• Top
  • Documentation
  • Books
  • Boolean-reasoning
  • Projects
  • Debugging
  • Std
    • Std/lists
    • Std/alists
    • Obags
    • Std/util
    • Std/strings
    • Std/osets
      • Omaps
        • Defomap
        • Update
        • Mapp
        • Assoc
        • Update*
        • Size
        • Keys
        • From-lists
        • Update-induction-on-maps
        • Compatiblep
        • Tail
        • Head
        • Restrict
        • Submap
        • Map
        • Rlookup
        • Emptyp
        • Rlookup*
        • Lookup*
        • Delete*
        • Values
        • In*
        • Lookup
        • Delete
        • Mfix
        • Head-val
        • Head-key
        • Omap-induction2
        • Omap-order-rules
      • All-by-membership
      • Defset
      • In
      • Primitives
      • Subset
      • Mergesort
      • Intersect
      • Union
      • Pick-a-point-subset-strategy
      • Delete
      • Double-containment
      • Difference
      • Cardinality
      • Set
      • Intersectp
    • Std/io
    • Std/basic
    • Std/system
    • Std/typed-lists
    • Std/bitsets
    • Std/testing
    • Std/typed-alists
    • Std/stobjs
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
  • Math
  • Testing-utilities

• 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)