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• Struct-declon-list

Struct-declon-list-equiv

Basic equivalence relation for struct-declon-list structures.

Definitions and Theorems

Function: struct-declon-list-equiv$inline

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

Theorem: struct-declon-list-equiv-is-an-equivalence

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

Theorem: struct-declon-list-equiv-implies-equal-struct-declon-list-fix-1

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

Theorem: struct-declon-list-fix-under-struct-declon-list-equiv

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

Theorem: equal-of-struct-declon-list-fix-1-forward-to-struct-declon-list-equiv

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

Theorem: equal-of-struct-declon-list-fix-2-forward-to-struct-declon-list-equiv

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

Theorem: struct-declon-list-equiv-of-struct-declon-list-fix-1-forward

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

Theorem: struct-declon-list-equiv-of-struct-declon-list-fix-2-forward

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