Section FORALL2.
Variable A: Type.
Variable B: Type.
Variable P: A -> B -> Prop.
Inductive list_forall2: list A -> list B -> Prop :=
| list_forall2_nil:
list_forall2 nil nil
| list_forall2_cons:
forall a1 al b1 bl,
P a1 b1 ->
list_forall2 al bl ->
list_forall2 (a1 :: al) (b1 :: bl).

Lemma list_forall2_in_left:
forall x1 l1 l2,
list_forall2 l1 l2 -> In x1 l1 -> exists x2, In x2 l2 /\ P x1 x2.