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• Node-list

Node-list-equiv

Basic equivalence relation for node-list structures.

Definitions and Theorems

Function: node-list-equiv$inline

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

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

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

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

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

Theorem: node-list-fix-under-node-list-equiv

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

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

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

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

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

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

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

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

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