		Search-engine friendly clone of the ACL2 documentation.

Dirabsdeclor

Dirabsdeclor-equiv

Basic equivalence relation for dirabsdeclor structures.

Definitions and Theorems

Function: dirabsdeclor-equiv$inline

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

Theorem: dirabsdeclor-equiv-is-an-equivalence

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

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

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

Theorem: dirabsdeclor-fix-under-dirabsdeclor-equiv

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

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

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

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

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

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

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

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

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