Inductive rel_map(A B : Type)(r : A -> B -> Prop) : list A -> list B -> Prop :=
| rm_nil : 
  rel_map r nil nil
| rm_step : 
  forall lsa lsb a b,
    rel_map r lsa lsb ->
    r a b ->
    rel_map r (a :: lsa) (b :: lsb).Inductive rel_map(A B : Type)(r : A -> B -> Prop) : list A -> list B -> Prop :=
| rm_nil : 
  rel_map r nil nil
| rm_step : 
  forall lsa lsb a b,
    rel_map r lsa lsb ->
    r a b ->
    rel_map r (a :: lsa) (b :: lsb).Inductive rel_map(A B : Type)(r : A -> B -> Prop) : list A -> list B -> Prop :=
| rm_nil : 
  rel_map r nil nil
| rm_step : 
  forall lsa lsb a b,
    rel_map r lsa lsb ->
    r a b ->
    rel_map r (a :: lsa) (b :: lsb).

Lemma rel_map_func : forall (A B : Type) (rel : A -> B -> Prop) lsa lsb1, rel_map rel lsa lsb1 -> forall lsb2, rel_map rel lsa lsb2 -> (forall a b1 b2, In a lsa -> rel a b1 -> rel a b2 -> b1 = b2) -> lsb1 = lsb2.