Set Bullet Behavior "Strict Subproofs".
(** * Results on joining lists and the necessary algebras *)
Fixpoint joinlist {A} {JA : Join A} (l : list A) (x : A) :=
match l with
| nil => identity x
| h :: l => exists y, joinlist l y /\ join h y x
end.
Instance Permutation_length' A {JA : Join A} {PA : Perm_alg A} :
Proper (@Permutation A ==> @eq A ==> Logic.iff) joinlist | 10.
(** * Other list functions *)
Fixpoint listoption_inv {A} (l : list (option A)) : list A :=
match l with
| Some x :: l => x :: listoption_inv l
| None :: l => listoption_inv l
| nil => nil
end.
Fixpoint all_but (i : nat) {A} (l : list A) : list A :=
match i with
O => tl l
| S i =>
match l with
nil => nil
| x :: l => x :: all_but i l
end
end.

Lemma permutation_all_but {A} i (l : list A) x :
nth_error l i = Some x ->
Permutation l (x :: all_but i l).