fact stringlengths 8 1.54k | type stringclasses 19
values | library stringclasses 8
values | imports listlengths 1 10 | filename stringclasses 98
values | symbolic_name stringlengths 1 42 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
dir_iso3l:= [:: id3; s05; s14; s23; r05; r14; r23; r50; r41;
r32; r024; r042; r012; r021; r031; r013; r043; r034;
s1; s2; s3; s4; s5; s6]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | dir_iso3l | |
S0:= [:: F5; F4; F3; F2; F1; F0]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | S0 | |
S0f(sc : cube) : cube := tnth [tuple of S0] sc. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | S0f | |
S0_inv: involutive S0f.
Proof. by move=> z; apply/eqP; case: z; do 6?case. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | S0_inv | |
s0:= (perm (inv_inj S0_inv)). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | s0 | |
is_iso3(p : {perm cube}) := forall fi, p (s0 fi) = s0 (p fi). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | is_iso3 | |
dir_iso_iso3: forall p, p \in dir_iso3 -> is_iso3 p.
Proof.
move=> p; rewrite inE.
by do ?case/orP; move/eqP=> <- a; rewrite !permE; case: a; do 6?case.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | dir_iso_iso3 | |
iso3_ndir: forall p, p \in dir_iso3 -> is_iso3 (s0 * p).
Proof.
move=> p; rewrite inE.
by do ?case/orP; move/eqP=> <- a; rewrite !(permM, permE); case: a; do 6?case.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | iso3_ndir | |
sop(p : {perm cube}) : seq cube := fgraph (val p). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | sop | |
sop_inj: injective sop.
Proof. by move=> p1 p2 /val_inj/(can_inj fgraphK)/val_inj. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | sop_inj | |
prod_tuple(t1 t2 : seq cube) :=
map (fun n : 'I_6 => nth F0 t2 n) t1. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | prod_tuple | |
sop_specx (n0 : 'I_6): nth F0 (sop x) n0 = x n0.
Proof. by rewrite nth_fgraph_ord pvalE. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | sop_spec | |
prod_t_correct: forall (x y : {perm cube}) (i : cube),
(x * y) i = nth F0 (prod_tuple (sop x) (sop y)) i.
Proof.
move=> x y i; rewrite permM -!sop_spec [RHS](nth_map F0) // size_tuple /=.
by rewrite card_ord ltn_ord.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | prod_t_correct | |
sop_morph: {morph sop : x y / x * y >-> prod_tuple x y}.
Proof.
move=> x y; apply: (@eq_from_nth _ F0) => [|/= i].
by rewrite size_map !size_tuple.
rewrite size_tuple card_ord => lti6.
by rewrite -[i]/(val (Ordinal lti6)) sop_spec -prod_t_correct.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | sop_morph | |
ecubes: seq cube := [:: F0; F1; F2; F3; F4; F5]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | ecubes | |
ecubes_def: ecubes = enum (@predT cube).
Proof. by apply: (inj_map val_inj); rewrite val_enum_ord. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | ecubes_def | |
seq_iso_L:= [::
[:: F0; F1; F2; F3; F4; F5];
S05; S14; S23; R05; R14; R23; R50; R41; R32;
R024; R042; R012; R021; R031; R013; R043; R034;
S1; S2; S3; S4; S5; S6]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | seq_iso_L | |
seqs1: forall f injf, sop (@perm _ f injf) = map f ecubes.
Proof.
move=> f ?; rewrite ecubes_def /sop /= -codom_ffun pvalE.
by apply: eq_codom; apply: permE.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | seqs1 | |
Lcorrect: seq_iso_L == map sop [:: id3; s05; s14; s23; r05; r14; r23;
r50; r41; r32; r024; r042; r012; r021; r031; r013; r043; r034;
s1; s2; s3; s4; s5; s6].
Proof. by rewrite /= !seqs1. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | Lcorrect | |
iso0_1: dir_iso3 =i dir_iso3l.
Proof. by move=> p; rewrite /= !inE /= -!(eq_sym p). Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | iso0_1 | |
L_iso: forall p, (p \in dir_iso3) = (sop p \in seq_iso_L).
