fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
solvable_AltF: 4 < #|T| -> solvable 'Alt_T = false.
Proof.
move=> card_T; apply/negP => Alt_solvable.
have/simple_Alt5 Alt_simple := card_T.
have := simple_sol_prime Alt_solvable Alt_simple.
have lt_T n : n <= 4 -> n < #|T| by move/leq_ltn_trans; apply.
have -> : #|('Alt_T)%G| = #|T|`! %/ 2 by rewrite -card_Alt ?mulKn ?lt_T.
move/even_prime => [/eqP|]; apply/negP.
rewrite neq_ltn leq_divRL // mulnC -[2 * 3]/(3`!).
by apply/orP; right; apply/ltnW/ltn_fact/lt_T.
by rewrite -dvdn2 dvdn_divRL dvdn_fact //=; apply/ltnW/lt_T.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action cyclic pgroup gseries sylow",
"From mathcomp Require Import primitive_action nilpotent maximal"
] |
solvable/alt.v
|
solvable_AltF
| |
solvable_SymF: 4 < #|T| -> solvable 'Sym_T = false.
Proof. by rewrite (series_sol (Alt_normal T)) => /solvable_AltF->. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype tuple tuple bigop prime finset ssralg",
"From mathcomp Require Import zmodp fingroup morphism perm automorphism quotient",
"From mathcomp Require Import action cyclic pgroup gseries sylow",
"From mathcomp Require Import primitive_action nilpotent maximal"
] |
solvable/alt.v
|
solvable_SymF
| |
burnside_formula: forall (gT : finGroupType) s (G : {group gT}),
uniq s -> s =i G ->
forall (sT : finType) (to : {action gT &-> sT}),
(#|orbit to G @: setT| * size s)%N = \sum_(p <- s) #|'Fix_to[p]|.
Proof.
move=> gT s G Us sG sT to.
rewrite big_uniq // -(card_uniqP Us) (eq_card sG) -Frobenius_Cauchy.
by apply: eq_big => // p _; rewrite setTI.
by apply/actsP=> ? _ ?; rewrite !inE.
Qed.
Arguments burnside_formula {gT}.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
burnside_formula
| |
colors:= 'I_n.
HB.instance Definition _ := Finite.on colors.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
colors
| |
square:= 'I_4.
HB.instance Definition _ := SubType.on square.
HB.instance Definition _ := Finite.on square.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
square
| |
mksquarei : square := Sub (i %% _) (ltn_mod i 4).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
mksquare
| |
c0:= mksquare 0.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
c0
| |
c1:= mksquare 1.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
c1
| |
c2:= mksquare 2.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
c2
| |
c3:= mksquare 3.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
c3
| |
R1(sc : square) : square := tnth [tuple c1; c2; c3; c0] sc.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
R1
| |
R2(sc : square) : square := tnth [tuple c2; c3; c0; c1] sc.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
R2
| |
R3(sc : square) : square := tnth [tuple c3; c0; c1; c2] sc.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
R3
| |
get_invelt l :=
match l with
| (_, (elt, ?x)) => x
| (elt, ?x) => x
| (?x, _) => get_inv elt x
end.
|
Ltac
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
get_inv
| |
rot_inv:= ((R1, R3), (R2, R2), (R3, R1)).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
rot_inv
| |
inj_tac:=
move: (erefl rot_inv); unfold rot_inv;
match goal with |- ?X = _ -> injective ?Y =>
move=> _; let x := get_inv Y X in
apply: (can_inj (g:=x)); move=> [val H1]
end.
|
Ltac
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
inj_tac
| |
R1_inj: injective R1.
Proof. by inj_tac; repeat (destruct val => //=; first by apply/eqP). Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
R1_inj
| |
R2_inj: injective R2.
Proof. by inj_tac; repeat (destruct val => //=; first by apply/eqP). Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
R2_inj
| |
R3_inj: injective R3.
