(* Definition of the opposite for nat *)
Definition Natopp := (fun x:nat => 0%nat).(* Definition of the opposite for N *)
Definition Nopp := (fun x:N => 0%N).(* Auxillary functions for Z *)

Definition is_Z0 := (Zeq_bool 0).Definition is_Z1 := (Zeq_bool 1).Definition is_Zpos := (Zle_bool 0).Definition is_Zdiv :=
  fun x y => if Zeq_bool x Z0 then false else Zeq_bool Z0 (Zmod y x).Definition Zgcd :=
 fun x y => (if (is_Zdiv x y) then x else if (is_Zdiv y x) then y else 1%Z).(* Check if a nat is a number *)
Ltac is_NatCst p :=
  match p with
  | O => constr:(true)
  | S ?p' => is_NatCst p'
  | _ => constr:(false)
end.(* Convert a Z into a nat if it is a number *)
Ltac NatCst t :=
  match is_NatCst t with
  | false => constr:(false)
  | _ => let res := eval compute in (Z_of_nat t) in constr:(res)
end.(* Check if a number is a positive *)
Ltac is_PCst p :=
  match p with
  | xH => constr:(true)
  | xO ?p' => is_PCst p'
  | xI ?p' => is_PCst p'
  | _ => constr:(false)
end.(* Check if a N is a number *)
Ltac is_NCst p :=
  match p with
  | N0 => constr:(true)
  | Npos ?p' => is_PCst p'
  | _ => constr:(false)
end.(* Convert a Z into a nat if it is a number *)
Ltac NCst t :=
  match is_NCst t with
  | false => constr:(false)
  | _ => let res := eval compute in (Z_of_N t) in constr:(res)
end.(* If a number is an integer return itself otherwise false *)
Ltac ZCst t :=
  match t with
  | Z0 => constr:(t)
  | Zpos ?p => match is_PCst p with
               | false => constr:(false)
               | _ => constr:(t)
               end
  | Zneg ?p => match is_PCst p with
               | false => constr:(false)
               | _ => constr:(t)
               end
  | _ => constr:(false)
  end.(* Check if a number is an integer *)
Ltac is_ZCst t := match t with
                | Z0 => constr:(true)
                | Zpos ?p => is_PCst p
                | Zneg ?p => is_PCst p
                | _ => constr:(false) end.(* Turn a positive into a real *)
Fixpoint P2R (z: positive) {struct z}: R :=
  match z with
     xH => 1%R
  | (xO xH) => 2%R
  | (xI xH) => 3%R
  | (xO z1) => (2*(P2R z1))%R
  | (xI z1) => (1+2*(P2R z1))%R
 end.(* Turn an integer into a real *)
Definition Z2R (z: Z): R :=
  match z with
     Z0 => 0%R
  | (Zpos z1) => (P2R z1)%R
  | (Zneg z1) => (-(P2R z1))%R
 end.(* Turn a R when possible into a Z *)
Ltac RCst t :=
  match t with
   | R0 => constr:(Z0)
   | R1 => constr:(Zpos xH)
   | Rplus ?e1 ?e2 =>
       match (RCst e1) with
        false => constr:(false)
      | ?e3 => match (RCst e2) with
                 false => constr:(false)
              |  ?e4 =>  eval vm_compute in (Zplus e3  e4)
              end
      end
   | Rminus ?e1 ?e2 =>
       match (RCst e1) with
        false => constr:(false)
      | ?e3 => match (RCst e2) with
                 false => constr:(false)
              |  ?e4 => eval vm_compute in (Zminus e3  e4)
              end
      end
   | Rmult ?e1 ?e2 =>
       match (RCst e1) with
        false => constr:(false)
      | ?e3 => match (RCst e2) with
                 false => constr:(false)
              |  ?e4 => eval vm_compute in (Zmult e3  e4)
              end
      end
   | Ropp ?e1 =>
       match (RCst e1) with
        false => constr:(false)
      | ?e3 => eval vm_compute in (Z.opp e3)
      end
   | IZR ?e1 =>
       match (ZCst e1) with
        false => constr:(false)
      | ?e3 => e3
      end

   | _ => constr:(false)
 end.(* Remove the Z.abs_nat of a number, unfortunately stops at
   the first Z.abs_nat x where x is not a number *)

Ltac clean_zabs term :=
  match term with
   context id [(Z.abs_nat ?X)] =>
     match is_ZCst X with
       true =>
         let x := eval vm_compute in (Z.abs_nat X) in
         let y := context id [x] in
           clean_zabs y
     | false => term
     end
    | _ => term
  end.(* Remove the Z.abs_N of a number, unfortunately stops at
   the first Z.abs_nat x where x is not a number *)

Ltac clean_zabs_N term :=
  match term with
   context id [(Z.abs_N ?X)] =>
     match is_ZCst X with
       true =>
         let x := eval vm_compute in (Z.abs_N X) in
         let y := context id [x] in
           clean_zabs_N y
     | false => term
     end
    | _ => term
  end.(* Equality test for Ltac *)

Ltac eqterm t1 t2 :=
  match constr:((t1,t2)) with (?X, ?X) => true | _ => false end.

Theorem Z2R_le: forall p q, (p <= q)%Z -> (Z2R p <= Z2R q)%R.