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• Maybe-nat-list

Maybe-nat-list-equiv

Basic equivalence relation for maybe-nat-list structures.

Definitions and Theorems

Function: maybe-nat-list-equiv$inline

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

Theorem: maybe-nat-list-equiv-is-an-equivalence

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

Theorem: maybe-nat-list-equiv-implies-equal-maybe-nat-list-fix-1

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

Theorem: maybe-nat-list-fix-under-maybe-nat-list-equiv

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

Theorem: equal-of-maybe-nat-list-fix-1-forward-to-maybe-nat-list-equiv

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

Theorem: equal-of-maybe-nat-list-fix-2-forward-to-maybe-nat-list-equiv

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

Theorem: maybe-nat-list-equiv-of-maybe-nat-list-fix-1-forward

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

Theorem: maybe-nat-list-equiv-of-maybe-nat-list-fix-2-forward

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