Definition codePA : nat -> bool :=
  orRel 1 (beq_nat (codeFormula PA6))
    (orRel 1 (beq_nat (codeFormula PA5))
       (orRel 1 (beq_nat (codeFormula PA4))
          (orRel 1 (beq_nat (codeFormula PA3))
             (orRel 1 (beq_nat (codeFormula PA2))
                (orRel 1 (beq_nat (codeFormula PA1)) codeInductionSchema))))).

Lemma codePAcorrect2 : forall f : Formula, ~ mem Formula PA f -> codePA (codeFormula f) = false.