(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the GNU General Public License as published by  *)
(*  the Free Software Foundation, either version 2 of the License, or  *)
(*  (at your option) any later version.  This file is also distributed *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)
(** This file collects a number of definitions and theorems that are used throughout the development.  It complements the Coq standard library. *)
Require Export String.
Require Export ZArith.
Require Export Znumtheory.
Require Export List.
Require Export Bool.
Global Set Asymmetric Patterns.
(** * Useful tactics *)
Ltac inv H := inversion H; clear H; subst.
Ltac predSpec pred predspec x y :=
generalize (predspec x y); case (pred x y); intro.
Ltac caseEq name :=
generalize (eq_refl name); pattern name at -1 in |- *; case name.
Ltac destructEq name :=
destruct name eqn:?.
Ltac decEq :=
match goal with
| [ |- _ = _ ] => f_equal
| [ |- (?X ?A <> ?X ?B) ] =>
cut (A <> B); [intro; congruence | try discriminate]
end.
Ltac byContradiction :=
cut False; [contradiction|idtac].
Ltac omegaContradiction :=
cut False; [contradiction|omega].

Lemma peq_false:
forall (A: Type) (x y: positive) (a b: A), x <> y -> (if peq x y then a else b) = b.