Section NoDup_gen_map.
Variable A B : Type.
Variable ea : A -> A -> Prop.
Variable eb : B -> B -> Prop.
Variable f : A -> B.
Hypothesis ea_symm :
forall a1 a2,
ea a1 a2 -> ea a2 a1.
Hypothesis ea_trans :
forall a1 a2 a3,
ea a1 a2 ->
ea a2 a3 ->
ea a1 a3.
Hypothesis ea_func :
forall a1 a2, ea a1 a2 -> eb (f a1) (f a2).
Hypothesis eb_symm :
forall b1 b2,
eb b1 b2 -> eb b2 b1.
Hypothesis eb_trans :
forall b1 b2 b3,
eb b1 b2 ->
eb b2 b3 ->
eb b1 b3.
Hypothesis ea_refl :
forall a y,
eb (f a) y ->
ea a a .
Hypothesis eb_func :
forall a1 a2,
eb (f a1) (f a2) -> ea a1 a2.
induction l; intuition; simpl in *.
intuition.
destruct H.
intuition.
intuition.
econstructor.
intuition.
eapply eb_symm.
eauto.
left.
eapply ea_refl.
eapply eb_symm.
eauto.
specialize (IHl y).
intuition.
destruct H2.
intuition.
econstructor.
split.
eauto.
intuition.
destruct H.
intuition.
left.
eapply eb_trans.
eapply eb_symm.
eauto.
eapply ea_func.
trivial.
right.
specialize (IHl y).
intuition.
eapply H2.
econstructor.
intuition.
eauto.
trivial.
Qed.

Theorem map_NoDup_gen :
forall (ls : list A),
NoDup_gen ea ls ->
NoDup_gen eb (map f ls).