Proof. by inj_tac; repeat (destruct val => //=; first by apply/eqP). Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
R3_inj
| |
r1:= (perm R1_inj).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
r1
| |
r2:= (perm R2_inj).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
r2
| |
r3:= (perm R3_inj).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
r3
| |
id1:= (1 : {perm square}).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
id1
| |
is_rot(r : {perm _}) := (r * r1 == r1 * r).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
is_rot
| |
rot:= [set r | is_rot r].
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
rot
| |
group_set_rot: group_set rot.
Proof.
apply/group_setP; split; first by rewrite /rot inE /is_rot mulg1 mul1g.
move=> x1 y; rewrite /rot !inE /= /is_rot; move/eqP => hx1; move/eqP => hy.
by rewrite -mulgA hy !mulgA hx1.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
group_set_rot
| |
rot_group:= Group group_set_rot.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
rot_group
| |
rotations:= [set id1; r1; r2; r3].
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
rotations
| |
rot_eq_c0: forall r s : {perm square},
is_rot r -> is_rot s -> r c0 = s c0 -> r = s.
Proof.
rewrite /is_rot => r s; move/eqP => hr; move/eqP=> hs hrs; apply/permP => a.
have ->: a = (r1 ^+ a) c0
by apply/eqP; case: a; do 4?case=> //=; rewrite ?permM !permE.
by rewrite -!permM -!commuteX // !permM hrs.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
rot_eq_c0
| |
rot_r1: forall r, is_rot r -> r = r1 ^+ (r c0).
Proof.
move=> r hr; apply: rot_eq_c0 => //; apply/eqP.
by symmetry; apply: commuteX.
by case: (r c0); do 4?case=> //=; rewrite ?permM !permE /=.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
rot_r1
| |
rotations_is_rot: forall r, r \in rotations -> is_rot r.
Proof.
move=> r Dr; apply/eqP; apply/permP => a; rewrite !inE -!orbA !permM in Dr *.
by case/or4P: Dr; move/eqP->; rewrite !permE //; case: a; do 4?case.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
rotations_is_rot
| |
rot_is_rot: rot = rotations.
Proof.
apply/setP=> r; apply/idP/idP => [|/rotations_is_rot] /[!inE]// h.
have -> : r = r1 ^+ (r c0) by apply: rot_eq_c0; rewrite // -rot_r1.
have e2: 2 = r2 c0 by rewrite permE /=.
have e3: 3 = r3 c0 by rewrite permE /=.
case (r c0); do 4?[case] => // ?; rewrite ?(expg1, eqxx, orbT) //.
by rewrite [nat_of_ord _]/= e2 -rot_r1 ?(eqxx, orbT, rotations_is_rot, inE).
by rewrite [nat_of_ord _]/= e3 -rot_r1 ?(eqxx, orbT, rotations_is_rot, inE).
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
rot_is_rot
| |
Sh(sc : square) : square := tnth [tuple c1; c0; c3; c2] sc.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
Sh
| |
Sh_inj: injective Sh.
Proof.
by apply: (can_inj (g:= Sh)); case; do 4?case=> //=; move=> H; apply/eqP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
Sh_inj
| |
sh:= (perm Sh_inj).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
sh
| |
sh_inv: sh^-1 = sh.
Proof.
apply: (mulIg sh); rewrite mulVg; apply/permP.
by case; do 4?case=> //=; move=> H; rewrite !permE /= !permE; apply/eqP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
sh_inv
| |
Sv(sc : square) : square := tnth [tuple c3; c2; c1; c0] sc.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
Sv
| |
Sv_inj: injective Sv.
Proof.
by apply: (can_inj (g:= Sv)); case; do 4?case=> //=; move=> H; apply/eqP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
Sv_inj
| |
sv:= (perm Sv_inj).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
sv
| |
sv_inv: sv^-1 = sv.
Proof.
apply: (mulIg sv); rewrite mulVg; apply/permP.
by case; do 4?case=> //=; move=> H; rewrite !permE /= !permE; apply/eqP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
sv_inv
| |
Sd1(sc : square) : square := tnth [tuple c0; c3; c2; c1] sc.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
Sd1
| |
Sd1_inj: injective Sd1.
