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• ACL2-number-list

ACL2-number-list-equiv

Basic equivalence relation for ACL2-number-list structures.

Definitions and Theorems

Function: acl2-number-list-equiv$inline

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

Theorem: acl2-number-list-equiv-is-an-equivalence

(defthm acl2-number-list-equiv-is-an-equivalence
  (and (booleanp (acl2-number-list-equiv x y))
       (acl2-number-list-equiv x x)
       (implies (acl2-number-list-equiv x y)
                (acl2-number-list-equiv y x))
       (implies (and (acl2-number-list-equiv x y)
                     (acl2-number-list-equiv y z))
                (acl2-number-list-equiv x z)))
  :rule-classes (:equivalence))

Theorem: acl2-number-list-equiv-implies-equal-acl2-number-list-fix-1

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

Theorem: acl2-number-list-fix-under-acl2-number-list-equiv

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

Theorem: equal-of-acl2-number-list-fix-1-forward-to-acl2-number-list-equiv

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

Theorem: equal-of-acl2-number-list-fix-2-forward-to-acl2-number-list-equiv

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

Theorem: acl2-number-list-equiv-of-acl2-number-list-fix-1-forward

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

Theorem: acl2-number-list-equiv-of-acl2-number-list-fix-2-forward

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