Proof.
by move=> p; rewrite (eqP Lcorrect) mem_map ?iso0_1 //; apply: sop_inj.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | L_iso | |
stable: forall x y,
x \in dir_iso3 -> y \in dir_iso3 -> x * y \in dir_iso3.
Proof.
move=> x y; rewrite !L_iso sop_morph => Hx Hy.
by move/sop: y Hy; apply/allP; move/sop: x Hx; apply/allP; vm_compute.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | stable | |
iso_eq_F0_F1: forall r s : {perm cube}, r \in dir_iso3 ->
s \in dir_iso3 -> r F0 = s F0 -> r F1 = s F1 -> r = s.
Proof.
move=> r s; rewrite !L_iso => hr hs hrs0 hrs1; apply: sop_inj; apply/eqP.
move/eqP: hrs0; apply/implyP; move/eqP: hrs1; apply/implyP; rewrite -!sop_spec.
by move/sop: r hr; apply/allP; move/sop: s h... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | iso_eq_F0_F1 | |
ndir_s0p: forall p, p \in dir_iso3 -> s0 * p \notin dir_iso3.
Proof.
move=> p; rewrite !L_iso sop_morph seqs1.
by move/sop: p; apply/allP; vm_compute.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | ndir_s0p | |
indir_iso3l:= map (mul s0) dir_iso3l. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | indir_iso3l | |
iso3l:= dir_iso3l ++ indir_iso3l. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | iso3l | |
seq_iso3_L:= map sop iso3l. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | seq_iso3_L | |
eqperm: forall p1 p2 : {perm cube},
(p1 == p2) = all (fun s => p1 s == p2 s) ecubes.
Proof.
move=> p1 p2; apply/eqP/allP=> [-> // | Ep12]; apply/permP=> x.
by apply/eqP; rewrite Ep12 // ecubes_def mem_enum.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | eqperm | |
iso_eq_F0_F1_F2: forall r s : {perm cube}, is_iso3 r ->
is_iso3 s -> r F0 = s F0 -> r F1 = s F1 -> r F2 = s F2 -> r = s.
Proof.
move=> r s hr hs hrs0 hrs1 hrs2.
have:= hrs0; have:= hrs1; have:= hrs2.
have e23: F2 = s0 F3 by apply/eqP; rewrite permE /S0f (tnth_nth F0).
have e14: F1 = s0 F4 by apply/eqP; rewrite perm... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | iso_eq_F0_F1_F2 | |
iso_tac:=
let a := fresh "a" in apply/permP => a;
apply/eqP; rewrite !permM !permE; case: a; do 6?case. | Ltac | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | iso_tac | |
inv_tac:=
apply: esym (etrans _ (mul1g _)); apply: canRL (mulgK _) _; iso_tac. | Ltac | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | inv_tac | |
dir_s0p: forall p, (s0 * p) \in dir_iso3 -> p \notin dir_iso3.
Proof.
move=> p Hs0p; move: (ndir_s0p Hs0p); rewrite mulgA.
have e: (s0^-1=s0) by inv_tac.
by rewrite -{1}e mulVg mul1g.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | dir_s0p | |
is_iso3bp := (p * s0 == s0 * p). | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | is_iso3b | |
iso3:= [set p | is_iso3b p]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | iso3 | |
is_iso3P: forall p, reflect (is_iso3 p) (p \in iso3).
Proof.
move=> p; apply: (iffP idP); rewrite inE /iso3 /is_iso3b /is_iso3 => e.
by move=> fi; rewrite -!permM (eqP e).
by apply/eqP; apply/permP=> z; rewrite !permM (e z).
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | is_iso3P | |
group_set_iso3: group_set iso3.
Proof.
apply/group_setP; split.
by apply/is_iso3P => fi; rewrite -!permM mulg1 mul1g.
move=> x1 y; rewrite /iso3 !inE /= /is_iso3.
rewrite /is_iso3b.
rewrite -mulgA.
move/eqP => hx1; move/eqP => hy.
rewrite hy !mulgA. by rewrite -hx1.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | group_set_iso3 | |
iso_group3:= Group group_set_iso3. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | iso_group3 | |
group_set_diso3: group_set dir_iso3.