Proof.
by apply: can_inj Sd1 _; case; do 4?case=> //=; move=> H; apply/eqP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
Sd1_inj
| |
sd1:= (perm Sd1_inj).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
sd1
| |
sd1_inv: sd1^-1 = sd1.
Proof.
apply: (mulIg sd1); rewrite mulVg; apply/permP.
by case; do 4?case=> //=; move=> H; rewrite !permE /= !permE; apply/eqP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
sd1_inv
| |
Sd2(sc : square) : square := tnth [tuple c2; c1; c0; c3] sc.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
Sd2
| |
Sd2_inj: injective Sd2.
Proof.
by apply: can_inj Sd2 _; case; do 4?case=> //=; move=> H; apply/eqP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
Sd2_inj
| |
sd2:= (perm Sd2_inj).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
sd2
| |
sd2_inv: sd2^-1 = sd2.
Proof.
apply: (mulIg sd2); rewrite mulVg; apply/permP.
by case; do 4?case=> //=; move=> H; rewrite !permE /= !permE; apply/eqP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
sd2_inv
| |
ord_enum4: enum 'I_4 = [:: c0; c1; c2; c3].
Proof. by apply: (inj_map val_inj); rewrite val_enum_ord. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
ord_enum4
| |
diff_id_sh: 1 != sh.
Proof.
by apply/eqP; move/(congr1 (fun p : {perm square} => p c0)); rewrite !permE.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
diff_id_sh
| |
isometries2:= [set 1; sh].
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
isometries2
| |
card_iso2: #|isometries2| = 2.
Proof. by rewrite cards2 diff_id_sh. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
card_iso2
| |
group_set_iso2: group_set isometries2.
Proof.
apply/group_setP; split => [|x y]; rewrite !inE ?eqxx //.
do 2![case/orP; move/eqP->]; rewrite ?(mul1g, mulg1) ?eqxx ?orbT//.
by rewrite -/sh -{1}sh_inv mulVg eqxx.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
group_set_iso2
| |
iso2_group:= Group group_set_iso2.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
iso2_group
| |
isometries:=
[set p | [|| p == 1, p == r1, p == r2, p == r3,
p == sh, p == sv, p == sd1 | p == sd2 ]].
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
isometries
| |
opp(sc : square) := tnth [tuple c2; c3; c0; c1] sc.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
opp
| |
is_iso(p : {perm square}) := forall ci, p (opp ci) = opp (p ci).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
is_iso
| |
isometries_iso: forall p, p \in isometries -> is_iso p.
Proof.
move=> p; rewrite inE.
by do ?case/orP; move/eqP=> -> a; rewrite !permE; case: a; do 4?case.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
isometries_iso
| |
non_injp a1 a2 heq1 heq2 :=
let h1:= fresh "h1" in
(absurd (p a1 = p a2); first (by red => - h1; move: (perm_inj h1));
by rewrite heq1 heq2; apply/eqP).
|
Ltac
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
non_inj
| |
is_isoPtacp f e0 e1 e2 e3 :=
suff ->: p = f by [rewrite inE eqxx ?orbT];
let e := fresh "e" in apply/permP;
(do 5?[case] => // ?; [move: e0 | move: e1 | move: e2 | move: e3]) => e;
apply: etrans (congr1 p _) (etrans e _); apply/eqP; rewrite // permE.
|
Ltac
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
is_isoPtac
| |
is_isoP: forall p, reflect (is_iso p) (p \in isometries).
Proof.
move=> p; apply: (iffP idP) => [|iso_p]; first exact: isometries_iso.
move e1: (p c1) (iso_p c1) => k1; move e0: (p c0) (iso_p c0) k1 e1 => k0.
case: k0 e0; do 4?[case] => //= ? e0 e2; do 5?[case] => //= ? e1 e3;
try by [non_inj p c0 c1 e0 e1 | non_inj p c0 c3 e0 e3].
by is_isoPtac p id1 e0 e1 e2 e3.
by is_isoPtac p sd1 e0 e1 e2 e3.
by is_isoPtac p sh e0 e1 e2 e3.
by is_isoPtac p r1 e0 e1 e2 e3.
by is_isoPtac p sd2 e0 e1 e2 e3.
by is_isoPtac p r2 e0 e1 e2 e3.
by is_isoPtac p r3 e0 e1 e2 e3.
by is_isoPtac p sv e0 e1 e2 e3.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
is_isoP
| |
group_set_iso: group_set isometries.
