Fixpoint LNN2LNT_term (t : fol.Term LNN) : Term :=
match t with
| fol.var v => var v
| apply f ts => apply LNT f (LNN2LNT_terms _ ts)
end
with LNN2LNT_terms (n : nat) (ts : fol.Terms LNN n) {struct ts} :
Terms n :=
match ts in (fol.Terms _ n0) return (Terms n0) with
| Tnil => Tnil LNT
| Tcons m s ss => Tcons LNT m (LNN2LNT_term s) (LNN2LNT_terms m ss)
end.
Definition LTFormula :=
existH 2 (equal (Plus (var 0) (Succ (var 2))) (var 1)).
Definition translateLT (ts : fol.Terms LNN (arity LNN (inl _ LT))) : Formula.
simpl in ts.
induction (consTerms _ _ ts).
induction x as (a, b).
induction (consTerms _ _ b).
induction x as (a0, b0).
set (x := LNN2LNT_term a) in *.
set (y := LNN2LNT_term a0) in *.
apply (subAllFormula LNT LTFormula).
intro.
induction H as [| H HrecH].
exact x.
induction H as [| H HrecH0].
exact y.
exact (var H).
Defined.
Fixpoint LNN2LNT_formula (f : fol.Formula LNN) : Formula :=
match f with
| fol.equal t1 t2 => equal (LNN2LNT_term t1) (LNN2LNT_term t2)
| atomic r ts =>
match
r as l return (fol.Terms LNN (arity LNN (inl _ l)) -> Formula)
with
| LT => fun t0 : fol.Terms LNN (arity LNN (inl _ LT)) => translateLT t0
end ts
| fol.impH A B => impH (LNN2LNT_formula A) (LNN2LNT_formula B)
| fol.notH A => notH (LNN2LNT_formula A)
| fol.forallH v A => forallH v (LNN2LNT_formula A)
end.

Lemma LNN2LNT_subFormula :
forall (T : System) (f : fol.Formula LNN) (v : nat) (s : fol.Term LNN),
SysPrf T
(iffH (LNN2LNT_formula (substituteFormula LNN f v s))
(substituteFormula LNT (LNN2LNT_formula f) v (LNN2LNT_term s))).