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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 (emptyp map))))
  (let ((__function__ 'head))
    (declare (ignorable __function__))
    (let ((pair (car (mfix map))))
      (mv (car pair) (cdr pair)))))

Theorem: head-key-when-emptyp

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

Theorem: head-value-when-emptyp

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

Theorem: head-key-count

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

Theorem: head-of-mfix-map

(defthm head-of-mfix-map
  (equal (head (mfix map)) (head map)))

Theorem: head-mequiv-congruence-on-map

(defthm head-mequiv-congruence-on-map
  (implies (mequiv map map-equiv)
           (equal (head map) (head map-equiv)))
  :rule-classes :congruence)