Ltac 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.

Lemma F_r3 : 'Fix_to[r3] =
[set x | (coin0 x == coin1 x)&&(coin1 x == coin2 x)&&(coin2 x == coin3 x)].