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• Sllong-integer-list

Sllong-integer-list-equiv

Basic equivalence relation for sllong-integer-list structures.

Definitions and Theorems

Function: sllong-integer-list-equiv$inline

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

Theorem: sllong-integer-list-equiv-is-an-equivalence

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

Theorem: sllong-integer-list-equiv-implies-equal-sllong-integer-list-fix-1

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

Theorem: sllong-integer-list-fix-under-sllong-integer-list-equiv

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

Theorem: equal-of-sllong-integer-list-fix-1-forward-to-sllong-integer-list-equiv

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

Theorem: equal-of-sllong-integer-list-fix-2-forward-to-sllong-integer-list-equiv

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

Theorem: sllong-integer-list-equiv-of-sllong-integer-list-fix-1-forward

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

Theorem: sllong-integer-list-equiv-of-sllong-integer-list-fix-2-forward

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