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Structures

31bits

An 31-bit unsigned bitstruct type.

This is a bitstruct type introduced by defbitstruct, represented as a unsigned 31-bit integer.

This is an atomic/empty structure; it has no fields.

Definitions and Theorems

Function: 31bits-p

(defun 31bits-p (x)
  (declare (xargs :guard t))
  (let ((__function__ '31bits-p))
    (declare (ignorable __function__))
    (mbe :logic (unsigned-byte-p 31 x)
         :exec (and (natp x) (< x 2147483648)))))

Theorem: 31bits-p-when-unsigned-byte-p

(defthm 31bits-p-when-unsigned-byte-p
  (implies (unsigned-byte-p 31 x)
           (31bits-p x)))

Theorem: unsigned-byte-p-when-31bits-p

(defthm unsigned-byte-p-when-31bits-p
  (implies (31bits-p x)
           (unsigned-byte-p 31 x)))

Theorem: 31bits-p-compound-recognizer

(defthm 31bits-p-compound-recognizer
  (implies (31bits-p x) (natp x))
  :rule-classes :compound-recognizer)

Function: 31bits-fix

(defun 31bits-fix (x)
  (declare (xargs :guard (31bits-p x)))
  (let ((__function__ '31bits-fix))
    (declare (ignorable __function__))
    (mbe :logic (loghead 31 x) :exec x)))

Theorem: 31bits-p-of-31bits-fix

(defthm 31bits-p-of-31bits-fix
  (b* ((fty::fixed (31bits-fix x)))
    (31bits-p fty::fixed))
  :rule-classes :rewrite)

Theorem: 31bits-fix-when-31bits-p

(defthm 31bits-fix-when-31bits-p
  (implies (31bits-p x)
           (equal (31bits-fix x) x)))

Function: 31bits-equiv$inline

(defun 31bits-equiv$inline (x y)
  (declare (xargs :guard (and (31bits-p x) (31bits-p y))))
  (equal (31bits-fix x) (31bits-fix y)))

Theorem: 31bits-equiv-is-an-equivalence

(defthm 31bits-equiv-is-an-equivalence
  (and (booleanp (31bits-equiv x y))
       (31bits-equiv x x)
       (implies (31bits-equiv x y)
                (31bits-equiv y x))
       (implies (and (31bits-equiv x y)
                     (31bits-equiv y z))
                (31bits-equiv x z)))
  :rule-classes (:equivalence))

Theorem: 31bits-equiv-implies-equal-31bits-fix-1

(defthm 31bits-equiv-implies-equal-31bits-fix-1
  (implies (31bits-equiv x x-equiv)
           (equal (31bits-fix x)
                  (31bits-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: 31bits-fix-under-31bits-equiv

(defthm 31bits-fix-under-31bits-equiv
  (31bits-equiv (31bits-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Subtopics

31bits-fix: Fixing function for 31bits bit structures.
31bits-p: Recognizer for 31bits bit structures.