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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)