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• Obags

Tail

Rest of a non-empty obag after removing an occurrence of its smallest element.

Signature

(tail bag) → bag1

Arguments

bag — Guard (bagp bag).

Returns

bag1 — Type (bagp bag1).

This is similar to set::tail on osets.

Definitions and Theorems

Function: tail

(defun tail (bag)
  (declare (xargs :guard (bagp bag)))
  (declare (xargs :guard (not (emptyp bag))))
  (let ((__function__ 'tail))
    (declare (ignorable __function__))
    (mbe :logic (cdr (bfix bag))
         :exec (cdr bag))))

Theorem: bagp-of-tail

(defthm bagp-of-tail
  (b* ((bag1 (tail bag))) (bagp bag1))
  :rule-classes :rewrite)

Theorem: tail-when-emptyp

(defthm tail-when-emptyp
  (implies (emptyp bag)
           (equal (tail bag) nil))
  :rule-classes (:rewrite :type-prescription))

Theorem: tail-count

(defthm tail-count
  (implies (not (emptyp bag))
           (< (acl2-count (tail bag))
              (acl2-count bag)))
  :rule-classes (:rewrite :linear))

Theorem: tail-count-built-in

(defthm tail-count-built-in
  (implies (not (emptyp bag))
           (o< (acl2-count (tail bag))
               (acl2-count bag)))
  :rule-classes :built-in-clause)