Proof.
apply/group_setP; split; first by rewrite inE eqxx /=.
by move=> x y hx hy; apply/is_isoP => ci; rewrite !permM !isometries_iso.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
group_set_iso
| |
iso_group:= Group group_set_iso.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
iso_group
| |
card_rot: #|rot| = 4.
Proof.
rewrite -[4]/(size [:: id1; r1; r2; r3]) -(card_uniqP _).
by apply: eq_card => x; rewrite rot_is_rot !inE -!orbA.
by apply: map_uniq (fun p : {perm square} => p c0) _ _; rewrite /= !permE.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
card_rot
| |
group_set_rotations: group_set rotations.
Proof. by rewrite -rot_is_rot group_set_rot. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
group_set_rotations
| |
rotations_group:= Group group_set_rotations.
|
Canonical
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
rotations_group
| |
col_squares:= {ffun square -> colors}.
|
Notation
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
col_squares
| |
act_f(sc : col_squares) (p : {perm square}) : col_squares :=
[ffun z => sc (p^-1 z)].
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
act_f
| |
act_f_1: forall k, act_f k 1 = k.
Proof. by move=> k; apply/ffunP=> a; rewrite ffunE invg1 permE. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
act_f_1
| |
act_f_morph: forall k x y, act_f k (x * y) = act_f (act_f k x) y.
Proof. by move=> k x y; apply/ffunP=> a; rewrite !ffunE invMg permE. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
act_f_morph
| |
to:= TotalAction act_f_1 act_f_morph.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
to
| |
square_coloring_number2:= #|orbit to isometries2 @: setT|.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
square_coloring_number2
| |
square_coloring_number4:= #|orbit to rotations @: setT|.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
square_coloring_number4
| |
square_coloring_number8:= #|orbit to isometries @: setT|.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
square_coloring_number8
| |
Fid: 'Fix_to(1) = setT.
Proof. by apply/setP=> x /=; rewrite in_setT; apply/afix1P; apply: act1. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
Fid
| |
card_Fid: #|'Fix_to(1)| = (n ^ 4)%N.
Proof.
rewrite -[4]card_ord -[n]card_ord -card_ffun_on Fid cardsE.
by symmetry; apply: eq_card => f; apply/ffun_onP.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
card_Fid
| |
coin0(sc : col_squares) : colors := sc c0.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
coin0
| |
coin1(sc : col_squares) : colors := sc c1.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
coin1
| |
coin2(sc : col_squares) : colors := sc c2.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
coin2
| |
coin3(sc : col_squares) : colors := sc c3.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
coin3
| |
eqperm_map: forall p1 p2 : col_squares,
(p1 == p2) = all (fun s => p1 s == p2 s) [:: c0; c1; c2; c3].
Proof.
move=> p1 p2; apply/eqP/allP=> [-> // | Ep12]; apply/ffunP=> x.
by apply/eqP; apply Ep12; case: x; do 4!case=> //.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
eqperm_map
| |
F_Sh:
'Fix_to[sh] = [set x | (coin0 x == coin1 x) && (coin2 x == coin3 x)].
Proof.
apply/setP=> x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f sh_inv !ffunE !permE /=.
by rewrite eq_sym (eq_sym (x c3)) andbT andbA !andbb.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
F_Sh
| |
F_Sv:
'Fix_to[sv] = [set x | (coin0 x == coin3 x) && (coin2 x == coin1 x)].
Proof.
apply/setP=> x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f sv_inv !ffunE !permE /=.
by rewrite eq_sym andbT andbC (eq_sym (x c1)) andbA -andbA !andbb andbC.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
F_Sv
| |
inv_tac:=
apply: esym (etrans _ (mul1g _)); apply: canRL (mulgK _) _;
let a := fresh "a" in apply/permP => a;
apply/eqP; rewrite permM !permE; case: a; do 4?case.
|
Ltac
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
inv_tac
| |
r1_inv: r1^-1 = r3.
