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• Bitsets

Bitset-insert

(bitset-insert a x) constructs the set X U {a}.

Signature

(bitset-insert a x) → bitset

Arguments

a — Guard (natp a).

x — Guard (natp x).

Returns

bitset — Type (natp bitset).

This looks pretty efficient, but keep in mind that it still has to construct a new set and hence is linear in the size of the set. You should probably avoid calling this in a loop if possible; instead consider functions like bitset-union.

Definitions and Theorems

Function: bitset-insert$inline

(defun acl2::bitset-insert$inline (a x)
  (declare (type unsigned-byte a)
           (type unsigned-byte x))
  (declare (xargs :guard (and (natp a) (natp x))))
  (declare (xargs :split-types t))
  (let ((__function__ 'bitset-insert))
    (declare (ignorable __function__))
    (the unsigned-byte
         (logior (the unsigned-byte (ash 1 (lnfix a)))
                 (the unsigned-byte (lnfix x))))))

Theorem: natp-of-bitset-insert

(defthm acl2::natp-of-bitset-insert
  (b* ((bitset (acl2::bitset-insert$inline a x)))
    (natp bitset))
  :rule-classes :type-prescription)

Theorem: bitset-insert-when-not-natp-left

(defthm bitset-insert-when-not-natp-left
  (implies (not (natp a))
           (equal (bitset-insert a x)
                  (bitset-insert 0 x))))

Theorem: bitset-insert-when-not-natp-right

(defthm bitset-insert-when-not-natp-right
  (implies (not (natp x))
           (equal (bitset-insert a x)
                  (bitset-singleton a))))

Theorem: bitset-insert-of-nfix-left

(defthm bitset-insert-of-nfix-left
  (equal (bitset-insert (nfix a) x)
         (bitset-insert a x)))

Theorem: bitset-insert-of-nfix-right

(defthm bitset-insert-of-nfix-right
  (equal (bitset-insert a (nfix x))
         (bitset-insert a x)))

Theorem: bitset-members-of-bitset-insert

(defthm set::bitset-members-of-bitset-insert
  (equal (bitset-members (bitset-insert a x))
         (insert (nfix a) (bitset-members x))))