Section Code_Term_Formula_Proof.
Variable L : Language.
Variable codeF : Functions L -> nat.
Variable codeR : Relations L -> nat.
Hypothesis codeFInj : forall f g : Functions L, codeF f = codeF g -> f = g.
Hypothesis codeRInj : forall R S : Relations L, codeR R = codeR S -> R = S.
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.
Let Prf := Prf L.
Let SysPrf := SysPrf L.
Fixpoint codeTerm (t : Term) : nat :=
match t with
| fol.var n => cPair 0 n
| fol.apply f ts => cPair (S (codeF f)) (codeTerms _ ts)
end
with codeTerms (n : nat) (ts : Terms n) {struct ts} : nat :=
match ts with
| Tnil => 0
| Tcons n t ss => S (cPair (codeTerm t) (codeTerms n ss))
end.
Fixpoint codeFormula (f : Formula) : nat :=
match f with
| fol.equal t1 t2 => cPair 0 (cPair (codeTerm t1) (codeTerm t2))
| fol.impH f1 f2 => cPair 1 (cPair (codeFormula f1) (codeFormula f2))
| fol.notH f1 => cPair 2 (codeFormula f1)
| fol.forallH n f1 => cPair 3 (cPair n (codeFormula f1))
| fol.atomic R ts => cPair (4+(codeR R)) (codeTerms _ ts)
end.
Fixpoint codePrf (Z : Formulas) (f : Formula) (prf : Prf Z f) {struct prf} :
nat :=
match prf with
| AXM A => cPair 0 (codeFormula A)
| MP Axm1 Axm2 A B rec1 rec2 =>
cPair 1
(cPair
(cPair (cPair 1 (cPair (codeFormula A) (codeFormula B)))
(codePrf _ _ rec1)) (cPair (codeFormula A) (codePrf _ _ rec2)))
| GEN Axm A v _ rec =>
cPair 2 (cPair v (cPair (codeFormula A) (codePrf _ _ rec)))
| IMP1 A B => cPair 3 (cPair (codeFormula A) (codeFormula B))
| IMP2 A B C =>
cPair 4 (cPair (codeFormula A) (cPair (codeFormula B) (codeFormula C)))
| CP A B => cPair 5 (cPair (codeFormula A) (codeFormula B))
| FA1 A v t => cPair 6 (cPair (codeFormula A) (cPair v (codeTerm t)))
| FA2 A v _ => cPair 7 (cPair (codeFormula A) v)
| FA3 A B v => cPair 8 (cPair (codeFormula A) (cPair (codeFormula B) v))
| EQ1 => cPair 9 0
| EQ2 => cPair 10 0
| EQ3 => cPair 11 0
| EQ4 r => cPair 12 (codeR r)
| EQ5 f => cPair 13 (codeF f)
end.

Lemma codePrfInjAxm :
forall (a b : Formula) (A B : Formulas) (p : Prf A a) (q : Prf B b),
codePrf A a p = codePrf B b q -> A = B.