Ltac generalize_all := 
   repeat match goal with
          H:_ |- _ => revert H
end.Ltac uniformize_term exp lvar :=
  let trm := interp_A lvar exp in 
  let trm2 :=constr:(interp_AF_to_F lvar (uniformize_areas trm)) in
(replace exp with trm2;[idtac|rewrite uniformize_areas_correct;trivial]).(* This is a metaification to collect the list of all points *)

Inductive formula : Type :=
  | f_imp : formula -> formula -> formula (* binary constructor *)
  | f_const : Prop -> formula (* contructor for constants *).(* This is a metaification to collect the list of all points *)

Inductive formula : Type :=
  | f_imp : formula -> formula -> formula (* binary constructor *)
  | f_const : Prop -> formula (* contructor for constants *).(* This is a metaification to collect the list of all points *)

Inductive formula : Type :=
  | f_imp : formula -> formula -> formula (* binary constructor *)
  | f_const : Prop -> formula (* contructor for constants *).Inductive formula2 : Type :=
  | f_imp2 : formula2 -> formula2 -> formula2 (* binary constructor *)
  | f_neq : AF -> AF -> formula2 
  | f_eq : AF -> AF -> formula2.Inductive formula2 : Type :=
  | f_imp2 : formula2 -> formula2 -> formula2 (* binary constructor *)
  | f_neq : AF -> AF -> formula2 
  | f_eq : AF -> AF -> formula2.Inductive formula2 : Type :=
  | f_imp2 : formula2 -> formula2 -> formula2 (* binary constructor *)
  | f_neq : AF -> AF -> formula2 
  | f_eq : AF -> AF -> formula2.Inductive formula2 : Type :=
  | f_imp2 : formula2 -> formula2 -> formula2 (* binary constructor *)
  | f_neq : AF -> AF -> formula2 
  | f_eq : AF -> AF -> formula2.Fixpoint uniformize_areas_formula2 (e:formula2) {struct e} : formula2 :=
match e with 
  | f_imp2 e1 e2 => 
    let s1 := (uniformize_areas_formula2 e1) in 
    let s2 := (uniformize_areas_formula2 e2) in 
    f_imp2 s1 s2
  | f_neq e1 e2 =>
    let s1 := (uniformize_areas e1) in 
    let s2 := (uniformize_areas e2) in 
    f_neq s1 s2
  | f_eq e1 e2 =>
    let s1 := (uniformize_areas e1) in 
    let s2 := (uniformize_areas e2) in 
    f_eq s1 s2
end.Fixpoint interp_formula2_to_prop (lvar:list (prodT AVars nat)) (e:formula2) {struct e} : Prop :=
  match e with
  | f_imp2 e1 e2 => 
        let s1 := interp_formula2_to_prop lvar e1 in
        let s2 := interp_formula2_to_prop lvar e2 in
      s1 -> s2
  | f_neq e1 e2 => 
        let s1 := interp_AF_to_F lvar e1 in
        let s2 := interp_AF_to_F lvar e2 in
        s1 <> s2
 | f_eq e1 e2 => 
        let s1 := interp_AF_to_F lvar e1 in
        let s2 := interp_AF_to_F lvar e2 in
        s1 = s2
end.

Lemma uniformize_areas_formula2_correct : forall (e:formula2) (lvar:list (prodT AVars nat)), interp_formula2_to_prop lvar (uniformize_areas_formula2 e) -> interp_formula2_to_prop lvar e.