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Attrib

Attrib-equiv

Basic equivalence relation for attrib structures.

Definitions and Theorems

Function: attrib-equiv$inline

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

Theorem: attrib-equiv-is-an-equivalence

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

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

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

Theorem: attrib-fix-under-attrib-equiv

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

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

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

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

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

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

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

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

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