Proof. by inv_tac. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
r1_inv
| |
r2_inv: r2^-1 = r2.
Proof. by inv_tac. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
r2_inv
| |
r3_inv: r3^-1 = r1.
Proof. by inv_tac. Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
r3_inv
| |
F_r2:
'Fix_to[r2] = [set x | (coin0 x == coin2 x) && (coin1 x == coin3 x)].
Proof.
apply/setP=> x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f r2_inv !ffunE !permE /=.
by rewrite eq_sym andbT andbCA andbC (eq_sym (x c3)) andbA -andbA !andbb andbC.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
F_r2
| |
F_r1: 'Fix_to[r1] =
[set x | (coin0 x == coin1 x)&&(coin1 x == coin2 x)&&(coin2 x == coin3 x)].
Proof.
apply/setP=> x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f r1_inv !ffunE !permE andbC.
by do 3![case E: {+}(_ == _); rewrite // {E}(eqP E)]; rewrite eqxx.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
F_r1
| |
F_r3: 'Fix_to[r3] =
[set x | (coin0 x == coin1 x)&&(coin1 x == coin2 x)&&(coin2 x == coin3 x)].
Proof.
apply/setP=> x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f r3_inv !ffunE !permE /=.
by do 3![case: eqVneq=> // <-].
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
F_r3
| |
card_n2: forall x y z t : square, uniq [:: x; y; z; t] ->
#|[set p : col_squares | (p x == p y) && (p z == p t)]| = (n ^ 2)%N.
Proof.
move=> x y z t Uxt; rewrite -[n]card_ord.
pose f (p : col_squares) := (p x, p z); rewrite -(@card_in_image _ _ f).
rewrite -mulnn -card_prod; apply: eq_card => [] [c d] /=; apply/imageP.
rewrite (cat_uniq [::x; y]) in Uxt; case/and3P: Uxt => _.
rewrite /= !orbF !andbT => /norP[] /[!inE] nxzt nyzt _.
exists [ffun i => if pred2 x y i then c else d].
by rewrite inE !ffunE /= !eqxx orbT (negbTE nxzt) (negbTE nyzt) !eqxx.
by rewrite {}/f !ffunE /= eqxx (negbTE nxzt).
move=> p1 p2 /[!inE] /andP[p1y p1t] /andP[p2y p2t] [px pz].
have eqp12: all (fun i => p1 i == p2 i) [:: x; y; z; t].
by rewrite /= -(eqP p1y) -(eqP p1t) -(eqP p2y) -(eqP p2t) px pz !eqxx.
apply/ffunP=> i; apply/eqP; apply: (allP eqp12).
by rewrite (subset_cardP _ (subset_predT _)) // (card_uniqP Uxt) card_ord.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
card_n2
| |
card_n:
#|[set x | (coin0 x == coin1 x)&&(coin1 x == coin2 x)&& (coin2 x == coin3 x)]|
= n.
Proof.
rewrite -[n]card_ord /coin0 /coin1 /coin2 /coin3.
pose f (p : col_squares) := p c3; rewrite -(@card_in_image _ _ f).
apply: eq_card => c /=; apply/imageP.
exists ([ffun => c] : col_squares); last by rewrite /f ffunE.
by rewrite /= inE !ffunE !eqxx.
move=> p1 p2; rewrite /= !inE /f -!andbA => eqp1 eqp2 eqp12.
apply/eqP; rewrite eqperm_map /= andbT.
case/and3P: eqp1; do 3!move/eqP->; case/and3P: eqp2; do 3!move/eqP->.
by rewrite !andbb eqp12.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
card_n
| |
burnside_app2: (square_coloring_number2 * 2 = n ^ 4 + n ^ 2)%N.
Proof.
rewrite (burnside_formula [:: id1; sh]) => [||p]; last first.
- by rewrite !inE.
