Require Vector.
Section Code_Substitute_Formula.
Variable L : Language.
Variable codeF : Functions L -> nat.
Variable codeR : Relations L -> nat.
Notation Formula := (Formula L) (only parsing).
Notation Formulas := (Formulas L) (only parsing).
Notation System := (System L) (only parsing).
Notation Term := (Term L) (only parsing).
Notation Terms := (Terms L) (only parsing).
Let var := var L.
Let apply := apply L.
Let codeFormula := codeFormula L codeF codeR.
Let codeTerm := codeTerm L codeF.
Definition cTriple (a b c : nat) : nat := cPair a (cPair b c).
Definition cTriplePi1 (n : nat) : nat := cPairPi1 n.
Definition cTriplePi2 (n : nat) : nat := cPairPi1 (cPairPi2 n).
Definition cTriplePi3 (n : nat) : nat := cPairPi2 (cPairPi2 n).

(* Specifications: *)
Lemma cTripleIsPR : isPR 3 cTriple.
Lemma cTriplePi1IsPR : isPR 1 cTriplePi1.
Lemma cTriplePi2IsPR : isPR 1 cTriplePi2.
Lemma cTriplePi3IsPR : isPR 1 cTriplePi3.
Lemma cTripleProj1 : forall a b c : nat, cTriplePi1 (cTriple a b c) = a.
unfold cTriplePi1, cTriple in |- *.