Section Code_Free_Vars.
Variable L : Language.
Variable codeF : Functions L -> nat.
Variable codeR : Relations L -> nat.
Let Formula := Formula L.
Let Formulas := Formulas L.
Let System := System L.
Let Term := Term L.
Let Terms := Terms L.
Let var := var L.
Let apply := apply L.
Let equal := equal L.
Let atomic := atomic L.
Let impH := impH L.
Let notH := notH L.
Let forallH := forallH L.
Let orH := orH L.
Let andH := andH L.
Let existH := existH L.
Let iffH := iffH L.
Let ifThenElseH := ifThenElseH L.
Definition codeFreeVarTermTerms : nat -> nat :=
evalStrongRec 0
(fun t recs : nat =>
cPair
(switchPR (cPairPi1 t) (cPairPi2 (codeNth (t - S (cPairPi2 t)) recs))
(S (cPair (cPairPi2 t) 0)))
(switchPR t
(codeApp (cPairPi1 (codeNth (t - S (cPairPi1 (pred t))) recs))
(cPairPi2 (codeNth (t - S (cPairPi2 (pred t))) recs))) 0)).
Definition codeFreeVarTerm (t : nat) : nat :=
cPairPi1 (codeFreeVarTermTerms t).
Definition codeFreeVarTerms (t : nat) : nat :=
cPairPi2 (codeFreeVarTermTerms t).
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 codeFreeVarFormulaIsPR : isPR 1 codeFreeVarFormula.