phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
iso_group3	:= Group group_set_iso3.	Canonical	iso_group3	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "group_set_iso3" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_set_diso3 : group_set dir_iso3.	Proof. apply/group_setP; split; first by rewrite inE eqxx /=. by apply: stable. Qed.	Lemma	group_set_diso3	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "apply", "dir_iso3", "eqxx", "group_set", "group_setP", "inE", "split", "stable" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
diso_group3	:= Group group_set_diso3.	Canonical	diso_group3	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "group_set_diso3" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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 = r...	Lemma	gen_diso3	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "apply", "dir_iso3", "eqxx", "gen_subG", "group1", "groupMl", "inE", "iso_tac", "last", "mem_gen", "predU1P", "r012", "r013", "r021", "r024", "r031", "r034", "r042", "r043", "r05", "r14", "r23", "r32", "r41", "r50", "s05", "s1", "s14", "s2", "s23", "s3", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
col_cubes	:= {ffun cube -> colors}.	Notation	col_cubes	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "colors", "cube" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
act_g (sc : col_cubes) (p : {perm cube}) : col_cubes	:= [ffun z => sc (p^-1 z)].	Definition	act_g	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "col_cubes", "cube" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
act_g_1 : forall k, act_g k 1 = k.	Proof. by move=> k; apply/ffunP=> a; rewrite ffunE invg1 permE. Qed.	Lemma	act_g_1	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "ffunE", "ffunP", "invg1", "permE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	act_g_morph	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "ffunE", "ffunP", "invMg", "permE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
to_g	:= TotalAction act_g_1 act_g_morph.	Definition	to_g	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "TotalAction", "act_g_1", "act_g_morph" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cube_coloring_number24	:= #\|orbit to_g diso_group3 @: setT\|.	Definition	cube_coloring_number24	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "diso_group3", "orbit", "setT", "to_g" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fid3 : 'Fix_to_g[1] = setT.	Proof. by apply/setP=> x /=; rewrite (sameP afix1P eqP) !inE act1 eqxx. Qed.	Lemma	Fid3	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act1", "afix1P", "apply", "eqxx", "inE", "setP", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	card_Fid3	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "Fid3", "apply", "card_ffun_on", "card_ord", "cardsT", "eq_card", "ff", "ffun_onP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
col0 (sc : col_cubes) : colors	:= sc F0.	Definition	col0	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "F0", "col_cubes", "colors" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
col1 (sc : col_cubes) : colors	:= sc F1.	Definition	col1	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "F1", "col_cubes", "colors" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
col2 (sc : col_cubes) : colors	:= sc F2.	Definition	col2	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "F2", "col_cubes", "colors" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
col3 (sc : col_cubes) : colors	:= sc F3.	Definition	col3	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "F3", "col_cubes", "colors" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
col4 (sc : col_cubes) : colors	:= sc F4.	Definition	col4	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "F4", "col_cubes", "colors" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
col5 (sc : col_cubes) : colors	:= sc F5.	Definition	col5	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "F5", "col_cubes", "colors" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	eqperm_map2	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "F0", "F1", "F2", "F3", "F4", "F5", "all", "allP", "apply", "col_cubes", "ffunP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
infE	:= (sameP afix1P eqP).	Notation	infE	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "afix1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_s05	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eq_sym", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s05", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_s14	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eq_sym", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s14", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
r05_inv : r05^-1 = r50.	Proof. by inv_tac. Qed.	Lemma	r05_inv	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "inv_tac", "r05", "r50" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
r50_inv : r50^-1 = r05.	Proof. by inv_tac. Qed.	Lemma	r50_inv	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "inv_tac", "r05", "r50" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
r14_inv : r14^-1 = r41.	Proof. by inv_tac. Qed.	Lemma	r14_inv	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "inv_tac", "r14", "r41" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
r41_inv : r41^-1 = r14.	Proof. by inv_tac. Qed.	Lemma	r41_inv	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "inv_tac", "r14", "r41" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
