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Omaps

Head

Smallest key, and associated value, of a non-empty omap.

Signature
(head map) → (mv key val)
Arguments: map — Guard (mapp map).

This is similar to set::head for osets.

Definitions and Theorems

Function: head

(defun head (map)
       (declare (xargs :guard (mapp map)))
       (declare (xargs :guard (not (empty map))))
       (let ((__function__ 'head))
            (declare (ignorable __function__))
            (let ((pair (car (mfix map))))
                 (mv (car pair) (cdr pair)))))

Theorem: head-key-when-empty

(defthm head-key-when-empty
        (implies (empty map)
                 (equal (mv-nth 0 (head map)) nil))
        :rule-classes (:rewrite :type-prescription))

Theorem: head-value-when-empty

(defthm head-value-when-empty
        (implies (empty map)
                 (equal (mv-nth 1 (head map)) nil))
        :rule-classes (:rewrite :type-prescription))

Theorem: head-key-count

(defthm head-key-count
        (implies (not (empty map))
                 (< (acl2-count (mv-nth 0 (head map)))
                    (acl2-count map)))
        :rule-classes (:rewrite :linear))

Theorem: head-value-count

(defthm head-value-count
        (implies (not (empty map))
                 (< (acl2-count (mv-nth 1 (head map)))
                    (acl2-count map)))
        :rule-classes (:rewrite :linear))

Theorem: head-key-count-built-in

(defthm head-key-count-built-in
        (implies (not (empty map))
                 (o< (acl2-count (mv-nth 0 (head map)))
                     (acl2-count map)))
        :rule-classes :built-in-clause)

Theorem: head-value-count-built-in

(defthm head-value-count-built-in
        (implies (not (empty map))
                 (o< (acl2-count (mv-nth 1 (head map)))
                     (acl2-count map)))
        :rule-classes :built-in-clause)