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  • Math
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• Modr/m-decoding
• Structures

Modr/m-structures

Bitstruct definitions to store a ModR/M byte and its fields.

Definitions and Theorems

Function: modr/m-p$inline

(defun modr/m-p$inline (x)
  (declare (xargs :guard t))
  (mbe :logic (unsigned-byte-p 8 x)
       :exec (and (natp x) (< x 256))))

Theorem: modr/m-p-when-unsigned-byte-p

(defthm modr/m-p-when-unsigned-byte-p
  (implies (unsigned-byte-p 8 x)
           (modr/m-p x)))

Theorem: unsigned-byte-p-when-modr/m-p

(defthm unsigned-byte-p-when-modr/m-p
  (implies (modr/m-p x)
           (unsigned-byte-p 8 x)))

Theorem: modr/m-p-compound-recognizer

(defthm modr/m-p-compound-recognizer
  (implies (modr/m-p x) (natp x))
  :rule-classes :compound-recognizer)

Function: modr/m-fix$inline

(defun modr/m-fix$inline (x)
  (declare (xargs :guard (modr/m-p x)))
  (mbe :logic (loghead 8 x) :exec x))

Theorem: modr/m-p-of-modr/m-fix

(defthm modr/m-p-of-modr/m-fix
  (b* ((fty::fixed (modr/m-fix$inline x)))
    (modr/m-p fty::fixed))
  :rule-classes :rewrite)

Theorem: modr/m-fix-when-modr/m-p

(defthm modr/m-fix-when-modr/m-p
  (implies (modr/m-p x)
           (equal (modr/m-fix x) x)))

Function: modr/m-equiv$inline

(defun modr/m-equiv$inline (x y)
  (declare (xargs :guard (and (modr/m-p x) (modr/m-p y))))
  (equal (modr/m-fix x) (modr/m-fix y)))

Theorem: modr/m-equiv-is-an-equivalence

(defthm modr/m-equiv-is-an-equivalence
  (and (booleanp (modr/m-equiv x y))
       (modr/m-equiv x x)
       (implies (modr/m-equiv x y)
                (modr/m-equiv y x))
       (implies (and (modr/m-equiv x y)
                     (modr/m-equiv y z))
                (modr/m-equiv x z)))
  :rule-classes (:equivalence))

Theorem: modr/m-equiv-implies-equal-modr/m-fix-1

(defthm modr/m-equiv-implies-equal-modr/m-fix-1
  (implies (modr/m-equiv x x-equiv)
           (equal (modr/m-fix x)
                  (modr/m-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: modr/m-fix-under-modr/m-equiv

(defthm modr/m-fix-under-modr/m-equiv
  (modr/m-equiv (modr/m-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Function: modr/m$inline

(defun modr/m$inline (r/m reg mod)
  (declare (xargs :guard (and (3bits-p r/m)
                              (3bits-p reg)
                              (2bits-p mod))))
  (b* ((r/m (mbe :logic (3bits-fix r/m) :exec r/m))
       (reg (mbe :logic (3bits-fix reg) :exec reg))
       (mod (mbe :logic (2bits-fix mod) :exec mod)))
    (logapp 3 r/m (logapp 3 reg mod))))

Theorem: modr/m-p-of-modr/m

(defthm modr/m-p-of-modr/m
  (b* ((modr/m (modr/m$inline r/m reg mod)))
    (modr/m-p modr/m))
  :rule-classes :rewrite)

Theorem: modr/m$inline-of-3bits-fix-r/m

(defthm modr/m$inline-of-3bits-fix-r/m
  (equal (modr/m$inline (3bits-fix r/m) reg mod)
         (modr/m$inline r/m reg mod)))

Theorem: modr/m$inline-3bits-equiv-congruence-on-r/m

(defthm modr/m$inline-3bits-equiv-congruence-on-r/m
  (implies (3bits-equiv r/m r/m-equiv)
           (equal (modr/m$inline r/m reg mod)
                  (modr/m$inline r/m-equiv reg mod)))
  :rule-classes :congruence)

Theorem: modr/m$inline-of-3bits-fix-reg

(defthm modr/m$inline-of-3bits-fix-reg
  (equal (modr/m$inline r/m (3bits-fix reg) mod)
         (modr/m$inline r/m reg mod)))

