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Heap

Heap-equiv

Basic equivalence relation for heap structures.

Definitions and Theorems

Function: heap-equiv$inline

(defun heap-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (heapp acl2::x) (heapp acl2::y))))
  (equal (heap-fix acl2::x)
         (heap-fix acl2::y)))

Theorem: heap-equiv-is-an-equivalence

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

Theorem: heap-equiv-implies-equal-heap-fix-1

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

Theorem: heap-fix-under-heap-equiv

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

Theorem: equal-of-heap-fix-1-forward-to-heap-equiv

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

Theorem: equal-of-heap-fix-2-forward-to-heap-equiv

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

Theorem: heap-equiv-of-heap-fix-1-forward

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

Theorem: heap-equiv-of-heap-fix-2-forward

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