s23_inv : s23^-1 = s23.	Proof. by inv_tac. Qed.	Lemma	s23_inv	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "inv_tac", "s23" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_s23	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eq_sym", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "permE", "s23", "s23_inv", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r05	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "permE", "r05", "r05_inv", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r50	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col1", "col2", "col3", "col4", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "permE", "r50", "r50_inv", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r23	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r23", "r32", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r32	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r23", "r32", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r14	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col2", "col3", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "permE", "r14", "r14_inv", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r41	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col2", "col3", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "permE", "r41", "r41_inv", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r024	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r024", "r042", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r042	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r024", "r042", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r012	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r012", "r021", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r021	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r012", "r021", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r031	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r013", "r031", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r013	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r013", "r031", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r043	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r034", "r043", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_r034	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r034", "r043", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_s1	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s1", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_s2	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s2", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_s3	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s3", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_s4	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s4", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_s5	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s5", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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	F_s6	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s6", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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...	Lemma	uniq4_uniq6	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "addn0", "addnI", "apply", "cardC", "card_ord", "card_size", "card_uniq_tuple", "cat_uniq", "cube", "existsP", "leq_ltn_trans", "lt0n", "ltn_add2l", "mem_cat", "rev_uniq", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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). 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 /=...	Lemma	card_n4	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "all", "allP", "apply", "card_in_image", "card_ord", "card_prod", "card_uniqP", "cat_uniq", "col_cubes", "cube", "eq_card", "eqxx", "ff", "ffunE", "ffunP", "imageP", "inE", "mulnA", "pred2", "subset_cardP", "subset_predT", "uniq", "uniq4_uniq6" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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). 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. h...	Lemma	card_n3_3	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "all", "allP", "apply", "card_in_image", "card_ord", "card_prod", "card_uniqP", "cat_uniq", "col_cubes", "cube", "eq_card", "eqxx", "ff", "ffunE", "ffunP", "imageP", "inE", "mulnA", "subset_cardP", "subset_predT", "uniq", "uniq4_uniq6" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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). 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 =>...	Lemma	card_n2_3	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "all", "allP", "apply", "card_in_image", "card_ord", "card_prod", "card_uniqP", "cat_uniq", "col_cubes", "cube", "eq_card", "eqxx", "ff", "ffunE", "ffunP", "imageP", "inE", "mulnn", "subset_cardP", "subset_predT", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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). 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...	Lemma	card_n3s	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "all", "allP", "apply", "card_in_image", "card_ord", "card_prod", "card_uniqP", "cat_uniq", "col_cubes", "cube", "eq_card", "eqxx", "ff", "ffunE", "ffunP", "imageP", "inE", "mulnA", "subset_cardP", "subset_predT", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
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). - apply: map_uniq (fun p : {perm cube} => (p F0, p F1)) _ _. have bsr : (fun p : {perm c...	Lemma	burnside_app_iso3	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "ACl", "F0", "F1", "F_r012", "F_r013", "F_r021", "F_r024", "F_r031", "F_r034", "F_r042", "F_r043", "F_r05", "F_r14", "F_r23", "F_r32", "F_r41", "F_r50", "F_s05", "F_s1", "F_s14", "F_s2", "F_s23", "F_s3", "F_s4", "F_s5", "F_s6", "Lcorrect", "addn", "addn0", "...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
burnside_app_iso_3_3col: cube_coloring_number24 3 = 57.	Proof. by apply/eqP; rewrite -(@eqn_pmul2r 24) // burnside_app_iso3. Qed.	Corollary	burnside_app_iso_3_3col	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "apply", "burnside_app_iso3", "cube_coloring_number24", "eqn_pmul2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
