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• Sparseint-impl

Sparseint$-plus-int-width

Signature

(sparseint$-plus-int-width width offset x x.height y cin) 
  → 
(mv sum height cout)

Arguments

width — Guard (posp width).

offset — Guard (natp offset).

x — Guard (sparseint$-p x).

x.height — Guard (natp x.height).

y — Guard (integerp y).

cin — Guard (bitp cin).

Returns

sum — Type (sparseint$-p sum).

height — Type (equal height (sparseint$-height sum)).

cout — Type (bitp cout).

Definitions and Theorems

Function: sparseint$-plus-int-width

(defun sparseint$-plus-int-width (width offset x x.height y cin)
 (declare (xargs :guard (and (posp width)
                             (natp offset)
                             (sparseint$-p x)
                             (natp x.height)
                             (integerp y)
                             (bitp cin))))
 (declare (xargs :guard (and (sparseint$-height-correctp x)
                             (equal x.height (sparseint$-height x))
                             (< (integer-length y) width))))
 (let ((__function__ 'sparseint$-plus-int-width))
  (declare (ignorable __function__))
  (b* ((width (lposfix width))
       (offset (lnfix offset))
       (y (mbe :logic (logext width y) :exec y))
       (cin (lbfix cin))
       (x.height (mbe :logic (sparseint$-height x)
                      :exec x.height)))
   (sparseint$-case
       x :leaf
       (b* ((xval (bignum-logext width (logtail offset x.val)))
            (sum (bignum-logext width (sum-with-cin cin xval y)))
            (cout (carry-out-bit (logbit (1- width) xval)
                                 (logbit (1- width) y)
                                 (logbit (1- width) sum))))
         (mv (sparseint$-leaf sum) 0 cout))
       :concat
       (b* ((x.msbs.height
                 (mbe :logic (sparseint$-height x.msbs)
                      :exec (- x.height (if x.lsbs-taller 2 1))))
            ((when (<= x.width offset))
             (sparseint$-plus-int-width width (- offset x.width)
                                        x.msbs x.msbs.height y cin))
            (width1 (- x.width offset))
            (x.lsbs.height
                 (mbe :logic (sparseint$-height x.lsbs)
                      :exec (- x.height (if x.msbs-taller 2 1))))
            ((when (<= width width1))
             (sparseint$-plus-int-width
                  width
                  offset x.lsbs x.lsbs.height y cin))
            ((mv lsbs-sum lsbs-sum-height lsbs-cout)
             (sparseint$-plus-int-width
                  width1 offset x.lsbs
                  x.lsbs.height (bignum-logext width1 y)
                  cin))
            ((mv msbs-sum msbs-sum-height msbs-cout)
             (sparseint$-plus-int-width
                  (- width width1)
                  0
                  x.msbs x.msbs.height (logtail width1 y)
                  lsbs-cout))
            ((mv sum-concat sum-height)
             (sparseint$-concatenate-rebalance
                  width1 lsbs-sum lsbs-sum-height
                  msbs-sum msbs-sum-height)))
         (mv sum-concat sum-height msbs-cout))))))

Theorem: sparseint$-p-of-sparseint$-plus-int-width.sum

(defthm sparseint$-p-of-sparseint$-plus-int-width.sum
  (b* (((mv ?sum ?height ?cout)
        (sparseint$-plus-int-width width offset x x.height y cin)))
    (sparseint$-p sum))
  :rule-classes :rewrite)

Theorem: return-type-of-sparseint$-plus-int-width.height

(defthm return-type-of-sparseint$-plus-int-width.height
  (b* (((mv ?sum ?height ?cout)
        (sparseint$-plus-int-width width offset x x.height y cin)))
    (equal height (sparseint$-height sum)))
  :rule-classes :rewrite)

Theorem: bitp-of-sparseint$-plus-int-width.cout

(defthm bitp-of-sparseint$-plus-int-width.cout
  (b* (((mv ?sum ?height ?cout)
        (sparseint$-plus-int-width width offset x x.height y cin)))
    (bitp cout))
  :rule-classes :type-prescription)

Theorem: sparseint$-height-correctp-of-sparseint$-plus-int-width

(defthm sparseint$-height-correctp-of-sparseint$-plus-int-width
  (b* (((mv ?sum ?height ?cout)
        (sparseint$-plus-int-width width offset x x.height y cin)))
    (implies (sparseint$-height-correctp x)
             (sparseint$-height-correctp sum))))

