Inductive AVars : Type :=  
  | PVar : Point -> AVars
  | FVar : F -> AVars.Inductive AVars : Type :=  
  | PVar : Point -> AVars
  | FVar : F -> AVars.Inductive AVars : Type :=  
  | PVar : Point -> AVars
  | FVar : F -> AVars.Inductive AP : Type := 
  | APvar : nat -> AP.Inductive AP : Type := 
  | APvar : nat -> AP.Inductive AF : Type :=
  | AS : nat -> nat -> nat -> AF
  | AD : nat -> nat -> AF
  | AFvar : nat -> AF
  | AF0 : AF
  | AF1 : AF
  | AFplus : AF -> AF -> AF
  | AFmult : AF -> AF -> AF
  | AFopp : AF -> AF
  | AFinv : AF -> AF.Inductive AF : Type :=
  | AS : nat -> nat -> nat -> AF
  | AD : nat -> nat -> AF
  | AFvar : nat -> AF
  | AF0 : AF
  | AF1 : AF
  | AFplus : AF -> AF -> AF
  | AFmult : AF -> AF -> AF
  | AFopp : AF -> AF
  | AFinv : AF -> AF.Inductive AF : Type :=
  | AS : nat -> nat -> nat -> AF
  | AD : nat -> nat -> AF
  | AFvar : nat -> AF
  | AF0 : AF
  | AF1 : AF
  | AFplus : AF -> AF -> AF
  | AFmult : AF -> AF -> AF
  | AFopp : AF -> AF
  | AFinv : AF -> AF.Inductive AF : Type :=
  | AS : nat -> nat -> nat -> AF
  | AD : nat -> nat -> AF
  | AFvar : nat -> AF
  | AF0 : AF
  | AF1 : AF
  | AFplus : AF -> AF -> AF
  | AFmult : AF -> AF -> AF
  | AFopp : AF -> AF
  | AFinv : AF -> AF.Inductive AF : Type :=
  | AS : nat -> nat -> nat -> AF
  | AD : nat -> nat -> AF
  | AFvar : nat -> AF
  | AF0 : AF
  | AF1 : AF
  | AFplus : AF -> AF -> AF
  | AFmult : AF -> AF -> AF
  | AFopp : AF -> AF
  | AFinv : AF -> AF.Inductive AF : Type :=
  | AS : nat -> nat -> nat -> AF
  | AD : nat -> nat -> AF
  | AFvar : nat -> AF
  | AF0 : AF
  | AF1 : AF
  | AFplus : AF -> AF -> AF
  | AFmult : AF -> AF -> AF
  | AFopp : AF -> AF
  | AFinv : AF -> AF.Inductive AF : Type :=
  | AS : nat -> nat -> nat -> AF
  | AD : nat -> nat -> AF
  | AFvar : nat -> AF
  | AF0 : AF
  | AF1 : AF
  | AFplus : AF -> AF -> AF
  | AFmult : AF -> AF -> AF
  | AFopp : AF -> AF
  | AFinv : AF -> AF.Inductive AF : Type :=
  | AS : nat -> nat -> nat -> AF
  | AD : nat -> nat -> AF
  | AFvar : nat -> AF
  | AF0 : AF
  | AF1 : AF
  | AFplus : AF -> AF -> AF
  | AFmult : AF -> AF -> AF
  | AFopp : AF -> AF
  | AFinv : AF -> AF.Inductive AF : Type :=
  | AS : nat -> nat -> nat -> AF
  | AD : nat -> nat -> AF
  | AFvar : nat -> AF
  | AF0 : AF
  | AF1 : AF
  | AFplus : AF -> AF -> AF
  | AFmult : AF -> AF -> AF
  | AFopp : AF -> AF
  | AFinv : AF -> AF.Inductive AF : Type :=
  | AS : nat -> nat -> nat -> AF
  | AD : nat -> nat -> AF
  | AFvar : nat -> AF
  | AF0 : AF
  | AF1 : AF
  | AFplus : AF -> AF -> AF
  | AFmult : AF -> AF -> AF
  | AFopp : AF -> AF
  | AFinv : AF -> AF.(* Checks if var is in lvar *)

Ltac List_mem var lvar :=
  match constr:(lvar) with
  | nil => constr:(false)
  | cons ?X1 ?X2 =>
      match constr:(X1 = var) with
      | (?X1 = ?X1) => constr:(true)
      | _ => List_mem var X2
      end
  end.(* Build the list of variables  *)