burnside_app_iso_2_4col: square_coloring_number8 4 = 55.	Proof. by apply/eqP; rewrite -(@eqn_pmul2r 8) // burnside_app_iso. Qed.	Corollary	burnside_app_iso_2_4col	solvable	solvable/burnside_app.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]	[ "apply", "burnside_app_iso", "eqn_pmul2r", "square_coloring_number8" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center (A : {set gT})	:= 'C_A(A).	Definition	center	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_group (G : {group gT}) : {group gT}	:= Eval hnf in [group of center G].	Canonical	center_group	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "center", "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Z' ( A )"	:= (center A) : group_scope.	Notation	''Z' ( A )	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "center" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Z' ( H )"	:= (center_group H) : Group_scope.	Notation	''Z' ( H )	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "center_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_center : GFunctor.pcontinuous (@center).	Proof. by move=> gT rT G D f; apply: morphim_subcent. Qed.	Lemma	morphim_center	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "apply", "center", "gT", "morphim_subcent", "pcontinuous" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_igFun	:= [igFun by fun _ _ => subsetIl _ _ & morphim_center].	Canonical	center_igFun	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "morphim_center", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_gFun	:= [gFun by morphim_center].	Canonical	center_gFun	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "morphim_center" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_pgFun	:= [pgFun by morphim_center].	Canonical	center_pgFun	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "morphim_center" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcentP A 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	subcentP	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "apply", "centP", "centralises", "inE", "last" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent_sub A B : 'C_A(B) \subset 'N_A(B).	Proof. by rewrite setIS ?cent_sub. Qed.	Lemma	subcent_sub	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "cent_sub", "setIS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent_norm G B : 'N_G(B) \subset 'N('C_G(B)).	Proof. by rewrite normsI ?subIset ?normG // orbC cent_norm. Qed.	Lemma	subcent_norm	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "cent_norm", "normG", "normsI", "subIset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent_normal G B : 'C_G(B) <\| 'N_G(B).	Proof. by rewrite /normal subcent_sub subcent_norm. Qed.	Lemma	subcent_normal	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "normal", "subcent_norm", "subcent_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent_char G H K : H \char G -> K \char G -> 'C_H(K) \char G.	Proof. case/charP=> sHG chHG /charP[sKG chKG]; apply/charP. split=> [\|f injf Gf]; first by rewrite subIset ?sHG. by rewrite injm_subcent ?chHG ?chKG. Qed.	Lemma	subcent_char	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "Gf", "apply", "char", "charP", "injf", "injm_subcent", "sHG", "sKG", "split", "subIset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centerP A x : reflect (x \in A /\ centralises x A) (x \in 'Z(A)).	Proof. exact: subcentP. Qed.	Lemma	centerP	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "centralises", "subcentP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_sub A : 'Z(A) \subset A.	Proof. exact: subsetIl. Qed.	Lemma	center_sub	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center1 : 'Z(1) = 1 :> {set gT}.	Proof. exact: gF1. Qed.	Lemma	center1	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "gF1", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
centerC A : {in A, centralised 'Z(A)}.	Proof. by apply/centsP; rewrite centsC subsetIr. Qed.	Lemma	centerC	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "apply", "centralised", "centsC", "centsP", "subsetIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_normal G : 'Z(G) <\| G.	Proof. exact: gFnormal. Qed.	Lemma	center_normal	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "gFnormal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_center_normal H G : H \subset 'Z(G) -> H <\| G.	Proof. by rewrite subsetI centsC /normal => /andP[-> /cents_norm]. Qed.	Lemma	sub_center_normal	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "centsC", "cents_norm", "normal", "subsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_abelian G : abelian 'Z(G).	Proof. by rewrite /abelian subIset // centsC subIset // subxx orbT. Qed.	Lemma	center_abelian	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "abelian", "centsC", "subIset", "subxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_char G : 'Z(G) \char G.	Proof. exact: gFchar. Qed.	Lemma	center_char	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "char", "gFchar" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_idP A : reflect ('Z(A) = A) (abelian A).	Proof. exact: setIidPl. Qed.	Lemma	center_idP	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "abelian", "setIidPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_class_formula G : #\|G\| = #\|'Z(G)\| + \sum_(xG in [set x ^: G \| x in G :\: 'C(G)]) #\|xG\|.	Proof. by rewrite acts_sum_card_orbit ?cardsID // astabsJ normsD ?norms_cent ?normG. Qed.	Lemma	center_class_formula	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "acts_sum_card_orbit", "astabsJ", "cardsID", "normG", "normsD", "norms_cent" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent1P A x y : reflect (y \in A /\ commute x y) (y \in 'C_A[x]).	Proof. rewrite inE; case: (y \in A); last by right; case. by apply: (iffP cent1P) => [\|[]]. Qed.	Lemma	subcent1P	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "apply", "cent1P", "commute", "inE", "last" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent1_id x G : x \in G -> x \in 'C_G[x].	Proof. by move=> Gx; rewrite inE Gx; apply/cent1P. Qed.	Lemma	subcent1_id	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "apply", "cent1P", "inE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent1_sub x G : 'C_G[x] \subset G.	Proof. exact: subsetIl. Qed.	Lemma	subcent1_sub	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent1C x y G : x \in G -> y \in 'C_G[x] -> x \in 'C_G[y].	Proof. by move=> Gx /subcent1P[_ cxy]; apply/subcent1P. Qed.	Lemma	subcent1C	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "apply", "subcent1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent1_cycle_sub x G : x \in G -> <[x]> \subset 'C_G[x].	Proof. by move=> Gx; rewrite cycle_subG ?subcent1_id. Qed.	Lemma	subcent1_cycle_sub	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "cycle_subG", "subcent1_id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent1_cycle_norm x G : 'C_G[x] \subset 'N(<[x]>).	Proof. by rewrite cents_norm // cent_gen cent_set1 subsetIr. Qed.	Lemma	subcent1_cycle_norm	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "cent_gen", "cent_set1", "cents_norm", "subsetIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subcent1_cycle_normal x G : x \in G -> <[x]> <\| 'C_G[x].	