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Sparseint$-rightshift-rec

Signature

(sparseint$-rightshift-rec shift x x.height) 
  → 
(mv rightshift height)

Arguments

shift — Guard (natp shift).

x — Guard (sparseint$-p x).

Returns

rightshift — Type (sparseint$-p rightshift).

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

Definitions and Theorems

Function: sparseint$-rightshift-rec

(defun sparseint$-rightshift-rec (shift x x.height)
 (declare (xargs :guard (and (natp shift) (sparseint$-p x))))
 (declare
      (xargs :guard (and (sparseint$-height-correctp x)
                         (equal x.height (sparseint$-height x)))))
 (let ((__function__ 'sparseint$-rightshift-rec))
  (declare (ignorable __function__))
  (sparseint$-case
   x :leaf
   (mv (sparseint$-leaf (logtail shift x.val))
       0)
   :concat
   (b*
    ((shift (lnfix shift))
     (x.height (mbe :logic (sparseint$-height x)
                    :exec x.height))
     ((when (eql shift 0))
      (mv (sparseint$-fix x) x.height))
     (x.msbs.height (mbe :logic (sparseint$-height x.msbs)
                         :exec (- x.height (if x.lsbs-taller 2 1))))
     ((when (<= x.width shift))
      (sparseint$-rightshift-rec (- shift x.width)
                                 x.msbs x.msbs.height))
     (x.lsbs.height (mbe :logic (sparseint$-height x.lsbs)
                         :exec (- x.height (if x.msbs-taller 2 1))))
     ((mv lsbs-shift lsbs-shift-height)
      (sparseint$-rightshift-rec shift x.lsbs x.lsbs.height)))
    (sparseint$-concatenate-rebalance (- x.width shift)
                                      lsbs-shift lsbs-shift-height
                                      x.msbs x.msbs.height)))))

Theorem: sparseint$-p-of-sparseint$-rightshift-rec.rightshift

(defthm sparseint$-p-of-sparseint$-rightshift-rec.rightshift
  (b* (((mv ?rightshift ?height)
        (sparseint$-rightshift-rec shift x x.height)))
    (sparseint$-p rightshift))
  :rule-classes :rewrite)

Theorem: return-type-of-sparseint$-rightshift-rec.height

(defthm return-type-of-sparseint$-rightshift-rec.height
  (b* (((mv ?rightshift ?height)
        (sparseint$-rightshift-rec shift x x.height)))
    (equal height (sparseint$-height rightshift)))
  :rule-classes :rewrite)

Theorem: sparseint$-height-correctp-of-sparseint$-rightshift-rec

(defthm sparseint$-height-correctp-of-sparseint$-rightshift-rec
  (b* (((mv ?rightshift ?height)
        (sparseint$-rightshift-rec shift x x.height)))
    (implies (sparseint$-height-correctp x)
             (sparseint$-height-correctp rightshift))))

Theorem: sparseint$-rightshift-rec-correct

(defthm sparseint$-rightshift-rec-correct
  (b* (((mv ?rightshift ?height)
        (sparseint$-rightshift-rec shift x x.height)))
    (equal (sparseint$-val rightshift)
           (logtail shift (sparseint$-val x)))))

Theorem: height-of-sparseint$-rightshift-rec

(defthm height-of-sparseint$-rightshift-rec
  (b* (((mv ?rightshift ?height)
        (sparseint$-rightshift-rec shift x x.height)))
    (implies (sparseint$-height-correctp x)
             (<= (sparseint$-height rightshift)
                 (sparseint$-height x))))
  :rule-classes :linear)

Theorem: sparseint$-rightshift-rec-of-nfix-shift

(defthm sparseint$-rightshift-rec-of-nfix-shift
  (equal (sparseint$-rightshift-rec (nfix shift)
                                    x x.height)
         (sparseint$-rightshift-rec shift x x.height)))

Theorem: sparseint$-rightshift-rec-nat-equiv-congruence-on-shift

(defthm sparseint$-rightshift-rec-nat-equiv-congruence-on-shift
  (implies
       (nat-equiv shift shift-equiv)
       (equal (sparseint$-rightshift-rec shift x x.height)
              (sparseint$-rightshift-rec shift-equiv x x.height)))
  :rule-classes :congruence)

Theorem: sparseint$-rightshift-rec-of-sparseint$-fix-x

(defthm sparseint$-rightshift-rec-of-sparseint$-fix-x
  (equal (sparseint$-rightshift-rec shift (sparseint$-fix x)
                                    x.height)
         (sparseint$-rightshift-rec shift x x.height)))

Theorem: sparseint$-rightshift-rec-sparseint$-equiv-congruence-on-x

(defthm sparseint$-rightshift-rec-sparseint$-equiv-congruence-on-x
  (implies
       (sparseint$-equiv x x-equiv)
       (equal (sparseint$-rightshift-rec shift x x.height)
              (sparseint$-rightshift-rec shift x-equiv x.height)))
  :rule-classes :congruence)