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Axiom pair : nat -> nat -> nat -> Prop.
Axiom pl : (nat -> Prop) -> (nat -> Prop) -> (nat -> Prop).
Axiom plImpR : forall k P Q,
pl P Q k -> forall (Q':nat -> Prop),
(forall k', Q k' -> Q' k') ->
pl P Q' k.
Definition nexists (P:nat -> nat -> Prop) : nat -> Prop :=
fun k' => exists k, P k k'.
Goal forall a A m,
True ->
(pl A (nexists (fun x => (nexists
(fun y => pl (pair a (S x)) (pair a (S y))))))) m.
Proof.
intros.
eapply plImpR; [ | intros; econstructor; econstructor; eauto].
clear H;
match goal with
| |- (pl _ (pl (pair _ ?x) _)) _ => replace x with 0
end.
Admitted.
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