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Jlocvar

Jlocvar-equiv

Basic equivalence relation for jlocvar structures.

Definitions and Theorems

Function: jlocvar-equiv$inline

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

Theorem: jlocvar-equiv-is-an-equivalence

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

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

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

Theorem: jlocvar-fix-under-jlocvar-equiv

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

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

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

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

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

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

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

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

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