Fixpoint freshmax_expr (e: expr) : var :=
  match e with
  | Nil => minid
  | Var v => v
  end.Definition var_max (x y : var) : var :=
 match Ident.compare x y with Lt => y | _ => x end.Definition freshmax_pn_atom (a: pn_atom) : var :=
  match a with
  | Equ e1 e2 => var_max (freshmax_expr e1) (freshmax_expr e2)
  | Nequ e1 e2 => var_max (freshmax_expr e1) (freshmax_expr e2)
  end.Definition freshmax_space_atom (a: space_atom) : var :=
  match a with
  | Next e1 e2 =>  var_max (freshmax_expr e1) (freshmax_expr e2)
  | Lseg e1 e2 =>  var_max (freshmax_expr e1) (freshmax_expr e2)
 end.Definition freshmax_list {A} (f: A -> var) (l: list A) : var :=
  fold_right (fun a i => var_max (f a) i) minid l.Definition freshmax_assertion (a: assertion) : var :=
  match a with
  | Assertion pi sigma =>
     var_max (freshmax_list freshmax_pn_atom pi)
            (freshmax_list freshmax_space_atom sigma)
  end.Definition freshmax_pure_atom (a: pure_atom) : var :=
  match a with
  | Eqv e1 e2 => var_max (freshmax_expr e1) (freshmax_expr e2)
  end.Definition freshmax_clause (c : clause) : var :=
 match c with
 | PureClause pi pi' _ _ =>  var_max (freshmax_list freshmax_pure_atom pi)
                                     (freshmax_list freshmax_pure_atom pi')
 | PosSpaceClause pi pi' sigma =>
     var_max (var_max (freshmax_list freshmax_pure_atom pi)
                                     (freshmax_list freshmax_pure_atom pi'))
                       (freshmax_list freshmax_space_atom sigma)
 | NegSpaceClause pi sigma pi' =>
     var_max (var_max (freshmax_list freshmax_pure_atom pi)
                                     (freshmax_list freshmax_pure_atom pi'))
                       (freshmax_list freshmax_space_atom sigma)
 end.

Lemma Ile_freshmax_i3: forall s m, (forall x, M.In x s -> Ile (freshmax_clause x) m) -> Ile (freshmax_list freshmax_clause (M.elements s)) m.