Proof.
apply/group_setP; split; first by rewrite inE eqxx /=.
by apply: stable.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | group_set_diso3 | |
diso_group3:= Group group_set_diso3. | Canonical | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | diso_group3 | |
gen_diso3: dir_iso3 = <<[set r05; r14]>>.
Proof.
apply/setP/subset_eqP/andP; split; first last.
rewrite gen_subG; apply/subsetP.
by move=> x /[!inE] /orP[] /eqP->; rewrite !eqxx !orbT.
apply/subsetP => x /[!inE].
have -> : s05 = r05 * r05 by iso_tac.
have -> : s14 = r14 * r14 by iso_tac.
have -> : s23 = r14 * r1... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | gen_diso3 | |
col_cubes:= {ffun cube -> colors}. | Notation | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | col_cubes | |
act_g(sc : col_cubes) (p : {perm cube}) : col_cubes :=
[ffun z => sc (p^-1 z)]. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | act_g | |
act_g_1: forall k, act_g k 1 = k.
Proof. by move=> k; apply/ffunP=> a; rewrite ffunE invg1 permE. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | act_g_1 | |
act_g_morph: forall k x y, act_g k (x * y) = act_g (act_g k x) y.
Proof. by move=> k x y; apply/ffunP=> a; rewrite !ffunE invMg permE. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | act_g_morph | |
to_g:= TotalAction act_g_1 act_g_morph. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | to_g | |
cube_coloring_number24:= #|orbit to_g diso_group3 @: setT|. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | cube_coloring_number24 | |
Fid3: 'Fix_to_g[1] = setT.
Proof. by apply/setP=> x /=; rewrite (sameP afix1P eqP) !inE act1 eqxx. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | Fid3 | |
card_Fid3: #|'Fix_to_g[1]| = (n ^ 6)%N.
Proof.
rewrite -[6]card_ord -[n]card_ord -card_ffun_on Fid3 cardsT.
by symmetry; apply: eq_card => ff; apply/ffun_onP.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | card_Fid3 | |
col0(sc : col_cubes) : colors := sc F0. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | col0 | |
col1(sc : col_cubes) : colors := sc F1. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | col1 | |
col2(sc : col_cubes) : colors := sc F2. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | col2 | |
col3(sc : col_cubes) : colors := sc F3. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | col3 | |
col4(sc : col_cubes) : colors := sc F4. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | col4 | |
col5(sc : col_cubes) : colors := sc F5. | Definition | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | col5 | |
eqperm_map2: forall p1 p2 : col_cubes,
(p1 == p2) = all (fun s => p1 s == p2 s) [:: F0; F1; F2; F3; F4; F5].
Proof.
move=> p1 p2; apply/eqP/allP=> [-> // | Ep12]; apply/ffunP=> x.
by apply/eqP; apply Ep12; case: x; do 6?case.
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | eqperm_map2 | |
infE:= (sameP afix1P eqP). | Notation | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | infE | |
F_s05:
'Fix_to_g[s05] = [set x | (col1 x == col4 x) && (col2 x == col3 x)].
Proof.
have s05_inv: s05^-1=s05 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g s05_inv !ffunE !permE /=.
apply sym_equal; rewrite !eqxx /= andbT/col1/col2/col3/col4/col5/col0.
by do 2![rewrite eq_sym; case: {+}(_ == _)=>... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_s05 | |
F_s14:
'Fix_to_g[s14]= [set x | (col0 x == col5 x) && (col2 x == col3 x)].
Proof.
have s14_inv: s14^-1=s14 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g s14_inv !ffunE !permE /=.
apply sym_equal; rewrite !eqxx /= andbT/col1/col2/col3/col4/col5/col0.
by do 2![rewrite eq_sym; case: {+}(_ == _)=... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_s14 | |
r05_inv: r05^-1 = r50.
Proof. by inv_tac. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | r05_inv | |
r50_inv: r50^-1 = r05.
