Section List_Remove.
Variable A : Set.
Hypothesis Aeq_dec : forall a b : A, {a = b} + {a <> b}.
Definition list_remove (x : A) (l : list A) : list A :=
list_rec (fun _ => list A) nil
(fun (a : A) _ (recl : list A) =>
match Aeq_dec a x with
| left _ => recl
| right _ => a :: recl
end) l.

Lemma In_list_remove1 :
forall (a b : A) (l : list A), In a (list_remove b l) -> In a l.