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 hs; 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 | 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 permE /S0f (tnth_nth F0).
have e05: F0 = s0 F5 by apply/eqP; rewrite permE /S0f (tnth_nth F0).
rewrite e23 e14 e05; rewrite !hr !hs.
move/perm_inj=> hrs3; move/perm_inj=> hrs4; move/perm_inj=> hrs5.
by apply/eqP; rewrite eqperm /= hrs0 hrs1 hrs2 hrs3 hrs4 hrs5 !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 | 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 * r14 * r05 * r05 by iso_tac.
have -> : r23 = r05 * r14 * r05 * r14 * r14 by iso_tac.
have -> : r50 = r05 * r05 * r05 by iso_tac.
have -> : r41 = r14 * r14 * r14 by iso_tac.
have -> : r32 = r14 * r14 * r14 * r05* r14 by iso_tac.
have -> : r024 = r05 * r14 * r14 * r14 by iso_tac.
have -> : r042 = r14 * r05 * r05 * r05 by iso_tac.
have -> : r012 = r14 * r05 by iso_tac.
have -> : r021 = r05 * r14 * r05 * r05 by iso_tac.
have -> : r031 = r05 * r14 by iso_tac.
have -> : r013 = r05 * r05 * r14 * r05 by iso_tac.
have -> : r043 = r14 * r14 * r14 * r05 by iso_tac.
have -> : r034 = r05 * r05 * r05 * r14 by iso_tac.
have -> : s1 = r14 * r14 * r05 by iso_tac.
have -> : s2 = r05 * r14 * r14 by iso_tac.
have -> : s3 = r05 * r14 * r05 by iso_tac.
have -> : s4 = r05 * r14 * r14 * r14 * r05 by iso_tac.
have -> : s5 = r14 * r05 * r05 by iso_tac.
have -> : s6 = r05 * r05 * r14 by iso_tac.
by do ?case/predU1P=> [<-|]; first exact: group1; last (move/eqP<-);
rewrite ?groupMl ?mem_gen // !inE eqxx ?orbT.
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 | 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: {+}(_ == _)=> //= ].
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_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: {+}(_ == _)=> //= ].
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_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; 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_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 ?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_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/col4.
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_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/col4.
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_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/col2/col3/col4/col5.
by do 4![case: eqVneq=> E; rewrite ?andbF // ?{}E].
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_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/col2/col3/col4/col5.
by do 4![case: eqVneq=> E; rewrite ?andbF // ?{}E].
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_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/col2/col3/col4/col5.
by do 4![case: eqVneq=> E; rewrite ?andbF // ?{}E].
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_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/col2/col3/col4/col5.
do 4![case: eqVneq=> E; rewrite ?andbF // ?{}E].
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_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/col2/col3/col4/col5.
by do 4![case: eqVneq=> E; rewrite ?andbF // ?{}E].
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_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/col2/col3/col4/col5.
by do 4![case: eqVneq=> E; rewrite ?andbF // ?{}E].
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_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/col2/col3/col4/col5.
by do 4![case: eqVneq=> E; rewrite ?andbF // ?{}E].
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_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/col2/col3/col4/col5.
by do 4![case: eqVneq=> E; rewrite ?andbF // ?{}E].
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_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=> E; rewrite ?andbF // ?{}E].
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_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=> E; rewrite ?andbF // ?{}E].
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_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=> E; rewrite ?andbF // ?{}E].
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_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=> E; rewrite ?andbF // ?{}E].
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_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=> E; rewrite ?andbF // ?{}E].
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_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=> E; rewrite ?andbF // ?{}E].
