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Structures

2bits

An 2-bit unsigned bitstruct type.

This is a bitstruct type introduced by defbitstruct, represented as a unsigned 2-bit integer.

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

Definitions and Theorems

Function: 2bits-p

(defun 2bits-p (x)
  (declare (xargs :guard t))
  (let ((__function__ '2bits-p))
    (declare (ignorable __function__))
    (mbe :logic (unsigned-byte-p 2 x)
         :exec (and (natp x) (< x 4)))))

Theorem: 2bits-p-when-unsigned-byte-p

(defthm 2bits-p-when-unsigned-byte-p
  (implies (unsigned-byte-p 2 x)
           (2bits-p x)))

Theorem: unsigned-byte-p-when-2bits-p

(defthm unsigned-byte-p-when-2bits-p
  (implies (2bits-p x)
           (unsigned-byte-p 2 x)))

Theorem: 2bits-p-compound-recognizer

(defthm 2bits-p-compound-recognizer
  (implies (2bits-p x) (natp x))
  :rule-classes :compound-recognizer)

Function: 2bits-fix

(defun 2bits-fix (x)
  (declare (xargs :guard (2bits-p x)))
  (let ((__function__ '2bits-fix))
    (declare (ignorable __function__))
    (mbe :logic (loghead 2 x) :exec x)))

Theorem: 2bits-p-of-2bits-fix

(defthm 2bits-p-of-2bits-fix
  (b* ((fty::fixed (2bits-fix x)))
    (2bits-p fty::fixed))
  :rule-classes :rewrite)

Theorem: 2bits-fix-when-2bits-p

(defthm 2bits-fix-when-2bits-p
  (implies (2bits-p x)
           (equal (2bits-fix x) x)))

Function: 2bits-equiv$inline

(defun 2bits-equiv$inline (x y)
  (declare (xargs :guard (and (2bits-p x) (2bits-p y))))
  (equal (2bits-fix x) (2bits-fix y)))

Theorem: 2bits-equiv-is-an-equivalence

(defthm 2bits-equiv-is-an-equivalence
  (and (booleanp (2bits-equiv x y))
       (2bits-equiv x x)
       (implies (2bits-equiv x y)
                (2bits-equiv y x))
       (implies (and (2bits-equiv x y)
                     (2bits-equiv y z))
                (2bits-equiv x z)))
  :rule-classes (:equivalence))

Theorem: 2bits-equiv-implies-equal-2bits-fix-1

(defthm 2bits-equiv-implies-equal-2bits-fix-1
  (implies (2bits-equiv x x-equiv)
           (equal (2bits-fix x)
                  (2bits-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: 2bits-fix-under-2bits-equiv

(defthm 2bits-fix-under-2bits-equiv
  (2bits-equiv (2bits-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Subtopics

2bits-p: Recognizer for 2bits bit structures.
2bits-fix: Fixing function for 2bits bit structures.