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• Jbinop

Jbinop-equiv

Basic equivalence relation for jbinop structures.

Definitions and Theorems

Function: jbinop-equiv$inline

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

Theorem: jbinop-equiv-is-an-equivalence

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

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

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

Theorem: jbinop-fix-under-jbinop-equiv

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

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

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

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

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

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

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

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

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