Theorem: modr/m$inline-3bits-equiv-congruence-on-reg

(defthm modr/m$inline-3bits-equiv-congruence-on-reg
  (implies (3bits-equiv reg reg-equiv)
           (equal (modr/m$inline r/m reg mod)
                  (modr/m$inline r/m reg-equiv mod)))
  :rule-classes :congruence)

Theorem: modr/m$inline-of-2bits-fix-mod

(defthm modr/m$inline-of-2bits-fix-mod
  (equal (modr/m$inline r/m reg (2bits-fix mod))
         (modr/m$inline r/m reg mod)))

Theorem: modr/m$inline-2bits-equiv-congruence-on-mod

(defthm modr/m$inline-2bits-equiv-congruence-on-mod
  (implies (2bits-equiv mod mod-equiv)
           (equal (modr/m$inline r/m reg mod)
                  (modr/m$inline r/m reg mod-equiv)))
  :rule-classes :congruence)

Function: modr/m-equiv-under-mask$inline

(defun modr/m-equiv-under-mask$inline (x x1 mask)
  (declare (xargs :guard (and (modr/m-p x)
                              (modr/m-p x1)
                              (integerp mask))))
  (fty::int-equiv-under-mask (modr/m-fix x)
                             (modr/m-fix x1)
                             mask))

Function: modr/m->r/m$inline

(defun modr/m->r/m$inline (x)
  (declare (xargs :guard (modr/m-p x)))
  (mbe :logic
       (let ((x (modr/m-fix x)))
         (part-select x :low 0 :width 3))
       :exec (the (unsigned-byte 3)
                  (logand (the (unsigned-byte 3) 7)
                          (the (unsigned-byte 8) x)))))

Theorem: 3bits-p-of-modr/m->r/m

(defthm 3bits-p-of-modr/m->r/m
  (b* ((r/m (modr/m->r/m$inline x)))
    (3bits-p r/m))
  :rule-classes :rewrite)

Theorem: modr/m->r/m$inline-of-modr/m-fix-x

(defthm modr/m->r/m$inline-of-modr/m-fix-x
  (equal (modr/m->r/m$inline (modr/m-fix x))
         (modr/m->r/m$inline x)))

Theorem: modr/m->r/m$inline-modr/m-equiv-congruence-on-x

(defthm modr/m->r/m$inline-modr/m-equiv-congruence-on-x
  (implies (modr/m-equiv x x-equiv)
           (equal (modr/m->r/m$inline x)
                  (modr/m->r/m$inline x-equiv)))
  :rule-classes :congruence)

Theorem: modr/m->r/m-of-modr/m

(defthm modr/m->r/m-of-modr/m
  (equal (modr/m->r/m (modr/m r/m reg mod))
         (3bits-fix r/m)))

Theorem: modr/m->r/m-of-write-with-mask

(defthm modr/m->r/m-of-write-with-mask
 (implies
  (and
     (fty::bitstruct-read-over-write-hyps x modr/m-equiv-under-mask)
     (modr/m-equiv-under-mask x y fty::mask)
     (equal (logand (lognot fty::mask) 7) 0))
  (equal (modr/m->r/m x)
         (modr/m->r/m y))))

Function: modr/m->reg$inline

(defun modr/m->reg$inline (x)
 (declare (xargs :guard (modr/m-p x)))
 (mbe
     :logic
     (let ((x (modr/m-fix x)))
       (part-select x :low 3 :width 3))
     :exec (the (unsigned-byte 3)
                (logand (the (unsigned-byte 3) 7)
                        (the (unsigned-byte 5)
                             (ash (the (unsigned-byte 8) x) -3))))))

Theorem: 3bits-p-of-modr/m->reg

(defthm 3bits-p-of-modr/m->reg
  (b* ((reg (modr/m->reg$inline x)))
    (3bits-p reg))
  :rule-classes :rewrite)

Theorem: modr/m->reg$inline-of-modr/m-fix-x

(defthm modr/m->reg$inline-of-modr/m-fix-x
  (equal (modr/m->reg$inline (modr/m-fix x))
         (modr/m->reg$inline x)))

