(**************************************************************************)
(*       ___                                                              *)
(*      ||M||                                                             *)
(*      ||A||       A project by Andrea Asperti                           *)
(*      ||T||                                                             *)
(*      ||I||       Developers:                                           *)
(*      ||T||         The HELM team.                                      *)
(*      ||A||         http://helm.cs.unibo.it                             *)
(*      \   /                                                             *)
(*       \ /        This file is distributed under the terms of the       *)
(*        v         GNU General Public License Version 2                  *)
(*                                                                        *)
(**************************************************************************)

(* This file was automatically generated: do not edit *********************)

include "Coq.ma".

(*#***********************************************************************)

(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)

(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)

(*   \VV/  **************************************************************)

(*    //   *      This file is distributed under the terms of the       *)

(*         *       GNU Lesser General Public License Version 2.1        *)

(*#***********************************************************************)

(*i $Id: BinInt.v,v 1.5.2.1 2004/07/16 19:31:20 herbelin Exp $ i*)

(*#**********************************************************)

(*#* Binary Integers (Pierre Cr\233\gut, CNET, Lannion, France) *)

(*#**********************************************************)

include "NArith/BinPos.ma".

include "NArith/Pnat.ma".

include "NArith/BinNat.ma".

include "Arith/Plus.ma".

include "Arith/Mult.ma".

(*#*********************************************************************)

(*#* Binary integer numbers *)

inline procedural "cic:/Coq/ZArith/BinInt/Z.ind".

(*#* Declare Scope Z_scope with Key Z *)

(* UNEXPORTED
Delimit Scope Z_scope with Z.
*)

(*#* Automatically open scope positive_scope for the constructors of Z *)

(* UNEXPORTED
Bind Scope Z_scope with Z.
*)

(* UNEXPORTED
Arguments Scope Zpos [positive_scope].
*)

(* UNEXPORTED
Arguments Scope Zneg [positive_scope].
*)

(*#* Subtraction of positive into Z *)

inline procedural "cic:/Coq/ZArith/BinInt/Zdouble_plus_one.con" as definition.

inline procedural "cic:/Coq/ZArith/BinInt/Zdouble_minus_one.con" as definition.

inline procedural "cic:/Coq/ZArith/BinInt/Zdouble.con" as definition.

inline procedural "cic:/Coq/ZArith/BinInt/ZPminus.con" as definition.

(*#* Addition on integers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zplus.con" as definition.

(* NOTATION
Infix "+" := Zplus : Z_scope.
*)

(*#* Opposite *)

inline procedural "cic:/Coq/ZArith/BinInt/Zopp.con" as definition.

(* NOTATION
Notation "- x" := (Zopp x) : Z_scope.
*)

(*#* Successor on integers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zsucc.con" as definition.

(*#* Predecessor on integers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zpred.con" as definition.

(*#* Subtraction on integers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zminus.con" as definition.

(* NOTATION
Infix "-" := Zminus : Z_scope.
*)

(*#* Multiplication on integers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zmult.con" as definition.

(* NOTATION
Infix "*" := Zmult : Z_scope.
*)

(*#* Comparison of integers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zcompare.con" as definition.

(* NOTATION
Infix "?=" := Zcompare (at level 70, no associativity) : Z_scope.
*)

(* UNEXPORTED
Ltac elim_compare com1 com2 :=
  case (Dcompare (com1 ?= com2)%Z);
   [ idtac | let x := fresh "H" in
             (intro x; case x; clear x) ].
*)

(*#* Sign function *)

inline procedural "cic:/Coq/ZArith/BinInt/Zsgn.con" as definition.

(*#* Direct, easier to handle variants of successor and addition *)

inline procedural "cic:/Coq/ZArith/BinInt/Zsucc'.con" as definition.

inline procedural "cic:/Coq/ZArith/BinInt/Zpred'.con" as definition.

inline procedural "cic:/Coq/ZArith/BinInt/Zplus'.con" as definition.