Ltac build_var_point_list_aux lvar point :=
     let res := List_mem (PVar point) lvar in
      match constr:(res) with
      | true => lvar
      | false => constr:(cons (PVar point) lvar)
      end.Ltac build_var_list_aux lvar trm :=
  match constr:(trm) with
  | F0 => lvar
  | F1 => lvar
  | (Fplus ?X1 ?X2) =>
      let l2 := build_var_list_aux lvar X2 in
      build_var_list_aux  l2 X1
  | (Fmult ?X1 ?X2) =>
      let l2 := build_var_list_aux  lvar X2 in
      build_var_list_aux l2 X1
  | (Fopp ?X1) => build_var_list_aux lvar X1
  | (Finv ?X1) => build_var_list_aux lvar X1
  | (S ?X1 ?X2 ?X3) =>  
  let l1 := build_var_point_list_aux lvar X3 in
  let l2 := build_var_point_list_aux l1 X2 in
  build_var_point_list_aux l2 X1
  | (DSeg ?X1 ?X2) => 
  let l1 := build_var_point_list_aux lvar X2 in
  build_var_point_list_aux l1 X1
  | (?X1 = ?X2) => 
  let l2 := build_var_point_list_aux lvar X2 in
  build_var_point_list_aux l2 X1

  | ?X1 =>
    let res := List_mem (FVar X1) lvar in
      match constr:(res) with
      | true => lvar
      | false => constr:(cons (FVar X1) lvar)
      end
  end.Ltac build_var_list trm :=
  build_var_list_aux (@nil AVars) trm.Ltac List_assoc elt lst :=
  match constr:(lst) with
  | nil => fail
  | (cons (@pairT AVars nat ?X1 ?X2) ?X3) =>
      match constr:(elt = X1) with
      | (?X1 = ?X1) => constr:(X2)
      | _ => List_assoc elt X3
      end
  end.Ltac number_aux lvar cpt :=
  match constr:(lvar) with
  | nil => constr:(@nil (prodT AVars nat))
  | cons ?X2 ?X3 =>
      let l2 := number_aux X3 (Datatypes.S cpt) in
      constr:(cons (@pairT  AVars nat X2 cpt) l2)
  end.Ltac number lvar := number_aux lvar (Datatypes.O).Ltac build_numbered_var_list trm := let lvar := build_var_list trm in
                             number lvar.Definition list_assoc_inv :=
  (fix list_assoc_inv_rec (A:Type) (B:Set)
                     (eq_dec:forall e1 e2:B, {e1 = e2} + {e1 <> e2})
                     (lst:list (prodT A B)) {struct lst} :
   B -> A -> A :=
     fun (key:B) (default:A) =>
       match lst with
       | nil => default
       | cons (pairT v e) l =>
           match eq_dec e key with
           | left _ => v
           | right _ => list_assoc_inv_rec A B eq_dec l key default
           end
       end).(* Interpretation *)
Parameter SomeValue :AVars.Parameter SomePoint :Point.Definition interp_AP_to_Point (lvar:list (prodT AVars nat)) (e:nat) : Point :=
 match(list_assoc_inv  AVars nat eq_nat_dec lvar e SomeValue) with
    | PVar p => p
    | _ => SomePoint
  end
.Fixpoint interp_AF_to_F (lvar:list (prodT AVars nat)) (e:AF) {struct e} : F :=
  match e with
  | AF0 => F0
  | AF1 => F1
  | AS e1 e2 e3 => let idx1 := (interp_AP_to_Point lvar e1) in 
S idx1 (interp_AP_to_Point lvar e2) (interp_AP_to_Point lvar e3)
  | AD e1 e2 => DSeg (interp_AP_to_Point lvar e1) (interp_AP_to_Point lvar e2) 
  | AFplus e1 e2 => Fplus (interp_AF_to_F lvar e1) (interp_AF_to_F lvar e2)
  | AFmult e1 e2 => Fmult (interp_AF_to_F lvar e1) (interp_AF_to_F lvar e2)
  | AFopp e => Fopp (interp_AF_to_F lvar e)
  | AFinv e => Finv (interp_AF_to_F lvar e)
  | AFvar n => 
  match(list_assoc_inv  AVars nat eq_nat_dec lvar n SomeValue) with
    | FVar f => f
    | _ => F0
  end
  end.(* Metaification *)

Ltac interp_A lvar trm :=
  match constr:(trm) with
  | F0 => constr:(AF0)
  | F1 => constr:(AF1)
  | (Fplus ?X1 ?X2) =>
      let e1 := interp_A lvar X1 with e2 := interp_A lvar X2 in
      constr:(AFplus e1 e2)
  | (Fmult ?X1 ?X2) =>
      let e1 := interp_A lvar X1 with e2 := interp_A lvar X2 in
      constr:(AFmult e1 e2)
  | (Fopp ?X1) =>
      let e := interp_A lvar X1 in
      constr:(AFopp e)
  | (Finv ?X1) => 
      let e := interp_A lvar X1 in
      constr:(AFinv e)
 | (S ?X1 ?X2 ?X3) =>
   let idx1 := List_assoc (PVar X1) lvar with 
   idx2 := List_assoc (PVar X2) lvar with 
   idx3 := List_assoc (PVar X3) lvar in 
   constr:(AS idx1 idx2 idx3)
 | (DSeg ?X1 ?X2) =>
  let idx1 := List_assoc (PVar X1) lvar with
  idx2 := List_assoc (PVar X2) lvar in 
  constr:(AD idx1 idx2)
 | ?X1 => let idx := List_assoc (FVar X1) lvar in
           constr:(AFvar idx)
  end.Fixpoint nateq (n m:nat) {struct m} : bool :=
  match n, m with
  | O, O => true
  | Datatypes.S n', Datatypes.S m' => nateq n' m'
  | _, _ => false
  end.

(* Specifications: *)
Lemma nateq_prop : forall n m:nat, (Is_true (nateq n m))-> n=m.