Theorem: modr/m->reg$inline-modr/m-equiv-congruence-on-x

(defthm modr/m->reg$inline-modr/m-equiv-congruence-on-x
  (implies (modr/m-equiv x x-equiv)
           (equal (modr/m->reg$inline x)
                  (modr/m->reg$inline x-equiv)))
  :rule-classes :congruence)

Theorem: modr/m->reg-of-modr/m

(defthm modr/m->reg-of-modr/m
  (equal (modr/m->reg (modr/m r/m reg mod))
         (3bits-fix reg)))

Theorem: modr/m->reg-of-write-with-mask

(defthm modr/m->reg-of-write-with-mask
 (implies
  (and
     (fty::bitstruct-read-over-write-hyps x modr/m-equiv-under-mask)
     (modr/m-equiv-under-mask x y fty::mask)
     (equal (logand (lognot fty::mask) 56)
            0))
  (equal (modr/m->reg x)
         (modr/m->reg y))))

Function: modr/m->mod$inline

(defun modr/m->mod$inline (x)
 (declare (xargs :guard (modr/m-p x)))
 (mbe
     :logic
     (let ((x (modr/m-fix x)))
       (part-select x :low 6 :width 2))
     :exec (the (unsigned-byte 2)
                (logand (the (unsigned-byte 2) 3)
                        (the (unsigned-byte 2)
                             (ash (the (unsigned-byte 8) x) -6))))))

Theorem: 2bits-p-of-modr/m->mod

(defthm 2bits-p-of-modr/m->mod
  (b* ((mod (modr/m->mod$inline x)))
    (2bits-p mod))
  :rule-classes :rewrite)

Theorem: modr/m->mod$inline-of-modr/m-fix-x

(defthm modr/m->mod$inline-of-modr/m-fix-x
  (equal (modr/m->mod$inline (modr/m-fix x))
         (modr/m->mod$inline x)))

Theorem: modr/m->mod$inline-modr/m-equiv-congruence-on-x

(defthm modr/m->mod$inline-modr/m-equiv-congruence-on-x
  (implies (modr/m-equiv x x-equiv)
           (equal (modr/m->mod$inline x)
                  (modr/m->mod$inline x-equiv)))
  :rule-classes :congruence)

Theorem: modr/m->mod-of-modr/m

(defthm modr/m->mod-of-modr/m
  (equal (modr/m->mod (modr/m r/m reg mod))
         (2bits-fix mod)))

Theorem: modr/m->mod-of-write-with-mask

(defthm modr/m->mod-of-write-with-mask
 (implies
  (and
     (fty::bitstruct-read-over-write-hyps x modr/m-equiv-under-mask)
     (modr/m-equiv-under-mask x y fty::mask)
     (equal (logand (lognot fty::mask) 192)
            0))
  (equal (modr/m->mod x)
         (modr/m->mod y))))

Theorem: modr/m-fix-in-terms-of-modr/m

(defthm modr/m-fix-in-terms-of-modr/m
  (equal (modr/m-fix x)
         (change-modr/m x)))

Function: !modr/m->r/m$inline

(defun !modr/m->r/m$inline (r/m x)
  (declare (xargs :guard (and (3bits-p r/m) (modr/m-p x))))
  (mbe :logic
       (b* ((r/m (mbe :logic (3bits-fix r/m) :exec r/m))
            (x (modr/m-fix x)))
         (part-install r/m x :width 3 :low 0))
       :exec (the (unsigned-byte 8)
                  (logior (the (unsigned-byte 8)
                               (logand (the (unsigned-byte 8) x)
                                       (the (signed-byte 4) -8)))
                          (the (unsigned-byte 3) r/m)))))

Theorem: modr/m-p-of-!modr/m->r/m

(defthm modr/m-p-of-!modr/m->r/m
  (b* ((new-x (!modr/m->r/m$inline r/m x)))
    (modr/m-p new-x))
  :rule-classes :rewrite)

Theorem: !modr/m->r/m$inline-of-3bits-fix-r/m

(defthm !modr/m->r/m$inline-of-3bits-fix-r/m
  (equal (!modr/m->r/m$inline (3bits-fix r/m) x)
         (!modr/m->r/m$inline r/m x)))