(* UNEXPORTED
Open Local Scope Z_scope.
*)

(*#*********************************************************************)

(*#* Inductive specification of Z *)

inline procedural "cic:/Coq/ZArith/BinInt/Zind.con" as theorem.

(*#*********************************************************************)

(*#* Properties of opposite on binary integer numbers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zopp_neg.con" as theorem.

(*#* [opp] is involutive *)

inline procedural "cic:/Coq/ZArith/BinInt/Zopp_involutive.con" as theorem.

(*#* Injectivity of the opposite *)

inline procedural "cic:/Coq/ZArith/BinInt/Zopp_inj.con" as theorem.

(*#*********************************************************************)

(* Properties of the direct definition of successor and predecessor *)

inline procedural "cic:/Coq/ZArith/BinInt/Zpred'_succ'.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zsucc'_discr.con" as lemma.

(*#*********************************************************************)

(*#* Other properties of binary integer numbers *)

inline procedural "cic:/Coq/ZArith/BinInt/ZL0.con" as lemma.

(*#*********************************************************************)

(*#* Properties of the addition on integers *)

(*#* zero is left neutral for addition *)

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_0_l.con" as theorem.

(*#* zero is right neutral for addition *)

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_0_r.con" as theorem.

(*#* addition is commutative *)

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_comm.con" as theorem.

(*#* opposite distributes over addition *)

inline procedural "cic:/Coq/ZArith/BinInt/Zopp_plus_distr.con" as theorem.

(*#* opposite is inverse for addition *)

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_opp_r.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_opp_l.con" as theorem.

(* UNEXPORTED
Hint Local Resolve Zplus_0_l Zplus_0_r.
*)

(*#* addition is associative *)

inline procedural "cic:/Coq/ZArith/BinInt/weak_assoc.con" as lemma.

(* UNEXPORTED
Hint Local Resolve weak_assoc.
*)

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_assoc.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_assoc_reverse.con" as lemma.

(*#* Associativity mixed with commutativity *)

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_permute.con" as theorem.

(*#* addition simplifies *)

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_reg_l.con" as theorem.

(*#* addition and successor permutes *)

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_succ_l.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_succ_r.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_succ_comm.con" as lemma.

(*#* Misc properties, usually redundant or non natural *)

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_0_r_reverse.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_0_simpl_l.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_0_simpl_l_reverse.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_eq_compat.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_opp_expand.con" as lemma.

(*#*********************************************************************)

(*#* Properties of successor and predecessor on binary integer numbers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zsucc_discr.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zpos_succ_morphism.con" as theorem.

(*#* successor and predecessor are inverse functions *)

inline procedural "cic:/Coq/ZArith/BinInt/Zsucc_pred.con" as theorem.

(* UNEXPORTED
Hint Immediate Zsucc_pred: zarith.
*)

inline procedural "cic:/Coq/ZArith/BinInt/Zpred_succ.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zsucc_inj.con" as theorem.

(*#* Misc properties, usually redundant or non natural *)

inline procedural "cic:/Coq/ZArith/BinInt/Zsucc_eq_compat.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zsucc_inj_contrapositive.con" as lemma.

(*#*********************************************************************)

(*#* Properties of subtraction on binary integer numbers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zminus_0_r.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zminus_0_l_reverse.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zminus_diag.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zminus_diag_reverse.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_minus_eq.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zminus_plus.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_minus.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zminus_succ_l.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zminus_plus_simpl_l.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zminus_plus_simpl_l_reverse.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zminus_plus_simpl_r.con" as lemma.

(*#* Misc redundant properties *)

inline procedural "cic:/Coq/ZArith/BinInt/Zeq_minus.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zminus_eq.con" as lemma.

(*#*********************************************************************)

(*#* Properties of multiplication on binary integer numbers *)

(*#* One is neutral for multiplication *)

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_1_l.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_1_r.con" as theorem.

(*#* Zero property of multiplication *)

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_0_l.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_0_r.con" as theorem.

