply/eqP/allP=> [-> // | Ep12]; apply/ffunP=> x.
by apply/eqP; apply Ep12; case: x; do 4!case=> //.
Qed. | Lemma | eqperm_map | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"all",
"allP",
"apply",
"c0",
"c1",
"c2",
"c3",
"col_squares",
"ffunP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F_Sh :
'Fix_to[sh] = [set x | (coin0 x == coin1 x) && (coin2 x == coin3 x)]. | Proof.
apply/setP=> x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f sh_inv !ffunE !permE /=.
by rewrite eq_sym (eq_sym (x c3)) andbT andbA !andbb.
Qed. | Lemma | F_Sh | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"act_f",
"afix1P",
"apply",
"c3",
"coin0",
"coin1",
"coin2",
"coin3",
"eq_sym",
"eqperm_map",
"ffunE",
"inE",
"permE",
"setP",
"sh",
"sh_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F_Sv :
'Fix_to[sv] = [set x | (coin0 x == coin3 x) && (coin2 x == coin1 x)]. | Proof.
apply/setP=> x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f sv_inv !ffunE !permE /=.
by rewrite eq_sym andbT andbC (eq_sym (x c1)) andbA -andbA !andbb andbC.
Qed. | Lemma | F_Sv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"act_f",
"afix1P",
"apply",
"c1",
"coin0",
"coin1",
"coin2",
"coin3",
"eq_sym",
"eqperm_map",
"ffunE",
"inE",
"permE",
"setP",
"sv",
"sv_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_tac | :=
apply: esym (etrans _ (mul1g _)); apply: canRL (mulgK _) _;
let a := fresh "a" in apply/permP => a;
apply/eqP; rewrite permM !permE; case: a; do 4?case. | Ltac | inv_tac | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"mul1g",
"mulgK",
"permE",
"permM",
"permP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r1_inv : r1^-1 = r3. | Proof. by inv_tac. Qed. | Lemma | r1_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"inv_tac",
"r1",
"r3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r2_inv : r2^-1 = r2. | Proof. by inv_tac. Qed. | Lemma | r2_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"inv_tac",
"r2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r3_inv : r3^-1 = r1. | Proof. by inv_tac. Qed. | Lemma | r3_inv | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"inv_tac",
"r1",
"r3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F_r2 :
'Fix_to[r2] = [set x | (coin0 x == coin2 x) && (coin1 x == coin3 x)]. | Proof.
apply/setP=> x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f r2_inv !ffunE !permE /=.
by rewrite eq_sym andbT andbCA andbC (eq_sym (x c3)) andbA -andbA !andbb andbC.
Qed. | Lemma | F_r2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"act_f",
"afix1P",
"apply",
"c3",
"coin0",
"coin1",
"coin2",
"coin3",
"eq_sym",
"eqperm_map",
"ffunE",
"inE",
"permE",
"r2",
"r2_inv",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F_r1 : 'Fix_to[r1] =
[set x | (coin0 x == coin1 x)&&(coin1 x == coin2 x)&&(coin2 x == coin3 x)]. | Proof.
apply/setP=> x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f r1_inv !ffunE !permE andbC.
by do 3![case E: {+}(_ == _); rewrite // {E}(eqP E)]; rewrite eqxx.
Qed. | Lemma | F_r1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"act_f",
"afix1P",
"apply",
"coin0",
"coin1",
"coin2",
"coin3",
"eqperm_map",
"eqxx",
"ffunE",
"inE",
"permE",
"r1",
"r1_inv",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F_r3 : 'Fix_to[r3] =
[set x | (coin0 x == coin1 x)&&(coin1 x == coin2 x)&&(coin2 x == coin3 x)]. | Proof.
apply/setP=> x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f r3_inv !ffunE !permE /=.
by do 3![case: eqVneq=> // <-].
Qed. | Lemma | F_r3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"act_f",
"afix1P",
"apply",
"coin0",
"coin1",
"coin2",
"coin3",
"eqVneq",
"eqperm_map",
"ffunE",
"inE",
"permE",
"r3",
"r3_inv",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_n2 : forall x y z t : square, uniq [:: x; y; z; t] ->
#|[set p : col_squares | (p x == p y) && (p z == p t)]| = (n ^ 2)%N. | Proof.
move=> x y z t Uxt; rewrite -[n]card_ord.
pose f (p : col_squares) := (p x, p z).
rewrite -(@card_in_image _ _ f); last first.
rewrite -mulnn -card_prod; apply: eq_card => [] [c d] /=; apply/imageP.
rewrite (cat_uniq [::x; y]) in Uxt; case/and3P: Uxt => _.
