Section list_util.
Variables A B C : Type.
Hypothesis A_eq_dec : forall x y : A, {x = y} + {x <> y}.
Proof using.
intuition.
symmetry in H.
induction l;
now inversion H.
Qed.
Proof using.
induction xs; intros.
- intuition.
- simpl in *.
concludes.
intuition; break_if; subst; try congruence; intuition.
Qed.
Proof using.
induction xs; intros.
- auto.
- simpl in *. break_if; simpl in *; intuition.
Qed.

Lemma remove_partition :
forall xs (p : A) ys,
remove A_eq_dec p (xs ++ p :: ys) = remove A_eq_dec p (xs ++ ys).