Proof. by inv_tac. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | r50_inv | |
r14_inv: r14^-1 = r41.
Proof. by inv_tac. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | r14_inv | |
r41_inv: r41^-1 = r14.
Proof. by inv_tac. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | r41_inv | |
s23_inv: s23^-1 = s23.
Proof. by inv_tac. Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | s23_inv | |
F_s23:
'Fix_to_g[s23] = [set x | (col0 x == col5 x) && (col1 x == col4 x)].
Proof.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g s23_inv !ffunE !permE /=.
apply sym_equal; rewrite !eqxx /= andbT/col1/col2/col3/col4/col5/col0.
by do 2![rewrite eq_sym; case: {+}(_ == _)=> //=].
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_s23 | |
F_r05: 'Fix_to_g[r05]=
[set x | (col1 x == col2 x) && (col2 x == col3 x)
&& (col3 x == col4 x)].
Proof.
apply sym_equal.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r05_inv !ffunE !permE /=.
rewrite !eqxx /= !andbT /col1/col2/col3/col4/col5/col0.
by do 3![case: eqVneq; rew... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r05 | |
F_r50: 'Fix_to_g[r50]=
[set x | (col1 x == col2 x) && (col2 x == col3 x)
&& (col3 x == col4 x)].
Proof.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r50_inv !ffunE !permE /=.
apply sym_equal; rewrite !eqxx /= !andbT /col1/col2/col3/col4.
by do 3![case: eqVneq; rewrite ?andb... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r50 | |
F_r23: 'Fix_to_g[r23] =
[set x | (col0 x == col1 x) && (col1 x == col4 x)
&& (col4 x == col5 x)].
Proof.
have r23_inv: r23^-1 = r32 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r23_inv !ffunE !permE /=.
apply sym_equal; rewrite !eqxx /= !andbT /col1/col0/col5/co... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r23 | |
F_r32: 'Fix_to_g[r32] =
[set x | (col0 x == col1 x) && (col1 x == col4 x)
&& (col4 x == col5 x)].
Proof.
have r32_inv: r32^-1 = r23 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r32_inv !ffunE !permE /=.
apply sym_equal; rewrite !eqxx /= !andbT /col1/col0/col5/co... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r32 | |
F_r14: 'Fix_to_g[r14] =
[set x | (col0 x == col2 x) && (col2 x == col3 x) && (col3 x == col5 x)].
Proof.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r14_inv !ffunE !permE /=.
apply sym_equal; rewrite !eqxx /= !andbT /col2/col0/col5/col3.
by do 3![case: eqVneq; rewrite ?andbF // => <-].
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r14 | |
F_r41: 'Fix_to_g[r41] =
[set x | (col0 x == col2 x) && (col2 x == col3 x) && (col3 x == col5 x)].
Proof.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r41_inv !ffunE !permE /=.
apply sym_equal; rewrite !eqxx /= !andbT /col2/col0/col5/col3.
by do 3![case: eqVneq; rewrite ?andbF // => <-].
Qed. | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r41 | |
F_r024: 'Fix_to_g[r024] =
[set x | (col0 x == col4 x) && (col4 x == col2 x) && (col1 x == col3 x)
&& (col3 x == col5 x) ].
Proof.
have r024_inv: r024^-1 = r042 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r024_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/co... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r024 | |
F_r042: 'Fix_to_g[r042] =
[set x | (col0 x == col4 x) && (col4 x == col2 x) && (col1 x == col3 x)
&& (col3 x == col5 x)].
Proof.
have r042_inv: r042^-1 = r024 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r042_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r042 | |
F_r012: 'Fix_to_g[r012] =
[set x | (col0 x == col2 x) && (col2 x == col1 x) && (col3 x == col4 x)
&& (col4 x == col5 x)].
Proof.
have r012_inv: r012^-1 = r021 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r012_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r012 | |
F_r021: 'Fix_to_g[r021] =
[set x | (col0 x == col2 x) && (col2 x == col1 x) && (col3 x == col4 x)
&& (col4 x == col5 x)].
