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• Assignment-list

Assignment-list-equiv

Basic equivalence relation for assignment-list structures.

Definitions and Theorems

Function: assignment-list-equiv$inline

(defun assignment-list-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (assignment-listp acl2::x)
                              (assignment-listp acl2::y))))
  (equal (assignment-list-fix acl2::x)
         (assignment-list-fix acl2::y)))

Theorem: assignment-list-equiv-is-an-equivalence

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

Theorem: assignment-list-equiv-implies-equal-assignment-list-fix-1

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

Theorem: assignment-list-fix-under-assignment-list-equiv

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

Theorem: equal-of-assignment-list-fix-1-forward-to-assignment-list-equiv

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

Theorem: equal-of-assignment-list-fix-2-forward-to-assignment-list-equiv

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

Theorem: assignment-list-equiv-of-assignment-list-fix-1-forward

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

Theorem: assignment-list-equiv-of-assignment-list-fix-2-forward

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