(* UNEXPORTED
Hint Local Resolve Zmult_0_l Zmult_0_r.
*)

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_0_r_reverse.con" as lemma.

(*#* Commutativity of multiplication *)

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_comm.con" as theorem.

(*#* Associativity of multiplication *)

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_assoc.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_assoc_reverse.con" as lemma.

(*#* Associativity mixed with commutativity *)

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_permute.con" as theorem.

(*#* Z is integral *)

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_integral_l.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_integral.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_1_inversion_l.con" as lemma.

(*#* Multiplication and Opposite *)

inline procedural "cic:/Coq/ZArith/BinInt/Zopp_mult_distr_l.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zopp_mult_distr_r.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zopp_mult_distr_l_reverse.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_opp_comm.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_opp_opp.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zopp_eq_mult_neg_1.con" as theorem.

(*#* Distributivity of multiplication over addition *)

inline procedural "cic:/Coq/ZArith/BinInt/weak_Zmult_plus_distr_r.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_plus_distr_r.con" as theorem.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_plus_distr_l.con" as theorem.

(*#* Distributivity of multiplication over subtraction *)

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_minus_distr_r.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_minus_distr_l.con" as lemma.

(*#* Simplification of multiplication for non-zero integers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_reg_l.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_reg_r.con" as lemma.

(*#* Addition and multiplication by 2 *)

inline procedural "cic:/Coq/ZArith/BinInt/Zplus_diag_eq_mult_2.con" as lemma.

(*#* Multiplication and successor *)

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_succ_r.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_succ_r_reverse.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_succ_l.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zmult_succ_l_reverse.con" as lemma.

(*#* Misc redundant properties *)

inline procedural "cic:/Coq/ZArith/BinInt/Z_eq_mult.con" as lemma.

(*#*********************************************************************)

(*#* Relating binary positive numbers and binary integers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zpos_xI.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zpos_xO.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zneg_xI.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zneg_xO.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zpos_plus_distr.con" as lemma.

inline procedural "cic:/Coq/ZArith/BinInt/Zneg_plus_distr.con" as lemma.

(*#*********************************************************************)

(*#* Order relations *)

inline procedural "cic:/Coq/ZArith/BinInt/Zlt.con" as definition.

inline procedural "cic:/Coq/ZArith/BinInt/Zgt.con" as definition.

inline procedural "cic:/Coq/ZArith/BinInt/Zle.con" as definition.

inline procedural "cic:/Coq/ZArith/BinInt/Zge.con" as definition.

inline procedural "cic:/Coq/ZArith/BinInt/Zne.con" as definition.

(* NOTATION
Infix "<=" := Zle : Z_scope.
*)

(* NOTATION
Infix "<" := Zlt : Z_scope.
*)

(* NOTATION
Infix ">=" := Zge : Z_scope.
*)

(* NOTATION
Infix ">" := Zgt : Z_scope.
*)

(* NOTATION
Notation "x <= y <= z" := (x <= y /\ y <= z) : Z_scope.
*)

(* NOTATION
Notation "x <= y < z" := (x <= y /\ y < z) : Z_scope.
*)

(* NOTATION
Notation "x < y < z" := (x < y /\ y < z) : Z_scope.
*)

(* NOTATION
Notation "x < y <= z" := (x < y /\ y <= z) : Z_scope.
*)

(*#*********************************************************************)

(*#* Absolute value on integers *)

inline procedural "cic:/Coq/ZArith/BinInt/Zabs_nat.con" as definition.

inline procedural "cic:/Coq/ZArith/BinInt/Zabs.con" as definition.

(*#*********************************************************************)

(*#* From [nat] to [Z] *)

inline procedural "cic:/Coq/ZArith/BinInt/Z_of_nat.con" as definition.

include "NArith/BinNat.ma".

inline procedural "cic:/Coq/ZArith/BinInt/Zabs_N.con" as definition.

inline procedural "cic:/Coq/ZArith/BinInt/Z_of_N.con" as definition.