Proof.
have r021_inv: r021^-1 = r012 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r021_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r021 | |
F_r031: 'Fix_to_g[r031] =
[set x | (col0 x == col3 x) && (col3 x == col1 x) && (col2 x == col4 x)
&& (col4 x == col5 x)].
Proof.
have r031_inv: r031^-1 = r013 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r031_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r031 | |
F_r013: 'Fix_to_g[r013] =
[set x | (col0 x == col3 x) && (col3 x == col1 x) && (col2 x == col4 x)
&& (col4 x == col5 x)].
Proof.
have r013_inv: r013^-1 = r031 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r013_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r013 | |
F_r043: 'Fix_to_g[r043] =
[set x | (col0 x == col4 x) && (col4 x == col3 x) && (col1 x == col2 x)
&& (col2 x == col5 x)].
Proof.
have r043_inv: r043^-1 = r034 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r043_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r043 | |
F_r034: 'Fix_to_g[r034] =
[set x | (col0 x == col4 x) && (col4 x == col3 x) && (col1 x == col2 x)
&& (col2 x == col5 x)].
Proof.
have r034_inv: r034^-1 = r043 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g r034_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_r034 | |
F_s1: 'Fix_to_g[s1] =
[set x | (col0 x == col5 x) && (col1 x == col2 x) && (col3 x == col4 x)].
Proof.
have s1_inv: s1^-1 = s1 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g s1_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col2/col3/col4/col5.
by do 3![case: eqVneq=>... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_s1 | |
F_s2: 'Fix_to_g[s2] =
[set x | (col0 x == col5 x) && (col1 x == col3 x) && (col2 x == col4 x)].
Proof.
have s2_inv: s2^-1 = s2 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g s2_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col2/col3/col4/col5.
by do 3![case: eqVneq=>... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_s2 | |
F_s3: 'Fix_to_g[s3] =
[set x | (col0 x == col1 x) && (col2 x == col3 x) && (col4 x == col5 x)].
Proof.
have s3_inv: s3^-1 = s3 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g s3_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col2/col3/col4/col5.
by do 3![case: eqVneq=>... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_s3 | |
F_s4: 'Fix_to_g[s4] =
[set x | (col0 x == col4 x) && (col1 x == col5 x) && (col2 x == col3 x)].
Proof.
have s4_inv: s4^-1 = s4 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g s4_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col2/col3/col4/col5.
by do 3![case: eqVneq=>... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_s4 | |
F_s5: 'Fix_to_g[s5] =
[set x | (col0 x == col2 x) && (col1 x == col4 x) && (col3 x == col5 x)].
Proof.
have s5_inv: s5^-1 = s5 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g s5_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col2/col3/col4/col5.
by do 3![case: eqVneq=>... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_s5 | |
F_s6: 'Fix_to_g[s6] =
[set x | (col0 x == col3 x) && (col1 x == col4 x) && (col2 x == col5 x)].
Proof.
have s6_inv: s6^-1 = s6 by inv_tac.
apply/setP => x; rewrite infE !inE eqperm_map2 /= /act_g s6_inv !ffunE !permE /=.
apply sym_equal; rewrite ?eqxx /= !andbT /col0/col1/col2/col3/col4/col5.
by do 3![case: eqVneq=>... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | F_s6 | |
uniq4_uniq6: forall x y z t : cube,
uniq [:: x; y; z; t] -> exists u, exists v, uniq [:: x; y; z; t; u; v].
Proof.
move=> x y z t Uxt; move: (cardC [in [:: x; y; z; t]]).
rewrite card_ord (card_uniq_tuple Uxt) => hcard.
have hcard2: #|[predC [:: x; y; z; t]]| = 2.
by apply: (@addnI 4); rewrite /injective hcard.
h... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | uniq4_uniq6 | |
card_n4: forall x y z t : cube, uniq [:: x; y; z; t] ->
#|[set p : col_cubes | (p x == p y) && (p z == p t)]| = (n ^ 4)%N.
