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Heap

Heap-fix

(heap-fix x) is a usual ACL2::fty omap fixing function.

Signature
(heap-fix x) → *
Arguments: x — Guard (heapp x).

Definitions and Theorems

Function: heap-fix

(defun heap-fix (x)
  (declare (xargs :guard (heapp x)))
  (mbe :logic (if (heapp x) x nil)
       :exec x))

Theorem: heapp-of-heap-fix

(defthm heapp-of-heap-fix
  (heapp (heap-fix x)))

Theorem: heap-fix-when-heapp

(defthm heap-fix-when-heapp
  (implies (heapp x)
           (equal (heap-fix x) x)))

Theorem: emptyp-heap-fix

(defthm emptyp-heap-fix
  (implies (or (omap::emptyp x) (not (heapp x)))
           (omap::emptyp (heap-fix x))))

Theorem: emptyp-of-heap-fix-to-not-heap-or-emptyp

(defthm emptyp-of-heap-fix-to-not-heap-or-emptyp
  (equal (omap::emptyp (heap-fix x))
         (or (not (heapp x)) (omap::emptyp x))))

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)