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Sparseint$-sign-ext

Signature

(sparseint$-sign-ext n x x.height) → (mv ext height)

Arguments

n — Guard (posp n).

x — Guard (sparseint$-p x).

x.height — Guard (natp x.height).

Returns

ext — Type (sparseint$-p ext).

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

Definitions and Theorems

Function: sparseint$-sign-ext

(defun sparseint$-sign-ext (n x x.height)
  (declare (xargs :guard (and (posp n)
                              (sparseint$-p x)
                              (natp x.height))))
  (declare
       (xargs :guard (and (sparseint$-height-correctp x)
                          (equal x.height (sparseint$-height x)))))
  (let ((__function__ 'sparseint$-sign-ext))
    (declare (ignorable __function__))
    (b* ((n (lposfix n))
         (x.height (mbe :logic (sparseint$-height x)
                        :exec x.height)))
      (sparseint$-concatenate-rebalance
           n x x.height
           (sparseint$-leaf (- (sparseint$-bit (1- n) x)))
           0))))

Theorem: sparseint$-p-of-sparseint$-sign-ext.ext

(defthm sparseint$-p-of-sparseint$-sign-ext.ext
  (b* (((mv ?ext ?height)
        (sparseint$-sign-ext n x x.height)))
    (sparseint$-p ext))
  :rule-classes :rewrite)

Theorem: return-type-of-sparseint$-sign-ext.height

(defthm return-type-of-sparseint$-sign-ext.height
  (b* (((mv ?ext ?height)
        (sparseint$-sign-ext n x x.height)))
    (equal height (sparseint$-height ext)))
  :rule-classes :rewrite)

Theorem: sparseint$-sign-ext-height-correct

(defthm sparseint$-sign-ext-height-correct
  (b* (((mv ?ext ?height)
        (sparseint$-sign-ext n x x.height)))
    (implies (sparseint$-height-correctp x)
             (sparseint$-height-correctp ext))))

Theorem: sparseint$-sign-ext-correct

(defthm sparseint$-sign-ext-correct
  (b* (((mv ?ext ?height)
        (sparseint$-sign-ext n x x.height)))
    (equal (sparseint$-val ext)
           (logext n (sparseint$-val x)))))

Theorem: sparseint$-sign-ext-of-pos-fix-n

(defthm sparseint$-sign-ext-of-pos-fix-n
  (equal (sparseint$-sign-ext (pos-fix n)
                              x x.height)
         (sparseint$-sign-ext n x x.height)))

Theorem: sparseint$-sign-ext-pos-equiv-congruence-on-n

(defthm sparseint$-sign-ext-pos-equiv-congruence-on-n
  (implies (pos-equiv n n-equiv)
           (equal (sparseint$-sign-ext n x x.height)
                  (sparseint$-sign-ext n-equiv x x.height)))
  :rule-classes :congruence)

Theorem: sparseint$-sign-ext-of-sparseint$-fix-x

(defthm sparseint$-sign-ext-of-sparseint$-fix-x
  (equal (sparseint$-sign-ext n (sparseint$-fix x)
                              x.height)
         (sparseint$-sign-ext n x x.height)))

Theorem: sparseint$-sign-ext-sparseint$-equiv-congruence-on-x

(defthm sparseint$-sign-ext-sparseint$-equiv-congruence-on-x
  (implies (sparseint$-equiv x x-equiv)
           (equal (sparseint$-sign-ext n x x.height)
                  (sparseint$-sign-ext n x-equiv x.height)))
  :rule-classes :congruence)

Theorem: sparseint$-sign-ext-of-nfix-x.height

(defthm sparseint$-sign-ext-of-nfix-x.height
  (equal (sparseint$-sign-ext n x (nfix x.height))
         (sparseint$-sign-ext n x x.height)))

Theorem: sparseint$-sign-ext-nat-equiv-congruence-on-x.height

(defthm sparseint$-sign-ext-nat-equiv-congruence-on-x.height
  (implies (nat-equiv x.height x.height-equiv)
           (equal (sparseint$-sign-ext n x x.height)
                  (sparseint$-sign-ext n x x.height-equiv)))
  :rule-classes :congruence)