Theorem: !modr/m->r/m$inline-3bits-equiv-congruence-on-r/m

(defthm !modr/m->r/m$inline-3bits-equiv-congruence-on-r/m
  (implies (3bits-equiv r/m r/m-equiv)
           (equal (!modr/m->r/m$inline r/m x)
                  (!modr/m->r/m$inline r/m-equiv x)))
  :rule-classes :congruence)

Theorem: !modr/m->r/m$inline-of-modr/m-fix-x

(defthm !modr/m->r/m$inline-of-modr/m-fix-x
  (equal (!modr/m->r/m$inline r/m (modr/m-fix x))
         (!modr/m->r/m$inline r/m x)))

Theorem: !modr/m->r/m$inline-modr/m-equiv-congruence-on-x

(defthm !modr/m->r/m$inline-modr/m-equiv-congruence-on-x
  (implies (modr/m-equiv x x-equiv)
           (equal (!modr/m->r/m$inline r/m x)
                  (!modr/m->r/m$inline r/m x-equiv)))
  :rule-classes :congruence)

Theorem: !modr/m->r/m-is-modr/m

(defthm !modr/m->r/m-is-modr/m
  (equal (!modr/m->r/m r/m x)
         (change-modr/m x :r/m r/m)))

Theorem: modr/m->r/m-of-!modr/m->r/m

(defthm modr/m->r/m-of-!modr/m->r/m
  (b* ((?new-x (!modr/m->r/m$inline r/m x)))
    (equal (modr/m->r/m new-x)
           (3bits-fix r/m))))

Theorem: !modr/m->r/m-equiv-under-mask

(defthm !modr/m->r/m-equiv-under-mask
  (b* ((?new-x (!modr/m->r/m$inline r/m x)))
    (modr/m-equiv-under-mask new-x x -8)))

Function: !modr/m->reg$inline

(defun !modr/m->reg$inline (reg x)
 (declare (xargs :guard (and (3bits-p reg) (modr/m-p x))))
 (mbe
    :logic
    (b* ((reg (mbe :logic (3bits-fix reg) :exec reg))
         (x (modr/m-fix x)))
      (part-install reg x :width 3 :low 3))
    :exec (the (unsigned-byte 8)
               (logior (the (unsigned-byte 8)
                            (logand (the (unsigned-byte 8) x)
                                    (the (signed-byte 7) -57)))
                       (the (unsigned-byte 6)
                            (ash (the (unsigned-byte 3) reg) 3))))))

Theorem: modr/m-p-of-!modr/m->reg

(defthm modr/m-p-of-!modr/m->reg
  (b* ((new-x (!modr/m->reg$inline reg x)))
    (modr/m-p new-x))
  :rule-classes :rewrite)

Theorem: !modr/m->reg$inline-of-3bits-fix-reg

(defthm !modr/m->reg$inline-of-3bits-fix-reg
  (equal (!modr/m->reg$inline (3bits-fix reg) x)
         (!modr/m->reg$inline reg x)))

Theorem: !modr/m->reg$inline-3bits-equiv-congruence-on-reg

(defthm !modr/m->reg$inline-3bits-equiv-congruence-on-reg
  (implies (3bits-equiv reg reg-equiv)
           (equal (!modr/m->reg$inline reg x)
                  (!modr/m->reg$inline reg-equiv x)))
  :rule-classes :congruence)

Theorem: !modr/m->reg$inline-of-modr/m-fix-x

(defthm !modr/m->reg$inline-of-modr/m-fix-x
  (equal (!modr/m->reg$inline reg (modr/m-fix x))
         (!modr/m->reg$inline reg x)))

Theorem: !modr/m->reg$inline-modr/m-equiv-congruence-on-x

(defthm !modr/m->reg$inline-modr/m-equiv-congruence-on-x
  (implies (modr/m-equiv x x-equiv)
           (equal (!modr/m->reg$inline reg x)
                  (!modr/m->reg$inline reg x-equiv)))
  :rule-classes :congruence)

Theorem: !modr/m->reg-is-modr/m

(defthm !modr/m->reg-is-modr/m
  (equal (!modr/m->reg reg x)
         (change-modr/m x :reg reg)))

Theorem: modr/m->reg-of-!modr/m->reg

(defthm modr/m->reg-of-!modr/m->reg
  (b* ((?new-x (!modr/m->reg$inline reg x)))
    (equal (modr/m->reg new-x)
           (3bits-fix reg))))

