Inductive is_pwd : list (nat*nat) -> Prop :=
  nil_is_pwd : (is_pwd nil)
|cons_is_pwd : forall (p n:nat)(tail:list (nat*nat)),(is_pwd tail)->(forall (n:nat),~(In (p,n) tail))->(is_pwd ((p,n)::tail)).Inductive is_pwd : list (nat*nat) -> Prop :=
  nil_is_pwd : (is_pwd nil)
|cons_is_pwd : forall (p n:nat)(tail:list (nat*nat)),(is_pwd tail)->(forall (n:nat),~(In (p,n) tail))->(is_pwd ((p,n)::tail)).Inductive is_pwd : list (nat*nat) -> Prop :=
  nil_is_pwd : (is_pwd nil)
|cons_is_pwd : forall (p n:nat)(tail:list (nat*nat)),(is_pwd tail)->(forall (n:nat),~(In (p,n) tail))->(is_pwd ((p,n)::tail)).

Lemma pwd_impl_set : forall (l:list (nat*nat)),(is_pwd l)->(is_set (nat*nat) l).