Fixpoint substituteTerm (s : fol.Term L) (x : nat) 
 (t : fol.Term L) {struct s} : fol.Term L :=
  match s with
  | fol.var v =>
      match eq_nat_dec x v with
      | left _ => t
      | right _ => var v
      end
  | fol.apply f ts => apply f (substituteTerms _ ts x t)
  end
 
 with substituteTerms (n : nat) (ss : fol.Terms L n) 
 (x : nat) (t : fol.Term L) {struct ss} : fol.Terms L n :=
  match ss in (fol.Terms _ n0) return (fol.Terms L n0) with
  | Tnil => Tnil L
  | Tcons m s ts =>
      Tcons L m (substituteTerm s x t) (substituteTerms m ts x t)
  end.

(* Specifications: *)
Lemma subTermVar1 : forall (v : nat) (s : fol.Term L), substituteTerm (var v) v s = s.
Lemma subTermVar2 : forall (v x : nat) (s : fol.Term L), v <> x -> substituteTerm (var x) v s = var x.
Lemma subTermFunction : forall (f : Functions L) (ts : fol.Terms L (arity L (inr _ f))) (v : nat) (s : fol.Term L), substituteTerm (apply f ts) v s = apply f (substituteTerms _ ts v s).