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• Jaccess

Jaccess-equiv

Basic equivalence relation for jaccess structures.

Definitions and Theorems

Function: jaccess-equiv$inline

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

Theorem: jaccess-equiv-is-an-equivalence

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

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

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

Theorem: jaccess-fix-under-jaccess-equiv

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

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

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

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

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

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

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

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

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