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• Toplevel-list

Toplevel-list-equiv

Basic equivalence relation for toplevel-list structures.

Definitions and Theorems

Function: toplevel-list-equiv$inline

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

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

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

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

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

Theorem: toplevel-list-fix-under-toplevel-list-equiv

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

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

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

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

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

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

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

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

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