Remark maxLemma :
 forall a b c d : nat, a <= b -> c <= d -> max a c <= max b d.Remark maxLemma2 :
 forall a b : list nat, fold_right max 0 a <= fold_right max 0 (a ++ b).Remark maxLemma3 :
 forall a b : list nat, fold_right max 0 b <= fold_right max 0 (a ++ b).Remark maxApp :
 forall a b : list nat,
 {fold_right max 0 (a ++ b) = fold_right max 0 a} +
 {fold_right max 0 (a ++ b) = fold_right max 0 b}.

(* Specifications: *)
Lemma boundSubTerm : forall (t : fol.Term L) (v : nat) (s : fol.Term L), codeTerm (substituteTerm L t v s) <= ReplaceTermTerm (codeTerm t) (fold_right max 0 (codeTerm s :: freeVarTerm L t)).
Lemma boundSubTerms : forall (n : nat) (ts : fol.Terms L n) (v : nat) (s : fol.Term L), codeTerms L codeF n (substituteTerms L n ts v s) <= ReplaceTermsTerm (codeTerms L codeF n ts) (fold_right max 0 (codeTerm s :: freeVarTerms L n ts)).
Lemma ReplaceTermTermSub : forall (t : fol.Term L) (v w s2 : nat), ReplaceTermTerm (codeTerm (substituteTerm L t v (var w))) s2 = ReplaceTermTerm (codeTerm t) s2.
Lemma ReplaceTermsTermSub : forall (n : nat) (ts : fol.Terms L n) (v w s2 : nat), ReplaceTermsTerm (codeTerms L codeF n (substituteTerms L n ts v (var w))) s2 = ReplaceTermsTerm (codeTerms L codeF n ts) s2.