Proof.
move=> x y z t Uxt; rewrite -[n]card_ord.
case: (uniq4_uniq6 Uxt) => u [v Uxv].
pose ff (p : col_cubes) := (p x, p z, p u, p v).
rewrite -(@card_in_image _ _ ff); first last.
move=> p1 ... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | card_n4 | |
card_n3_3: forall x y z t: cube, uniq [:: x; y; z; t] ->
#|[set p : col_cubes | (p x == p y) && (p y == p z)&& (p z == p t)]|
= (n ^ 3)%N.
Proof.
move=> x y z t Uxt; rewrite -[n]card_ord.
case: (uniq4_uniq6 Uxt) => u [v Uxv].
pose ff (p : col_cubes) := (p x, p u, p v);
rewrite -(@card_in_image _ _ ff); firs... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | card_n3_3 | |
card_n2_3: forall x y z t u v: cube, uniq [:: x; y; z; t; u; v] ->
#|[set p : col_cubes | (p x == p y) && (p y == p z)&& (p t == p u )
&& (p u== p v)]| = (n ^ 2)%N.
Proof.
move=> x y z t u v Uxv; rewrite -[n]card_ord .
pose ff (p : col_cubes) := (p x, p t).
rewrite -(@card_in_image _ _ f... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | card_n2_3 | |
card_n3s: forall x y z t u v: cube, uniq [:: x; y; z; t; u; v] ->
#|[set p : col_cubes | (p x == p y) && (p z == p t)&& (p u == p v )]|
= (n ^ 3)%N.
Proof.
move=> x y z t u v Uxv; rewrite -[n]card_ord .
pose ff (p : col_cubes) := (p x, p z, p u).
rewrite -(@card_in_image _ _ ff); first last.
move=> p1 p2 /[!inE... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | card_n3s | |
burnside_app_iso3:
(cube_coloring_number24 * 24 =
n ^ 6 + 6 * n ^ 3 + 3 * n ^ 4 + 8 * (n ^ 2) + 6 * n ^ 3)%N.
Proof.
pose iso_list := [:: id3; s05; s14; s23; r05; r14; r23; r50; r41; r32;
r024; r042; r012; r021; r031; r013; r043; r034;
s1; s2; s3; s4; s5; s6].
rewrite (bu... | Lemma | solvable | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] | solvable/burnside_app.v | burnside_app_iso3 | |
center(A : {set gT}) := 'C_A(A). | Definition | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] | solvable/center.v | center | |
center_group(G : {group gT}) : {group gT} :=
Eval hnf in [group of center G]. | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] | solvable/center.v | center_group | |
morphim_center: GFunctor.pcontinuous (@center).
Proof. by move=> gT rT G D f; apply: morphim_subcent. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] | solvable/center.v | morphim_center | |
center_igFun:= [igFun by fun _ _ => subsetIl _ _ & morphim_center]. | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] | solvable/center.v | center_igFun | |
center_gFun:= [gFun by morphim_center]. | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] | solvable/center.v | center_gFun | |
center_pgFun:= [pgFun by morphim_center]. | Canonical | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] | solvable/center.v | center_pgFun | |
subcentPA B x : reflect (x \in A /\ centralises x B) (x \in 'C_A(B)).
Proof.
rewrite inE. case: (x \in A); last by right; case.
by apply: (iffP centP) => [|[]].
Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] | solvable/center.v | subcentP | |
subcent_subA B : 'C_A(B) \subset 'N_A(B).
Proof. by rewrite setIS ?cent_sub. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] | solvable/center.v | subcent_sub | |
subcent_normG B : 'N_G(B) \subset 'N('C_G(B)).
Proof. by rewrite normsI ?subIset ?normG // orbC cent_norm. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] | solvable/center.v | subcent_norm | |
subcent_normalG B : 'C_G(B) <| 'N_G(B).
Proof. by rewrite /normal subcent_sub subcent_norm. Qed. | Lemma | solvable | [
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import fintype bigop finset fingroup morphism perm",
"From mathcomp Require Import automorphism quotient action gproduct gfunctor",
"From mathcomp Require Import cyclic"
] | solvable/center.v | subcent_normal |
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