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(*                                                                          *)
(*                The formal system \lambda\delta                           *)
(*                                                                          *)
(*                Author: Ferruccio Guidi                                   *)
(*                                                                          *)
(*                This file is distributed under the terms of the           *)
(*                GNU General Public License Version 2                      *)
(*                                                                          *)
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Require Export bg_require.

      Tactic Definition Arith0 x :=
         Replace (S x) with (plus (1) x); XAuto.

      Tactic Definition Arith1 x :=
         Replace x with (plus x (0)); [ XAuto | Auto with arith core ].

      Tactic Definition Arith1In H x :=
         XReplaceIn H x '(plus x (0)).

      Tactic Definition Arith2 x :=
         Replace x with (plus (0) x); XAuto.

      Tactic Definition Arith3 x :=
         Replace (S x) with (S (plus (0) x)); XAuto.

      Tactic Definition Arith3In H x :=
         XReplaceIn H '(S x) '(S (plus (0) x)).

      Tactic Definition Arith4 x y :=
         Replace (S (plus x y)) with (plus (S x) y); XAuto.

      Tactic Definition Arith4In H x y :=
         XReplaceIn H '(S (plus x y)) '(plus (S x) y).

      Tactic Definition Arith4c x y :=
         Arith4 x y; Rewrite plus_sym.

      Tactic Definition Arith5 x y :=
         Replace (S (plus x y)) with (plus x (S y)); Auto with arith core.

      Tactic Definition Arith5In H x y :=
         XReplaceIn H '(S (plus x y)) '(plus x (S y)); Auto with arith core.

      Tactic Definition Arith5' x y :=
         Replace (plus x (S y)) with (S (plus x y)); Auto with arith core.

      Tactic Definition Arith5'In H x y :=
         XReplaceIn H '(plus x (S y)) '(S (plus x y)); Auto with arith core.

      Tactic Definition Arith5'c x y :=
         Arith5' x y; Rewrite plus_sym.

      Tactic Definition Arith6In H x y :=
         XReplaceIn H '(plus x (S y)) '(plus (1) (plus x y));
         [ Idtac | Simpl; Auto with arith core ].

      Tactic Definition Arith7 x :=
         Replace (S x) with (plus x (1));
         [ Idtac | Rewrite plus_sym; Auto with arith core ].

      Tactic Definition Arith7In H x :=
         XReplaceIn H '(S x) '(plus x (1)) ;
         [ Idtac | Rewrite plus_sym; Auto with arith core ].

      Tactic Definition Arith7' x :=
         Replace (plus x (1)) with (S x);
         [ Idtac | Rewrite plus_sym; Auto with arith core ].

      Tactic Definition Arith7'In H x :=
         XReplaceIn H '(plus x (1)) '(S x);
         [ Idtac | Rewrite plus_sym; Auto with arith core ].

      Tactic Definition Arith8 x y :=
         Replace x with (plus y (minus x y));
         [ Idtac | Auto with arith core ].

      Tactic Definition Arith8' x y :=
         Replace (plus y (minus x y)) with x;
         [ Idtac | Auto with arith core ].

      Tactic Definition Arith9'In H x :=
         XReplaceIn H '(S (plus x (0))) '(S x).

      Tactic Definition Arith10 x :=
         Replace x with (minus (S x) (1));
         [ Idtac | Simpl; Rewrite <- minus_n_O; Auto with arith core ].

      Tactic Definition Arith11 x y :=
         Arith5' x y; Rewrite plus_sym; Arith5 y x.

      Tactic Definition Arith12In H :=
         Rewrite minus_x_Sy in H; [ Idtac | XAuto ].