		Search-engine friendly clone of the ACL2 documentation.

Node

Node-equiv

Basic equivalence relation for node structures.

Definitions and Theorems

Function: node-equiv$inline

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

Theorem: node-equiv-is-an-equivalence

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

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

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

Theorem: node-fix-under-node-equiv

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

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

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

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

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

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

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

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

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