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• Bool

Bool-equiv

Basic equivalence relation for bool structures.

Definitions and Theorems

Function: bool-equiv$inline

(defun bool-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (boolp acl2::x) (boolp acl2::y))))
  (equal (bool-fix acl2::x)
         (bool-fix acl2::y)))

Theorem: bool-equiv-is-an-equivalence

(defthm bool-equiv-is-an-equivalence
  (and (booleanp (bool-equiv x y))
       (bool-equiv x x)
       (implies (bool-equiv x y)
                (bool-equiv y x))
       (implies (and (bool-equiv x y) (bool-equiv y z))
                (bool-equiv x z)))
  :rule-classes (:equivalence))

Theorem: bool-equiv-implies-equal-bool-fix-1

(defthm bool-equiv-implies-equal-bool-fix-1
  (implies (bool-equiv acl2::x x-equiv)
           (equal (bool-fix acl2::x)
                  (bool-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: bool-fix-under-bool-equiv

(defthm bool-fix-under-bool-equiv
  (bool-equiv (bool-fix acl2::x) acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-bool-fix-1-forward-to-bool-equiv

(defthm equal-of-bool-fix-1-forward-to-bool-equiv
  (implies (equal (bool-fix acl2::x) acl2::y)
           (bool-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-bool-fix-2-forward-to-bool-equiv

(defthm equal-of-bool-fix-2-forward-to-bool-equiv
  (implies (equal acl2::x (bool-fix acl2::y))
           (bool-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bool-equiv-of-bool-fix-1-forward

(defthm bool-equiv-of-bool-fix-1-forward
  (implies (bool-equiv (bool-fix acl2::x) acl2::y)
           (bool-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bool-equiv-of-bool-fix-2-forward

(defthm bool-equiv-of-bool-fix-2-forward
  (implies (bool-equiv acl2::x (bool-fix acl2::y))
           (bool-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)