Proof. by move=> Gx; rewrite /normal subcent1_cycle_norm subcent1_cycle_sub. Qed.	Lemma	subcent1_cycle_normal	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "normal", "subcent1_cycle_norm", "subcent1_cycle_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cyclic_center_factor_abelian G : cyclic (G / 'Z(G)) -> abelian G.	Proof. case/cyclicP=> a Ga; case: (cosetP a) => /= z Nz def_a. have G_Zz: G :=: 'Z(G) * <[z]>. rewrite -quotientK ?cycle_subG ?quotient_cycle //=. by rewrite -def_a -Ga quotientGK // center_normal. rewrite G_Zz abelianM cycle_abelian center_abelian centsC /= G_Zz. by rewrite subIset ?centS ?orbT ?mulG_subr. Qed.	Lemma	cyclic_center_factor_abelian	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "abelian", "abelianM", "centS", "center_abelian", "center_normal", "centsC", "cosetP", "cycle_abelian", "cycle_subG", "cyclic", "cyclicP", "mulG_subr", "quotientGK", "quotientK", "quotient_cycle", "subIset" ]	Gorenstein. 1.3.4	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cyclic_factor_abelian H G : H \subset 'Z(G) -> cyclic (G / H) -> abelian G.	Proof. move=> sHZ cycGH; apply: cyclic_center_factor_abelian. have /andP[_ nHG]: H <\| G := sub_center_normal sHZ. have [f <-]:= homgP (homg_quotientS nHG (gFnorm _ G) sHZ). exact: morphim_cyclic cycGH. Qed.	Lemma	cyclic_factor_abelian	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "abelian", "apply", "cyclic", "cyclic_center_factor_abelian", "gFnorm", "homgP", "homg_quotientS", "morphim_cyclic", "nHG", "sub_center_normal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_center G : G \subset D -> f @* 'Z(G) = 'Z(f @* G).	Proof. exact: injm_subcent. Qed.	Lemma	injm_center	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "injm_subcent" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_center (aT rT : finGroupType) (G : {group aT}) (H : {group rT}) : G \isog H -> 'Z(G) \isog 'Z(H).	Proof. exact: gFisog. Qed.	Lemma	isog_center	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "aT", "gFisog", "group", "isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_prod H K : K \subset 'C(H) -> 'Z(H) * 'Z(K) = 'Z(H * K).	Proof. move=> cHK; apply/setP=> z; rewrite {3}/center centM !inE. have cKH: H \subset 'C(K) by rewrite centsC. apply/imset2P/and3P=> [[x y /setIP[Hx cHx] /setIP[Ky cKy] ->{z}]\| []]. by rewrite imset2_f ?groupM // ?(subsetP cHK) ?(subsetP cKH). case/imset2P=> x y Hx Ky ->{z}. rewrite groupMr => [\|cHx]; first exact: su...	Lemma	center_prod	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "apply", "cKH", "centM", "center", "centsC", "groupM", "groupMl", "groupMr", "imset2P", "imset2_f", "inE", "setIP", "setP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_cprod A B G : A \* B = G -> 'Z(A) \* 'Z(B) = 'Z(G).	Proof. case/cprodP => [[H K -> ->] <- cHK]. rewrite cprodE ?center_prod //= subIset ?(subset_trans cHK) //. by rewrite centS ?center_sub. Qed.	Lemma	center_cprod	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "centS", "center_prod", "center_sub", "cprodE", "cprodP", "subIset", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_bigcprod I r P (F : I -> {set gT}) G : \big[cprod/1]_(i <- r \| P i) F i = G -> \big[cprod/1]_(i <- r \| P i) 'Z(F i) = 'Z(G).	Proof. elim/big_ind2: _ G => [_ <-\|A B C D IHA IHB G dG\|_ _ G ->]; rewrite ?center1 //. case/cprodP: dG IHA IHB (dG) => [[H K -> ->] _ _] IHH IHK dG. by rewrite (IHH H) // (IHK K) // (center_cprod dG). Qed.	Lemma	center_bigcprod	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "big_ind2", "center1", "center_cprod", "cprod", "cprodP", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cprod_center_id G : G \* 'Z(G) = G.	Proof. by rewrite cprodE ?subsetIr // mulGSid ?center_sub. Qed.	Lemma	cprod_center_id	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "center_sub", "cprodE", "mulGSid", "subsetIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_dprod A B G : A \x B = G -> 'Z(A) \x 'Z(B) = 'Z(G).	Proof. case/dprodP=> [[H1 H2 -> ->] defG cH12 trH12]. move: defG; rewrite -cprodE // => /center_cprod/cprodP[_ /= <- cZ12]. by apply: dprodE; rewrite //= setIAC setIA -setIA trH12 (setIidPl _) ?sub1G. Qed.	Lemma	center_dprod	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "apply", "center_cprod", "cprodE", "cprodP", "defG", "dprodE", "dprodP", "setIA", "setIAC", "setIidPl", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_bigdprod I r P (F: I -> {set gT}) G : \big[dprod/1]_(i <- r \| P i) F i = G -> \big[dprod/1]_(i <- r \| P i) 'Z(F i) = 'Z(G).	Proof. elim/big_ind2: _ G => [_ <-\|A B C D IHA IHB G dG\|_ _ G ->]; rewrite ?center1 //. case/dprodP: dG IHA IHB (dG) => [[H K -> ->] _ _ _] IHH IHK dG. by rewrite (IHH H) // (IHK K) // (center_dprod dG). Qed.	Lemma	center_bigdprod	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "big_ind2", "center1", "center_dprod", "dprod", "dprodP", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Aut_cprod_full G H K : H \* K = G -> 'Z(H) = 'Z(K) -> Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) -> Aut_in (Aut K) 'Z(K) \isog Aut 'Z(K) -> Aut_in (Aut G) 'Z(G) \isog Aut 'Z(G).	Proof. move=> defG eqZHK; have [_ defHK cHK] := cprodP defG. have defZ: 'Z(G) = 'Z(H) by rewrite -defHK -center_prod // eqZHK mulGid. have ziHK: H :&: K = 'Z(K). by apply/eqP; rewrite eqEsubset subsetI -{1 2}eqZHK !center_sub setIS. have AutZP := Aut_sub_fullP (@center_sub gT _). move/AutZP=> AutZHfull /AutZP AutZKfu...	Lemma	Aut_cprod_full	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "Aut", "AutZHfull", "Aut_in", "Aut_sub_fullP", "apply", "center_prod", "center_sub", "cfHK", "cprodP", "cprodmEl", "cprodm_morphism", "defG", "domP", "eqEsubset", "eq_fHK", "eqxx", "fH", "fK", "gH", "gK", "gT", "im_cprodm", "injm_center", "injm_cprodm", "isog", "mor...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_cprod_by & isom 'Z(H) 'Z(K) gz	:= [set xy \| let: (x, y) := xy in (x \in 'Z(H)) && (y == (gz x)^-1)].	Definition	ker_cprod_by	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isoZ : isom 'Z(H) 'Z(K) gz.		Hypothesis	isoZ	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
kerHK	:= ker_cprod_by isoZ.	Let	kerHK	solvable	solvable/center.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]	[ "isoZ", "ker_cprod_by" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

