Definition codeFreeVarFormula : nat -> nat :=
  evalStrongRec 0
    (fun f recs : nat =>
     switchPR (cPairPi1 f)
       (switchPR (pred (cPairPi1 f))
          (switchPR (pred (pred (cPairPi1 f)))
             (switchPR (pred (pred (pred (cPairPi1 f))))
                (codeFreeVarTerms (cPairPi2 f))
                (codeListRemove (cPairPi1 (cPairPi2 f))
                   (codeNth (f - S (cPairPi2 (cPairPi2 f))) recs)))
             (codeNth (f - S (cPairPi2 f)) recs))
          (codeApp (codeNth (f - S (cPairPi1 (cPairPi2 f))) recs)
             (codeNth (f - S (cPairPi2 (cPairPi2 f))) recs)))
       (codeApp (codeFreeVarTerm (cPairPi1 (cPairPi2 f)))
          (codeFreeVarTerm (cPairPi2 (cPairPi2 f))))).

Lemma codeFreeVarFormulaCorrect : forall f : Formula, codeFreeVarFormula (codeFormula L codeF codeR f) = codeList (freeVarFormula L f).