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_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.
have: #|[predC [:: x; y; z; t]]| != 0 by rewrite hcard2.
case/existsP=> u Hu; exists u.
move: (cardC [in [:: x; y; z; t; u]]); rewrite card_ord => hcard5.
have: #|[predC [:: x; y; z; t; u]]| !=0.
rewrite -lt0n -(ltn_add2l #|[:: x; y; z; t; u]|) hcard5 addn0.
by apply: (leq_ltn_trans (card_size [:: x; y; z; t; u])).
case/existsP => v; rewrite (mem_cat _ [:: _; _; _; _]) => /norP[Hv Huv].
exists v; rewrite (cat_uniq [:: x; y; z; t]) Uxt andTb -rev_uniq /= orbF.
by rewrite negb_or Hu Hv Huv.
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 | 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 p2 /[!inE] /andP[p1y p1t] /andP[p2y p2t] [px pz] pu pv.
have eqp12 : all (fun i => p1 i == p2 i) [:: x; y; z; t; u; v].
by rewrite /= -(eqP p1y) -(eqP p1t) -(eqP p2y) -(eqP p2t) px pz pu pv !eqxx.
apply/ffunP=> i; apply/eqP; apply: (allP eqp12).
by rewrite (subset_cardP _ (subset_predT _)) // (card_uniqP Uxv) card_ord.
have -> : forall n, (n ^ 4 = n * n * n * n)%N by move=> ?; rewrite -!mulnA.
rewrite -!card_prod; apply: eq_card => [] [[[c d] e] g] /=; apply/imageP => /=.
move: Uxv; rewrite (cat_uniq [:: x; y; z; t]) => /and3P[_]/=; rewrite orbF.
move=> /norP[] /[!inE] + + /andP[/negPf nuv _].
rewrite orbA => /norP[/negPf nxyu /negPf nztu].
rewrite orbA => /norP[/negPf nxyv /negPf nztv].
move: Uxt; rewrite (cat_uniq [::x; y]) => /and3P[_]/= /[!(andbT, orbF)].
move=> /norP[] /[!inE] /negPf nxyz /negPf nxyt _.
exists [ffun i => if pred2 x y i then c else if pred2 z t i then d
else if u == i then e else g].
by rewrite !(inE, ffunE, eqxx,orbT)//= nxyz nxyt.
by rewrite {}/ff !ffunE /= !eqxx /= nxyz nxyu nztu nxyv nztv nuv.
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_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); first last.
move=> p1 p2 /[!inE]; rewrite -!andbA.
move=> /and3P[/eqP p1xy /eqP p1yz /eqP p1zt].
move=> /and3P[/eqP p2xy /eqP p2yz /eqP p2zt] [px pu] pv.
have eqp12: all (fun i => p1 i == p2 i) [:: x; y; z; t; u; v].
by rewrite /= -p1zt -p2zt -p1yz -p2yz -p1xy -p2xy px pu pv !eqxx.
apply/ffunP=> i; apply/eqP; apply: (allP eqp12).
by rewrite (subset_cardP _ (subset_predT _)) // (card_uniqP Uxv) card_ord.
have -> : forall n, (n ^ 3 = n * n * n)%N by move=> ?; rewrite -!mulnA.
rewrite -!card_prod; apply: eq_card => [] [[c d] e] /=; apply/imageP.
move: Uxv; rewrite (cat_uniq [::x; y; z; t]) => /and3P[_ hasxt].
rewrite /uniq !inE !andbT => /negPf nuv.
exists [ffun i => if i \in [:: x; y; z; t] then c else if u == i then d else e].
by rewrite /= !(inE, ffunE, eqxx, orbT).
rewrite {}/ff !(ffunE, inE, eqxx) /=; move: hasxt; rewrite nuv.
by do 8![case E: ( _ == _ ); rewrite ?(eqP E)/= ?inE ?eqxx //= ?E {E}].
