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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Bool.
Require Export Ring_theory.
Require Export Ring_base.
Require Export InitialRing.
Require Export Ring_tac.
Lemma BoolTheory :
ring_theory false true xorb andb xorb (fun b:bool => b) (eq(A:=bool)).
split; simpl.
destruct x; reflexivity.
destruct x; destruct y; reflexivity.
destruct x; destruct y; destruct z; reflexivity.
reflexivity.
destruct x; destruct y; reflexivity.
destruct x; destruct y; reflexivity.
destruct x; destruct y; destruct z; reflexivity.
reflexivity.
destruct x; reflexivity.
Qed.
Definition bool_eq (b1 b2:bool) :=
if b1 then b2 else negb b2.
Lemma bool_eq_ok : forall b1 b2, bool_eq b1 b2 = true -> b1 = b2.
destruct b1; destruct b2; auto.
Qed.
Ltac bool_cst t :=
let t := eval hnf in t in
match t with
true => constr:true
| false => constr:false
| _ => constr:NotConstant
end.
Add Ring bool_ring : BoolTheory (decidable bool_eq_ok, constants [bool_cst]).
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