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Asgop

Asgop-equiv

Basic equivalence relation for asgop structures.

Definitions and Theorems

Function: asgop-equiv$inline

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

Theorem: asgop-equiv-is-an-equivalence

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

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

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

Theorem: asgop-fix-under-asgop-equiv

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

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

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

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

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

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

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

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

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