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• Symbol-list

Symbol-list-equiv

Basic equivalence relation for symbol-list structures.

Definitions and Theorems

Function: symbol-list-equiv$inline

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

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

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

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

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

Theorem: symbol-list-fix-under-symbol-list-equiv

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

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

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

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

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

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

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

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

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