iso_group3

:= Group group_set_iso3.

Canonical

iso_group3

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "group_set_iso3" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_set_diso3 : group_set dir_iso3.

Proof. apply/group_setP; split; first by rewrite inE eqxx /=. by apply: stable. Qed.

Lemma

group_set_diso3

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "apply", "dir_iso3", "eqxx", "group_set", "group_setP", "inE", "split", "stable" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

diso_group3

:= Group group_set_diso3.

Canonical

diso_group3

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "group_set_diso3" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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 = r...

Lemma

gen_diso3

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "apply", "dir_iso3", "eqxx", "gen_subG", "group1", "groupMl", "inE", "iso_tac", "last", "mem_gen", "predU1P", "r012", "r013", "r021", "r024", "r031", "r034", "r042", "r043", "r05", "r14", "r23", "r32", "r41", "r50", "s05", "s1", "s14", "s2", "s23", "s3", ...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

col_cubes

:= {ffun cube -> colors}.

Notation

col_cubes

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "colors", "cube" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

act_g (sc : col_cubes) (p : {perm cube}) : col_cubes

:= [ffun z => sc (p^-1 z)].

Definition

act_g

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "col_cubes", "cube" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

act_g_1 : forall k, act_g k 1 = k.

Proof. by move=> k; apply/ffunP=> a; rewrite ffunE invg1 permE. Qed.

Lemma

act_g_1

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "ffunE", "ffunP", "invg1", "permE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

act_g_morph

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "ffunE", "ffunP", "invMg", "permE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

to_g

:= TotalAction act_g_1 act_g_morph.

Definition

to_g

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "TotalAction", "act_g_1", "act_g_morph" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cube_coloring_number24

:= #|orbit to_g diso_group3 @: setT|.

Definition

cube_coloring_number24

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "diso_group3", "orbit", "setT", "to_g" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fid3 : 'Fix_to_g[1] = setT.

Proof. by apply/setP=> x /=; rewrite (sameP afix1P eqP) !inE act1 eqxx. Qed.

Lemma

Fid3

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act1", "afix1P", "apply", "eqxx", "inE", "setP", "setT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

card_Fid3

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "Fid3", "apply", "card_ffun_on", "card_ord", "cardsT", "eq_card", "ff", "ffun_onP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

col0 (sc : col_cubes) : colors

:= sc F0.

Definition

col0

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "F0", "col_cubes", "colors" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

col1 (sc : col_cubes) : colors

:= sc F1.

Definition

col1

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "F1", "col_cubes", "colors" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

col2 (sc : col_cubes) : colors

:= sc F2.

Definition

col2

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "F2", "col_cubes", "colors" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

col3 (sc : col_cubes) : colors

:= sc F3.

Definition

col3

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "F3", "col_cubes", "colors" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

col4 (sc : col_cubes) : colors

:= sc F4.

Definition

col4

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "F4", "col_cubes", "colors" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

col5 (sc : col_cubes) : colors

:= sc F5.

Definition

col5

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "F5", "col_cubes", "colors" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

eqperm_map2

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "F0", "F1", "F2", "F3", "F4", "F5", "all", "allP", "apply", "col_cubes", "ffunP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

infE

:= (sameP afix1P eqP).

Notation

infE

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "afix1P" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_s05

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eq_sym", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s05", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_s14

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eq_sym", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s14", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

r05_inv : r05^-1 = r50.

Proof. by inv_tac. Qed.

Lemma

r05_inv

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "inv_tac", "r05", "r50" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

r50_inv : r50^-1 = r05.

Proof. by inv_tac. Qed.

Lemma

r50_inv

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "inv_tac", "r05", "r50" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

r14_inv : r14^-1 = r41.

Proof. by inv_tac. Qed.

Lemma

r14_inv

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "inv_tac", "r14", "r41" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

r41_inv : r41^-1 = r14.

Proof. by inv_tac. Qed.

Lemma

r41_inv

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "inv_tac", "r14", "r41" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

s23_inv : s23^-1 = s23.

Proof. by inv_tac. Qed.

Lemma

s23_inv

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "inv_tac", "s23" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_s23

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eq_sym", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "permE", "s23", "s23_inv", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r05

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "permE", "r05", "r05_inv", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r50

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col1", "col2", "col3", "col4", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "permE", "r50", "r50_inv", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r23

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r23", "r32", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r32

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r23", "r32", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r14

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col2", "col3", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "permE", "r14", "r14_inv", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r41

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col2", "col3", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "permE", "r41", "r41_inv", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r024

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r024", "r042", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r042

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r024", "r042", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r012

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r012", "r021", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r021

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r012", "r021", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r031

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r013", "r031", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r013

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r013", "r031", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r043

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r034", "r043", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_r034

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "r034", "r043", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_s1

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s1", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_s2

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s2", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_s3

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s3", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_s4

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s4", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_s5

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s5", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

F_s6

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "act_g", "apply", "col0", "col1", "col2", "col3", "col4", "col5", "eqVneq", "eqperm_map2", "eqxx", "ffunE", "inE", "infE", "inv_tac", "permE", "s6", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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...

Lemma

uniq4_uniq6

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "addn0", "addnI", "apply", "cardC", "card_ord", "card_size", "card_uniq_tuple", "cat_uniq", "cube", "existsP", "leq_ltn_trans", "lt0n", "ltn_add2l", "mem_cat", "rev_uniq", "uniq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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). 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 /=...