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_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 _ _ ff); first last.
move=> p1 p2 /[!inE]; rewrite -!andbA.
move=> /and4P[/eqP p1xy /eqP p1yz /eqP p1tu /eqP p1uv].
move=> /and4P[/eqP p2xy/eqP p2yz /eqP p2tu /eqP p2uv] [px pu].
have eqp12: all (fun i => p1 i == p2 i) [:: x; y; z; t; u; v].
by rewrite /= -p1yz -p2yz -p1xy -p2xy -p1uv -p2uv -p1tu -p2tu px pu !eqxx.
apply/ffunP=> i; apply/eqP; apply: (allP eqp12).
by rewrite (subset_cardP _ (subset_predT _)) // (card_uniqP Uxv) card_ord.
rewrite -mulnn -!card_prod; apply: eq_card => [] [c d]/=; apply/imageP.
move: Uxv; rewrite (cat_uniq [::x; y; z]) => /= /and3P[Uxt + nuv].
move=> /[!orbF] /norP[] /[!inE] /negPf nxyzt /norP[/negPf nxyzu /negPf nxyzv].
exists [ffun i => if (i \in [:: x; y; z] ) then c else d].
by rewrite /= !(inE, ffunE, eqxx, orbT, nxyzt, nxyzu, nxyzv).
by rewrite {}/ff !ffunE !inE /= !eqxx /= nxyzt.
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_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]; rewrite -!andbA.
move=> /and3P[/eqP p1xy /eqP p1zt /eqP p1uv].
move=> /and3P[/eqP p2xy /eqP p2zt /eqP p2uv] [px pz] pu.
have eqp12: all (fun i => p1 i == p2 i) [:: x; y; z; t; u; v].
by rewrite /= -p1xy -p2xy -p1zt -p2zt -p1uv -p2uv px pz pu !eqxx.
apply/ffunP=> i; apply/eqP; apply: (allP eqp12).
by rewrite (subset_cardP _ (subset_predT _)) // (card_uniqP Uxv) card_ord.
have -> : forall n, (n ^ 3 = n * n * n)%N by move=> ?; rewrite -!mulnA.
rewrite -!card_prod; apply: eq_card => [] [[c d] e] /=; apply/imageP.
move: Uxv; rewrite (cat_uniq [::x; y; z; t]) => /and3P[Uxt + nuv].
move=> /= /[!orbF] /norP[] /[!inE].
rewrite orbA => /norP[/negPf nxyu /negPf nztu].
rewrite orbA => /norP[/negPf nxyv /negPf nztv].
move: Uxt; rewrite (cat_uniq [::x; y]) => /and3P[_].
rewrite /= !orbF !andbT => /norP[] /[!inE] /negPf nxyz /negPf nxyt _.
exists [ffun i => if i \in [:: x; y] then c
else if i \in [:: z; t] then d else e].
by rewrite !(inE, ffunE, eqxx,orbT)//= nxyz nxyt nxyu nztu nxyv nztv !eqxx.
by rewrite {}/ff !ffunE !inE /= !eqxx nxyz nxyu nztu.
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_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 (burnside_formula iso_list); last first.
- by move=> p; rewrite !inE /= !(eq_sym _ p).
- apply: map_uniq (fun p : {perm cube} => (p F0, p F1)) _ _.
have bsr : (fun p : {perm cube} => (p F0, p F1)) =1
(fun p => (nth F0 p F0, nth F0 p F1)) \o sop.
by move=> x; rewrite /= -2!sop_spec.
by rewrite (eq_map bsr) map_comp -(eqP Lcorrect); vm_compute.
rewrite !big_cons big_nil {1}card_Fid3 /= F_s05 F_s14 F_s23 F_r05 F_r14 F_r23
F_r50 F_r41 F_r32 F_r024 F_r042 F_r012 F_r021 F_r031 F_r013 F_r043 F_r034
F_s1 F_s2 F_s3 F_s4 F_s5 F_s6.
rewrite !card_n4 // !card_n3_3 // !card_n2_3 // !card_n3s //.
by rewrite [RHS]addn.[ACl 1 * 3 * 2 * 4 * 5] !addnA !addn0.
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 | 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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