Theorem: sparseint$-val-of-sparseint$-plus-int-width

(defthm sparseint$-val-of-sparseint$-plus-int-width
 (b* (((mv ?sum ?height ?cout)
       (sparseint$-plus-int-width width offset x x.height y cin)))
  (and
   (equal
       (sparseint$-val sum)
       (logext width
               (sum-with-cin cin (logtail offset (sparseint$-val x))
                             y)))
   (equal cout
          (carry-out width
                     cin (logtail offset (sparseint$-val x))
                     y)))))

Theorem: sparseint$-plus-int-width-of-pos-fix-width

(defthm sparseint$-plus-int-width-of-pos-fix-width
  (equal (sparseint$-plus-int-width (pos-fix width)
                                    offset x x.height y cin)
         (sparseint$-plus-int-width width offset x x.height y cin)))

Theorem: sparseint$-plus-int-width-pos-equiv-congruence-on-width

(defthm sparseint$-plus-int-width-pos-equiv-congruence-on-width
 (implies
  (pos-equiv width width-equiv)
  (equal
   (sparseint$-plus-int-width width offset x x.height y cin)
   (sparseint$-plus-int-width width-equiv offset x x.height y cin)))
 :rule-classes :congruence)

Theorem: sparseint$-plus-int-width-of-nfix-offset

(defthm sparseint$-plus-int-width-of-nfix-offset
  (equal (sparseint$-plus-int-width width (nfix offset)
                                    x x.height y cin)
         (sparseint$-plus-int-width width offset x x.height y cin)))

Theorem: sparseint$-plus-int-width-nat-equiv-congruence-on-offset

(defthm sparseint$-plus-int-width-nat-equiv-congruence-on-offset
 (implies
  (nat-equiv offset offset-equiv)
  (equal
   (sparseint$-plus-int-width width offset x x.height y cin)
   (sparseint$-plus-int-width width offset-equiv x x.height y cin)))
 :rule-classes :congruence)

Theorem: sparseint$-plus-int-width-of-sparseint$-fix-x

(defthm sparseint$-plus-int-width-of-sparseint$-fix-x
  (equal (sparseint$-plus-int-width width offset (sparseint$-fix x)
                                    x.height y cin)
         (sparseint$-plus-int-width width offset x x.height y cin)))

Theorem: sparseint$-plus-int-width-sparseint$-equiv-congruence-on-x

(defthm sparseint$-plus-int-width-sparseint$-equiv-congruence-on-x
 (implies
  (sparseint$-equiv x x-equiv)
  (equal
   (sparseint$-plus-int-width width offset x x.height y cin)
   (sparseint$-plus-int-width width offset x-equiv x.height y cin)))
 :rule-classes :congruence)

Theorem: sparseint$-plus-int-width-of-nfix-x.height

(defthm sparseint$-plus-int-width-of-nfix-x.height
  (equal (sparseint$-plus-int-width width offset x (nfix x.height)
                                    y cin)
         (sparseint$-plus-int-width width offset x x.height y cin)))

Theorem: sparseint$-plus-int-width-nat-equiv-congruence-on-x.height

(defthm sparseint$-plus-int-width-nat-equiv-congruence-on-x.height
 (implies
  (nat-equiv x.height x.height-equiv)
  (equal
   (sparseint$-plus-int-width width offset x x.height y cin)
   (sparseint$-plus-int-width width offset x x.height-equiv y cin)))
 :rule-classes :congruence)

Theorem: sparseint$-plus-int-width-of-ifix-y

(defthm sparseint$-plus-int-width-of-ifix-y
  (equal (sparseint$-plus-int-width width offset x x.height (ifix y)
                                    cin)
         (sparseint$-plus-int-width width offset x x.height y cin)))

Theorem: sparseint$-plus-int-width-int-equiv-congruence-on-y

(defthm sparseint$-plus-int-width-int-equiv-congruence-on-y
 (implies
  (int-equiv y y-equiv)
  (equal
   (sparseint$-plus-int-width width offset x x.height y cin)
   (sparseint$-plus-int-width width offset x x.height y-equiv cin)))
 :rule-classes :congruence)

Theorem: sparseint$-plus-int-width-of-bfix-cin

(defthm sparseint$-plus-int-width-of-bfix-cin
 (equal
    (sparseint$-plus-int-width width offset x x.height y (bfix cin))
    (sparseint$-plus-int-width width offset x x.height y cin)))

Theorem: sparseint$-plus-int-width-bit-equiv-congruence-on-cin

(defthm sparseint$-plus-int-width-bit-equiv-congruence-on-cin
 (implies
  (bit-equiv cin cin-equiv)
  (equal
   (sparseint$-plus-int-width width offset x x.height y cin)
   (sparseint$-plus-int-width width offset x x.height y cin-equiv)))
 :rule-classes :congruence)