rewrite /= !orbF !andbT => /norP[] /[!inE] nxzt ny... | Lemma | card_n2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"all",
"allP",
"apply",
"card_in_image",
"card_ord",
"card_prod",
"card_uniqP",
"cat_uniq",
"col_squares",
"eq_card",
"eqxx",
"ffunE",
"ffunP",
"imageP",
"inE",
"last",
"mulnn",
"pred2",
"square",
"subset_cardP",
"subset_predT",
"uniq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_n :
#|[set x | (coin0 x == coin1 x)&&(coin1 x == coin2 x)&& (coin2 x == coin3 x)]|
= n. | Proof.
rewrite -[n]card_ord /coin0 /coin1 /coin2 /coin3.
pose f (p : col_squares) := p c3; rewrite -(@card_in_image _ _ f); last first.
apply: eq_card => c /=; apply/imageP.
exists ([ffun => c] : col_squares); last by rewrite /f ffunE.
by rewrite /= inE !ffunE !eqxx.
move=> p1 p2; rewrite /= !inE /f -!andbA => eq... | Lemma | card_n | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"apply",
"c3",
"card_in_image",
"card_ord",
"coin0",
"coin1",
"coin2",
"coin3",
"col_squares",
"eq_card",
"eqperm_map",
"eqxx",
"ffunE",
"imageP",
"inE",
"last"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
burnside_app2 : (square_coloring_number2 * 2 = n ^ 4 + n ^ 2)%N. | Proof.
rewrite (burnside_formula [:: id1; sh]) => [|p|].
- by rewrite /= inE diff_id_sh.
- by rewrite !inE.
by rewrite 2!big_cons big_nil addn0 {1}card_Fid F_Sh card_n2.
Qed. | Lemma | burnside_app2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F_Sh",
"addn0",
"big_cons",
"big_nil",
"burnside_formula",
"card_Fid",
"card_n2",
"diff_id_sh",
"id1",
"inE",
"sh",
"square_coloring_number2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
burnside_app_rot :
(square_coloring_number4 * 4 = n ^ 4 + n ^ 2 + 2 * n)%N. | Proof.
rewrite (burnside_formula [:: id1; r1; r2; r3]) => [|p|].
- by apply: map_uniq (fun p : {perm square} => p c0) _ _; rewrite /= !permE.
- by rewrite !inE !orbA.
rewrite !big_cons big_nil /= addn0 {1}card_Fid F_r1 F_r2 F_r3.
by rewrite card_n card_n2 //= [n + _]addnC !addnA addn0.
Qed. | Lemma | burnside_app_rot | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F_r1",
"F_r2",
"F_r3",
"addn0",
"addnA",
"addnC",
"apply",
"big_cons",
"big_nil",
"burnside_formula",
"c0",
"card_Fid",
"card_n",
"card_n2",
"id1",
"inE",
"map_uniq",
"permE",
"r1",
"r2",
"r3",
"square",
"square_coloring_number4"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F_Sd1 : 'Fix_to[sd1] = [set x | coin1 x == coin3 x]. | Proof.
apply/setP => x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f sd1_inv !ffunE !permE /=.
by rewrite !eqxx !andbT eq_sym /= andbb.
Qed. | Lemma | F_Sd1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"act_f",
"afix1P",
"apply",
"coin1",
"coin3",
"eq_sym",
"eqperm_map",
"eqxx",
"ffunE",
"inE",
"permE",
"sd1",
"sd1_inv",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_n3 : forall x y : square, x != y ->
#|[set k : col_squares | k x == k y]| = (n ^ 3)%N. | Proof.
move=> x y nxy; apply/eqP; case: (posnP n) => [n0|].
by rewrite n0; apply/existsP=> [] [p _]; case: (p c0) => i; rewrite n0.
move/eqn_pmul2l <-; rewrite -expnS -card_Fid Fid cardsT.
rewrite -{1}[n]card_ord -cardX.
pose pk k := [ffun i => k (if i == y then x else i) : colors].
rewrite -(@card_image _ _ (fun k :... | Lemma | card_n3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Fid",
"apply",
"c0",
"cardX",
"card_Fid",
"card_image",
"card_ord",
"cardsT",
"col_squares",
"colors",
"eq_card",
"eqn_pmul2l",
"eqxx",
"existsP",
"expnS",
"ffunE",
"ffunP",
"imageP",
"inE",
"last",
"posnP",
"square"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F_Sd2 : 'Fix_to[sd2] = [set x | coin0 x == coin2 x]. | Proof.
apply/setP => x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
by rewrite /act_f sd2_inv !ffunE !permE /= !eqxx !andbT eq_sym /= andbb.
Qed. | Lemma | F_Sd2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"act_f",
"afix1P",
"apply",
"coin0",
"coin2",
"eq_sym",
"eqperm_map",
"eqxx",
"ffunE",
"inE",
"permE",
"sd2",
"sd2_inv",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
burnside_app_iso :
(square_coloring_number8 * 8 = n ^ 4 + 2 * n ^ 3 + 3 * n ^ 2 + 2 * n)%N. | Proof.
pose iso_list := [:: id1; r1; r2; r3; sh; sv; sd1; sd2].
rewrite (burnside_formula iso_list) => [|p|].
