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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 (empty bag))))
       (let ((__function__ 'head))
            (declare (ignorable __function__))
            (mbe :logic (car (bfix bag))
                 :exec (car bag))))

Theorem: head-when-empty

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

Theorem: head-count

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

Theorem: head-count-built-in

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