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• Lit-list

Lit-list-equiv

Basic equivalence relation for lit-list structures.

Definitions and Theorems

Function: lit-list-equiv$inline

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

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

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

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

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

Theorem: lit-list-fix-under-lit-list-equiv

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

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

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

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

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

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

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

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

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