- apply: map_uniq (fun p : {perm square} => (p c0, p c1)) _ _.
by rewrite /= !permE.
- by rewrite /= !inE.
rewrite !big_cons big_nil {1}card_Fid F_r1 F_r2 F_r3 F_Sh F_Sv F_Sd1 F_Sd2.
rewrite card_n !card_n3 /... | Lemma | burnside_app_iso | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"ACl",
"F_Sd1",
"F_Sd2",
"F_Sh",
"F_Sv",
"F_r1",
"F_r2",
"F_r3",
"addn",
"addn0",
"addnA",
"apply",
"big_cons",
"big_nil",
"burnside_formula",
"c0",
"c1",
"card_Fid",
"card_n",
"card_n2",
"card_n3",
"id1",
"inE",
"map_uniq",
"permE",
"r1",
"r2",
"r3",
"sd1",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cube | := 'I_6. | Definition | cube | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mkFcube i : cube | := Sub (i %% 6) (ltn_mod i 6). | Definition | mkFcube | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"Sub",
"cube",
"ltn_mod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F0 | := mkFcube 0. | Definition | F0 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"mkFcube"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F1 | := mkFcube 1. | Definition | F1 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"mkFcube"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F2 | := mkFcube 2. | Definition | F2 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"mkFcube"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F3 | := mkFcube 3. | Definition | F3 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"mkFcube"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F4 | := mkFcube 4. | Definition | F4 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"mkFcube"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
F5 | := mkFcube 5. | Definition | F5 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"mkFcube"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S05 | := [:: F0; F4; F3; F2; F1; F5]. | Definition | S05 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | axial symetries | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
S05f (sc : cube) : cube | := tnth [tuple of S05] sc. | Definition | S05f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S05",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S14 | := [:: F5; F1; F3; F2; F4; F0]. | Definition | S14 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S14f (sc : cube) : cube | := tnth [tuple of S14] sc. | Definition | S14f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S14",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S23 | := [:: F5; F4; F2; F3; F1; F0]. | Definition | S23 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
S23f (sc : cube) : cube | := tnth [tuple of S23] sc. | Definition | S23f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"S23",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R05 | := [:: F0; F2; F4; F1; F3; F5]. | Definition | R05 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | rotations 90 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
R05f (sc : cube) : cube | := tnth [tuple of R05] sc. | Definition | R05f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R05",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R50 | := [:: F0; F3; F1; F4; F2; F5]. | Definition | R50 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R50f (sc : cube) : cube | := tnth [tuple of R50] sc. | Definition | R50f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R50",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R14 | := [:: F3; F1; F0; F5; F4; F2]. | Definition | R14 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R14f (sc : cube) : cube | := tnth [tuple of R14] sc. | Definition | R14f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R14",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R41 | := [:: F2; F1; F5; F0; F4; F3]. | Definition | R41 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R41f (sc : cube) : cube | := tnth [tuple of R41] sc. | Definition | R41f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R41",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R23 | := [:: F1; F5; F2; F3; F0; F4]. | Definition | R23 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R23f (sc : cube) : cube | := tnth [tuple of R23] sc. | Definition | R23f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R23",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R32 | := [:: F4; F0; F2; F3; F5; F1]. | Definition | R32 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R32f (sc : cube) : cube | := tnth [tuple of R32] sc. | Definition | R32f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R32",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R024 | := [:: F2; F5; F4; F1; F0; F3]. | Definition | R024 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | rotations 120 | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
R024f (sc : cube) : cube | := tnth [tuple of R024] sc. | Definition | R024f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R024",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R042 | := [:: F4; F3; F0; F5; F2; F1]. | Definition | R042 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R042f (sc : cube) : cube | := tnth [tuple of R042] sc. | Definition | R042f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R042",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R012 | := [:: F1; F2; F0; F5; F3; F4]. | Definition | R012 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R012f (sc : cube) : cube | := tnth [tuple of R012] sc. | Definition | R012f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R012",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R021 | := [:: F2; F0; F1; F4; F5; F3]. | Definition | R021 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R021f (sc : cube) : cube | := tnth [tuple of R021] sc. | Definition | R021f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R021",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R031 | := [:: F3; F0; F4; F1; F5; F2]. | Definition | R031 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R031f (sc : cube) : cube | := tnth [tuple of R031] sc. | Definition | R031f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R031",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R013 | := [:: F1; F3; F5; F0; F2; F4]. | Definition | R013 | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"F0",
"F1",
"F2",
"F3",
"F4",
"F5"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
R013f (sc : cube) : cube | := tnth [tuple of R013] sc. | Definition | R013f | solvable | solvable/burnside_app.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"finset",
"fingroup",
"action",
"perm",
"primitive_action",
"ssrAC"
] | [
"R013",
"cube",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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