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• Obags

Head

Smallest element of a non-empty obag.

Signature

(head bag) → *

Arguments

bag — Guard (bagp bag).

This is similar to head for osets.

Definitions and Theorems

Function: head

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

Theorem: head-when-emptyp

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

Theorem: head-count

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

Theorem: head-count-built-in

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