Lemma

card_n4

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "all", "allP", "apply", "card_in_image", "card_ord", "card_prod", "card_uniqP", "cat_uniq", "col_cubes", "cube", "eq_card", "eqxx", "ff", "ffunE", "ffunP", "imageP", "inE", "mulnA", "pred2", "subset_cardP", "subset_predT", "uniq", "uniq4_uniq6" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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). 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. h...

Lemma

card_n3_3

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "all", "allP", "apply", "card_in_image", "card_ord", "card_prod", "card_uniqP", "cat_uniq", "col_cubes", "cube", "eq_card", "eqxx", "ff", "ffunE", "ffunP", "imageP", "inE", "mulnA", "subset_cardP", "subset_predT", "uniq", "uniq4_uniq6" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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). 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 =>...

Lemma

card_n2_3

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "all", "allP", "apply", "card_in_image", "card_ord", "card_prod", "card_uniqP", "cat_uniq", "col_cubes", "cube", "eq_card", "eqxx", "ff", "ffunE", "ffunP", "imageP", "inE", "mulnn", "subset_cardP", "subset_predT", "uniq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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). 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...

Lemma

card_n3s

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "all", "allP", "apply", "card_in_image", "card_ord", "card_prod", "card_uniqP", "cat_uniq", "col_cubes", "cube", "eq_card", "eqxx", "ff", "ffunE", "ffunP", "imageP", "inE", "mulnA", "subset_cardP", "subset_predT", "uniq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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). - apply: map_uniq (fun p : {perm cube} => (p F0, p F1)) _ _. have bsr : (fun p : {perm c...

Lemma

burnside_app_iso3

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "ACl", "F0", "F1", "F_r012", "F_r013", "F_r021", "F_r024", "F_r031", "F_r034", "F_r042", "F_r043", "F_r05", "F_r14", "F_r23", "F_r32", "F_r41", "F_r50", "F_s05", "F_s1", "F_s14", "F_s2", "F_s23", "F_s3", "F_s4", "F_s5", "F_s6", "Lcorrect", "addn", "addn0", "...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

burnside_app_iso_3_3col: cube_coloring_number24 3 = 57.

Proof. by apply/eqP; rewrite -(@eqn_pmul2r 24) // burnside_app_iso3. Qed.

Corollary

burnside_app_iso_3_3col

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "apply", "burnside_app_iso3", "cube_coloring_number24", "eqn_pmul2r" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

burnside_app_iso_2_4col: square_coloring_number8 4 = 55.

Proof. by apply/eqP; rewrite -(@eqn_pmul2r 8) // burnside_app_iso. Qed.

Corollary

burnside_app_iso_2_4col

solvable

solvable/burnside_app.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "tuple", "finfun", "bigop", "finset", "fingroup", "action", "perm", "primitive_action", "ssrAC" ]

[ "apply", "burnside_app_iso", "eqn_pmul2r", "square_coloring_number8" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center (A : {set gT})

:= 'C_A(A).

Definition

center

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "gT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_group (G : {group gT}) : {group gT}

:= Eval hnf in [group of center G].

Canonical

center_group

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "center", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''Z' ( A )"

:= (center A) : group_scope.

Notation

''Z' ( A )

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "center" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''Z' ( H )"

:= (center_group H) : Group_scope.

Notation

''Z' ( H )

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "center_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_center : GFunctor.pcontinuous (@center).

Proof. by move=> gT rT G D f; apply: morphim_subcent. Qed.

Lemma

morphim_center

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "apply", "center", "gT", "morphim_subcent", "pcontinuous" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_igFun

:= [igFun by fun _ _ => subsetIl _ _ & morphim_center].

Canonical

center_igFun

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "morphim_center", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_gFun

:= [gFun by morphim_center].

Canonical

center_gFun

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "morphim_center" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_pgFun

:= [pgFun by morphim_center].

Canonical

center_pgFun

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "morphim_center" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcentP A 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

subcentP

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "apply", "centP", "centralises", "inE", "last" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent_sub A B : 'C_A(B) \subset 'N_A(B).

Proof. by rewrite setIS ?cent_sub. Qed.

Lemma

subcent_sub

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "cent_sub", "setIS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent_norm G B : 'N_G(B) \subset 'N('C_G(B)).

Proof. by rewrite normsI ?subIset ?normG // orbC cent_norm. Qed.

Lemma

subcent_norm

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "cent_norm", "normG", "normsI", "subIset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent_normal G B : 'C_G(B) <| 'N_G(B).

Proof. by rewrite /normal subcent_sub subcent_norm. Qed.

Lemma

subcent_normal

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "normal", "subcent_norm", "subcent_sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent_char G H K : H \char G -> K \char G -> 'C_H(K) \char G.

Proof. case/charP=> sHG chHG /charP[sKG chKG]; apply/charP. split=> [|f injf Gf]; first by rewrite subIset ?sHG. by rewrite injm_subcent ?chHG ?chKG. Qed.

Lemma

subcent_char

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "Gf", "apply", "char", "charP", "injf", "injm_subcent", "sHG", "sKG", "split", "subIset" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centerP A x : reflect (x \in A /\ centralises x A) (x \in 'Z(A)).

Proof. exact: subcentP. Qed.

Lemma

centerP

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "centralises", "subcentP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_sub A : 'Z(A) \subset A.

Proof. exact: subsetIl. Qed.

Lemma

center_sub

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center1 : 'Z(1) = 1 :> {set gT}.

Proof. exact: gF1. Qed.

