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• Sparseint-impl

Sparseint$-plus-offset

Signature

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

Arguments

x — Guard (sparseint$-p x).

x.height — Guard (natp x.height).

y-offset — Guard (natp y-offset).

y — Guard (sparseint$-p y).

y.height — Guard (natp y.height).

cin — Guard (bitp cin).

Returns

sum — Type (sparseint$-p sum).

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

Definitions and Theorems

Function: sparseint$-plus-offset

(defun sparseint$-plus-offset (x x.height y-offset y y.height cin)
 (declare (xargs :guard (and (sparseint$-p x)
                             (natp x.height)
                             (natp y-offset)
                             (sparseint$-p y)
                             (natp y.height)
                             (bitp cin))))
 (declare
      (xargs :guard (and (sparseint$-height-correctp x)
                         (equal x.height (sparseint$-height x))
                         (sparseint$-height-correctp y)
                         (equal y.height (sparseint$-height y)))))
 (let ((__function__ 'sparseint$-plus-offset))
  (declare (ignorable __function__))
  (b* ((x.height (mbe :logic (sparseint$-height x)
                      :exec x.height))
       (y.height (mbe :logic (sparseint$-height y)
                      :exec y.height))
       (y-offset (lnfix y-offset)))
   (sparseint$-case
    x
    :leaf
    (sparseint$-case
        y
        :leaf
        (mv (sparseint$-leaf
                 (sum-with-cin cin x.val (logtail y-offset y.val)))
            0)
        :concat (sparseint$-plus-int y-offset y y.height x.val cin))
    :concat
    (sparseint$-case
      y :leaf
      (sparseint$-plus-int 0 x x.height (logtail y-offset y.val)
                           cin)
      :concat
      (b*
        ((y.msbs.height
              (mbe :logic (sparseint$-height y.msbs)
                   :exec (- y.height (if y.lsbs-taller 2 1))))
         ((when (<= y.width y-offset))
          (sparseint$-plus-offset x x.height (- y-offset y.width)
                                  y.msbs y.msbs.height cin))
         (x.lsbs.height
              (mbe :logic (sparseint$-height x.lsbs)
                   :exec (- x.height (if x.msbs-taller 2 1))))
         ((mv lsbs-sum lsbs-sum.height lsbs-cout)
          (sparseint$-plus-width
               x.width x.lsbs
               x.lsbs.height y-offset y y.height cin))
         (x.msbs.height
              (mbe :logic (sparseint$-height x.msbs)
                   :exec (- x.height (if x.lsbs-taller 2 1))))
         ((mv msbs-sum msbs-sum.height)
          (sparseint$-plus-offset x.msbs
                                  x.msbs.height (+ x.width y-offset)
                                  y y.height lsbs-cout)))
        (sparseint$-concatenate-rebalance
             x.width lsbs-sum lsbs-sum.height
             msbs-sum msbs-sum.height)))))))

Theorem: sparseint$-p-of-sparseint$-plus-offset.sum

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

Theorem: return-type-of-sparseint$-plus-offset.height

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

Theorem: sparseint$-height-correctp-of-sparseint$-plus-offset

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

Theorem: sparseint$-val-of-sparseint$-plus-offset

(defthm sparseint$-val-of-sparseint$-plus-offset
 (b* (((mv ?sum ?height)
       (sparseint$-plus-offset x x.height y-offset y y.height cin)))
   (equal (sparseint$-val sum)
          (sum-with-cin cin (sparseint$-val x)
                        (logtail y-offset (sparseint$-val y))))))

Theorem: sparseint$-plus-offset-of-sparseint$-fix-x

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

Theorem: sparseint$-plus-offset-sparseint$-equiv-congruence-on-x

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

Theorem: sparseint$-plus-offset-of-nfix-x.height

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

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

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

Theorem: sparseint$-plus-offset-of-nfix-y-offset

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

Theorem: sparseint$-plus-offset-nat-equiv-congruence-on-y-offset

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

Theorem: sparseint$-plus-offset-of-sparseint$-fix-y

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

Theorem: sparseint$-plus-offset-sparseint$-equiv-congruence-on-y

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

Theorem: sparseint$-plus-offset-of-nfix-y.height

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

Theorem: sparseint$-plus-offset-nat-equiv-congruence-on-y.height

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

Theorem: sparseint$-plus-offset-of-bfix-cin

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

Theorem: sparseint$-plus-offset-bit-equiv-congruence-on-cin

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