Theorem: !modr/m->reg-equiv-under-mask

(defthm !modr/m->reg-equiv-under-mask
  (b* ((?new-x (!modr/m->reg$inline reg x)))
    (modr/m-equiv-under-mask new-x x -57)))

Function: !modr/m->mod$inline

(defun !modr/m->mod$inline (mod x)
 (declare (xargs :guard (and (2bits-p mod) (modr/m-p x))))
 (mbe
    :logic
    (b* ((mod (mbe :logic (2bits-fix mod) :exec mod))
         (x (modr/m-fix x)))
      (part-install mod x :width 2 :low 6))
    :exec (the (unsigned-byte 8)
               (logior (the (unsigned-byte 8)
                            (logand (the (unsigned-byte 8) x)
                                    (the (signed-byte 9) -193)))
                       (the (unsigned-byte 8)
                            (ash (the (unsigned-byte 2) mod) 6))))))

Theorem: modr/m-p-of-!modr/m->mod

(defthm modr/m-p-of-!modr/m->mod
  (b* ((new-x (!modr/m->mod$inline mod x)))
    (modr/m-p new-x))
  :rule-classes :rewrite)

Theorem: !modr/m->mod$inline-of-2bits-fix-mod

(defthm !modr/m->mod$inline-of-2bits-fix-mod
  (equal (!modr/m->mod$inline (2bits-fix mod) x)
         (!modr/m->mod$inline mod x)))

Theorem: !modr/m->mod$inline-2bits-equiv-congruence-on-mod

(defthm !modr/m->mod$inline-2bits-equiv-congruence-on-mod
  (implies (2bits-equiv mod mod-equiv)
           (equal (!modr/m->mod$inline mod x)
                  (!modr/m->mod$inline mod-equiv x)))
  :rule-classes :congruence)

Theorem: !modr/m->mod$inline-of-modr/m-fix-x

(defthm !modr/m->mod$inline-of-modr/m-fix-x
  (equal (!modr/m->mod$inline mod (modr/m-fix x))
         (!modr/m->mod$inline mod x)))

Theorem: !modr/m->mod$inline-modr/m-equiv-congruence-on-x

(defthm !modr/m->mod$inline-modr/m-equiv-congruence-on-x
  (implies (modr/m-equiv x x-equiv)
           (equal (!modr/m->mod$inline mod x)
                  (!modr/m->mod$inline mod x-equiv)))
  :rule-classes :congruence)

Theorem: !modr/m->mod-is-modr/m

(defthm !modr/m->mod-is-modr/m
  (equal (!modr/m->mod mod x)
         (change-modr/m x :mod mod)))

Theorem: modr/m->mod-of-!modr/m->mod

(defthm modr/m->mod-of-!modr/m->mod
  (b* ((?new-x (!modr/m->mod$inline mod x)))
    (equal (modr/m->mod new-x)
           (2bits-fix mod))))

Theorem: !modr/m->mod-equiv-under-mask

(defthm !modr/m->mod-equiv-under-mask
  (b* ((?new-x (!modr/m->mod$inline mod x)))
    (modr/m-equiv-under-mask new-x x 63)))

Function: modr/m-debug$inline

(defun modr/m-debug$inline (x)
  (declare (xargs :guard (modr/m-p x)))
  (b* (((modr/m x)))
    (cons (cons 'r/m x.r/m)
          (cons (cons 'reg x.reg)
                (cons (cons 'mod x.mod) nil)))))

Theorem: return-type-of-modr/m->r/m-linear

(defthm return-type-of-modr/m->r/m-linear
  (< (modr/m->r/m modr/m) 8)
  :rule-classes :linear)

Theorem: return-type-of-modr/m->reg-linear

(defthm return-type-of-modr/m->reg-linear
  (< (modr/m->reg modr/m) 8)
  :rule-classes :linear)

Theorem: return-type-of-modr/m->mod-linear

(defthm return-type-of-modr/m->mod-linear
  (< (modr/m->mod modr/m) 4)
  :rule-classes :linear)

Subtopics

Modr/m

An 8-bit unsigned bitstruct type.

Modr/m-equiv-under-mask

Modr/m-debug