Lemma

center1

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "gF1", "gT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

centerC A : {in A, centralised 'Z(A)}.

Proof. by apply/centsP; rewrite centsC subsetIr. Qed.

Lemma

centerC

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "apply", "centralised", "centsC", "centsP", "subsetIr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_normal G : 'Z(G) <| G.

Proof. exact: gFnormal. Qed.

Lemma

center_normal

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "gFnormal" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_center_normal H G : H \subset 'Z(G) -> H <| G.

Proof. by rewrite subsetI centsC /normal => /andP[-> /cents_norm]. Qed.

Lemma

sub_center_normal

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "centsC", "cents_norm", "normal", "subsetI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_abelian G : abelian 'Z(G).

Proof. by rewrite /abelian subIset // centsC subIset // subxx orbT. Qed.

Lemma

center_abelian

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "abelian", "centsC", "subIset", "subxx" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_char G : 'Z(G) \char G.

Proof. exact: gFchar. Qed.

Lemma

center_char

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "char", "gFchar" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_idP A : reflect ('Z(A) = A) (abelian A).

Proof. exact: setIidPl. Qed.

Lemma

center_idP

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "abelian", "setIidPl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_class_formula G : #|G| = #|'Z(G)| + \sum_(xG in [set x ^: G | x in G :\: 'C(G)]) #|xG|.

Proof. by rewrite acts_sum_card_orbit ?cardsID // astabsJ normsD ?norms_cent ?normG. Qed.

Lemma

center_class_formula

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "acts_sum_card_orbit", "astabsJ", "cardsID", "normG", "normsD", "norms_cent" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent1P A x y : reflect (y \in A /\ commute x y) (y \in 'C_A[x]).

Proof. rewrite inE; case: (y \in A); last by right; case. by apply: (iffP cent1P) => [|[]]. Qed.

Lemma

subcent1P

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "apply", "cent1P", "commute", "inE", "last" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent1_id x G : x \in G -> x \in 'C_G[x].

Proof. by move=> Gx; rewrite inE Gx; apply/cent1P. Qed.

Lemma

subcent1_id

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "apply", "cent1P", "inE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent1_sub x G : 'C_G[x] \subset G.

Proof. exact: subsetIl. Qed.

Lemma

subcent1_sub

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent1C x y G : x \in G -> y \in 'C_G[x] -> x \in 'C_G[y].

Proof. by move=> Gx /subcent1P[_ cxy]; apply/subcent1P. Qed.

Lemma

subcent1C

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "apply", "subcent1P" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent1_cycle_sub x G : x \in G -> <[x]> \subset 'C_G[x].

Proof. by move=> Gx; rewrite cycle_subG ?subcent1_id. Qed.

Lemma

subcent1_cycle_sub

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "cycle_subG", "subcent1_id" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent1_cycle_norm x G : 'C_G[x] \subset 'N(<[x]>).

Proof. by rewrite cents_norm // cent_gen cent_set1 subsetIr. Qed.

Lemma

subcent1_cycle_norm

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "cent_gen", "cent_set1", "cents_norm", "subsetIr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subcent1_cycle_normal x G : x \in G -> <[x]> <| 'C_G[x].

Proof. by move=> Gx; rewrite /normal subcent1_cycle_norm subcent1_cycle_sub. Qed.

Lemma

subcent1_cycle_normal

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "normal", "subcent1_cycle_norm", "subcent1_cycle_sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cyclic_center_factor_abelian G : cyclic (G / 'Z(G)) -> abelian G.

Proof. case/cyclicP=> a Ga; case: (cosetP a) => /= z Nz def_a. have G_Zz: G :=: 'Z(G) * <[z]>. rewrite -quotientK ?cycle_subG ?quotient_cycle //=. by rewrite -def_a -Ga quotientGK // center_normal. rewrite G_Zz abelianM cycle_abelian center_abelian centsC /= G_Zz. by rewrite subIset ?centS ?orbT ?mulG_subr. Qed.

Lemma

cyclic_center_factor_abelian

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "abelian", "abelianM", "centS", "center_abelian", "center_normal", "centsC", "cosetP", "cycle_abelian", "cycle_subG", "cyclic", "cyclicP", "mulG_subr", "quotientGK", "quotientK", "quotient_cycle", "subIset" ]

Gorenstein. 1.3.4

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cyclic_factor_abelian H G : H \subset 'Z(G) -> cyclic (G / H) -> abelian G.

Proof. move=> sHZ cycGH; apply: cyclic_center_factor_abelian. have /andP[_ nHG]: H <| G := sub_center_normal sHZ. have [f <-]:= homgP (homg_quotientS nHG (gFnorm _ G) sHZ). exact: morphim_cyclic cycGH. Qed.

Lemma

cyclic_factor_abelian

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "abelian", "apply", "cyclic", "cyclic_center_factor_abelian", "gFnorm", "homgP", "homg_quotientS", "morphim_cyclic", "nHG", "sub_center_normal" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_center G : G \subset D -> f @* 'Z(G) = 'Z(f @* G).

Proof. exact: injm_subcent. Qed.

Lemma

injm_center

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "injm_subcent" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_center (aT rT : finGroupType) (G : {group aT}) (H : {group rT}) : G \isog H -> 'Z(G) \isog 'Z(H).

Proof. exact: gFisog. Qed.