- by rewrite /= inE diff_id_sh.
by rewrite 2!big_cons big_nil addn0 {1}card_Fid F_Sh card_n2.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
burnside_app2
| |
burnside_app_rot:
(square_coloring_number4 * 4 = n ^ 4 + n ^ 2 + 2 * n)%N.
Proof.
rewrite (burnside_formula [:: id1; r1; r2; r3]) => [||p]; last first.
- by rewrite !inE !orbA.
- by apply: map_uniq (fun p : {perm square} => p c0) _ _; rewrite /= !permE.
rewrite !big_cons big_nil /= addn0 {1}card_Fid F_r1 F_r2 F_r3.
by rewrite card_n card_n2 //= [n + _]addnC !addnA addn0.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
burnside_app_rot
| |
F_Sd1: 'Fix_to[sd1] = [set x | coin1 x == coin3 x].
Proof.
apply/setP => x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
rewrite /act_f sd1_inv !ffunE !permE /=.
by rewrite !eqxx !andbT eq_sym /= andbb.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
F_Sd1
| |
card_n3: forall x y : square, x != y ->
#|[set k : col_squares | k x == k y]| = (n ^ 3)%N.
Proof.
move=> x y nxy; apply/eqP; case: (posnP n) => [n0|].
by rewrite n0; apply/existsP=> [] [p _]; case: (p c0) => i; rewrite n0.
move/eqn_pmul2l <-; rewrite -expnS -card_Fid Fid cardsT.
rewrite -{1}[n]card_ord -cardX.
pose pk k := [ffun i => k (if i == y then x else i) : colors].
rewrite -(@card_image _ _ (fun k : col_squares => (k y, pk k))).
apply/eqP; apply: eq_card => ck /=; rewrite inE /= [_ \in _]inE.
apply/eqP/imageP; last first.
by case=> k _ -> /=; rewrite !ffunE if_same eqxx.
case: ck => c k /= kxy.
exists [ffun i => if i == y then c else k i]; first by rewrite inE.
rewrite !ffunE eqxx; congr (_, _); apply/ffunP=> i; rewrite !ffunE.
case Eiy: (i == y); last by rewrite Eiy.
by rewrite (negbTE nxy) (eqP Eiy).
move=> k1 k2 [Eky Epk]; apply/ffunP=> i.
have{Epk}: pk k1 i = pk k2 i by rewrite Epk.
by rewrite !ffunE; case: eqP => // ->.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
card_n3
| |
F_Sd2: 'Fix_to[sd2] = [set x | coin0 x == coin2 x].
Proof.
apply/setP => x; rewrite (sameP afix1P eqP) !inE eqperm_map /=.
by rewrite /act_f sd2_inv !ffunE !permE /= !eqxx !andbT eq_sym /= andbb.
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
F_Sd2
| |
burnside_app_iso:
(square_coloring_number8 * 8 = n ^ 4 + 2 * n ^ 3 + 3 * n ^ 2 + 2 * n)%N.
Proof.
pose iso_list := [:: id1; r1; r2; r3; sh; sv; sd1; sd2].
rewrite (burnside_formula iso_list) => [||p]; last first.
- by rewrite /= !inE.
- apply: map_uniq (fun p : {perm square} => (p c0, p c1)) _ _.
by rewrite /= !permE.
rewrite !big_cons big_nil {1}card_Fid F_r1 F_r2 F_r3 F_Sh F_Sv F_Sd1 F_Sd2.
rewrite card_n !card_n3 // !card_n2 //= !addnA !addn0.
by rewrite [LHS]addn.[ACl 1 * 7 * 8 * 3 * 5 * 6 * 2 * 4].
Qed.
|
Lemma
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
burnside_app_iso
| |
cube:= 'I_6.
HB.instance Definition _ := SubType.on cube.
HB.instance Definition _ := Finite.on cube.
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
cube
| |
mkFcubei : cube := Sub (i %% 6) (ltn_mod i 6).
|
Definition
|
solvable
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype tuple finfun bigop finset fingroup",
"From mathcomp Require Import action perm primitive_action ssrAC"
] |
solvable/burnside_app.v
|
mkFcube
|
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