Library Coq.setoid_ring.Ring

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)).

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.

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]).