Lemma

isog_center

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "aT", "gFisog", "group", "isog" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_prod H K : K \subset 'C(H) -> 'Z(H) * 'Z(K) = 'Z(H * K).

Proof. move=> cHK; apply/setP=> z; rewrite {3}/center centM !inE. have cKH: H \subset 'C(K) by rewrite centsC. apply/imset2P/and3P=> [[x y /setIP[Hx cHx] /setIP[Ky cKy] ->{z}]| []]. by rewrite imset2_f ?groupM // ?(subsetP cHK) ?(subsetP cKH). case/imset2P=> x y Hx Ky ->{z}. rewrite groupMr => [|cHx]; first exact: su...

Lemma

center_prod

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "apply", "cKH", "centM", "center", "centsC", "groupM", "groupMl", "groupMr", "imset2P", "imset2_f", "inE", "setIP", "setP", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_cprod A B G : A \* B = G -> 'Z(A) \* 'Z(B) = 'Z(G).

Proof. case/cprodP => [[H K -> ->] <- cHK]. rewrite cprodE ?center_prod //= subIset ?(subset_trans cHK) //. by rewrite centS ?center_sub. Qed.

Lemma

center_cprod

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "centS", "center_prod", "center_sub", "cprodE", "cprodP", "subIset", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_bigcprod I r P (F : I -> {set gT}) G : \big[cprod/1]_(i <- r | P i) F i = G -> \big[cprod/1]_(i <- r | P i) 'Z(F i) = 'Z(G).

Proof. elim/big_ind2: _ G => [_ <-|A B C D IHA IHB G dG|_ _ G ->]; rewrite ?center1 //. case/cprodP: dG IHA IHB (dG) => [[H K -> ->] _ _] IHH IHK dG. by rewrite (IHH H) // (IHK K) // (center_cprod dG). Qed.

Lemma

center_bigcprod

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "big_ind2", "center1", "center_cprod", "cprod", "cprodP", "gT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cprod_center_id G : G \* 'Z(G) = G.

Proof. by rewrite cprodE ?subsetIr // mulGSid ?center_sub. Qed.

Lemma

cprod_center_id

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "center_sub", "cprodE", "mulGSid", "subsetIr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_dprod A B G : A \x B = G -> 'Z(A) \x 'Z(B) = 'Z(G).

Proof. case/dprodP=> [[H1 H2 -> ->] defG cH12 trH12]. move: defG; rewrite -cprodE // => /center_cprod/cprodP[_ /= <- cZ12]. by apply: dprodE; rewrite //= setIAC setIA -setIA trH12 (setIidPl _) ?sub1G. Qed.

Lemma

center_dprod

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "apply", "center_cprod", "cprodE", "cprodP", "defG", "dprodE", "dprodP", "setIA", "setIAC", "setIidPl", "sub1G" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

center_bigdprod I r P (F: I -> {set gT}) G : \big[dprod/1]_(i <- r | P i) F i = G -> \big[dprod/1]_(i <- r | P i) 'Z(F i) = 'Z(G).

Proof. elim/big_ind2: _ G => [_ <-|A B C D IHA IHB G dG|_ _ G ->]; rewrite ?center1 //. case/dprodP: dG IHA IHB (dG) => [[H K -> ->] _ _ _] IHH IHK dG. by rewrite (IHH H) // (IHK K) // (center_dprod dG). Qed.

Lemma

center_bigdprod

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "big_ind2", "center1", "center_dprod", "dprod", "dprodP", "gT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Aut_cprod_full G H K : H \* K = G -> 'Z(H) = 'Z(K) -> Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) -> Aut_in (Aut K) 'Z(K) \isog Aut 'Z(K) -> Aut_in (Aut G) 'Z(G) \isog Aut 'Z(G).

Proof. move=> defG eqZHK; have [_ defHK cHK] := cprodP defG. have defZ: 'Z(G) = 'Z(H) by rewrite -defHK -center_prod // eqZHK mulGid. have ziHK: H :&: K = 'Z(K). by apply/eqP; rewrite eqEsubset subsetI -{1 2}eqZHK !center_sub setIS. have AutZP := Aut_sub_fullP (@center_sub gT _). move/AutZP=> AutZHfull /AutZP AutZKfu...

Lemma

Aut_cprod_full

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "Aut", "AutZHfull", "Aut_in", "Aut_sub_fullP", "apply", "center_prod", "center_sub", "cfHK", "cprodP", "cprodmEl", "cprodm_morphism", "defG", "domP", "eqEsubset", "eq_fHK", "eqxx", "fH", "fK", "gH", "gK", "gT", "im_cprodm", "injm_center", "injm_cprodm", "isog", "mor...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_cprod_by & isom 'Z(H) 'Z(K) gz

:= [set xy | let: (x, y) := xy in (x \in 'Z(H)) && (y == (gz x)^-1)].

Definition

ker_cprod_by

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "isom" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isoZ : isom 'Z(H) 'Z(K) gz.

Hypothesis

isoZ

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "isom" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

kerHK

:= ker_cprod_by isoZ.

Let

kerHK

solvable

solvable/center.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "fintype", "bigop", "finset", "fingroup", "morphism", "perm", "automorphism", "quotient", "action", "gproduct", "gfunctor", "cyclic", "finfun" ]

[ "isoZ", "ker_cprod_by" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d