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Sparseint

Sparseint-fix

Fixing function for sparseints

Signature
(sparseint-fix x) → new-x
Arguments: x — Guard (sparseint-p x).
Returns: new-x — Type (sparseint-p new-x).

Definitions and Theorems

Function: sparseint-fix$inline

(defun sparseint-fix$inline (x)
  (declare (xargs :guard (sparseint-p x)))
  (let ((__function__ 'sparseint-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((x (sparseint$-fix x))
              ((when (sparseint$-height-correctp x))
               x)
              ((sparseint$-concat x)))
           (sparseint$-concatenate x.width (sparseint-fix x.lsbs)
                                   (sparseint-fix x.msbs)))
         :exec x)))

Theorem: sparseint-p-of-sparseint-fix

(defthm sparseint-p-of-sparseint-fix
  (b* ((new-x (sparseint-fix$inline x)))
    (sparseint-p new-x))
  :rule-classes :rewrite)

Theorem: sparseint-fix-when-sparseint-p

(defthm sparseint-fix-when-sparseint-p
  (implies (sparseint-p x)
           (equal (sparseint-fix x) x)))

Function: sparseint-equiv$inline

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

Theorem: sparseint-equiv-is-an-equivalence

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

Theorem: sparseint-equiv-implies-equal-sparseint-fix-1

(defthm sparseint-equiv-implies-equal-sparseint-fix-1
  (implies (sparseint-equiv x x-equiv)
           (equal (sparseint-fix x)
                  (sparseint-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sparseint-fix-under-sparseint-equiv

(defthm sparseint-fix-under-sparseint-equiv
  (sparseint-equiv (sparseint-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sparseint-fix-1-forward-to-sparseint-equiv

(defthm equal-of-sparseint-fix-1-forward-to-sparseint-equiv
  (implies (equal (sparseint-fix x) y)
           (sparseint-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-sparseint-fix-2-forward-to-sparseint-equiv

(defthm equal-of-sparseint-fix-2-forward-to-sparseint-equiv
  (implies (equal x (sparseint-fix y))
           (sparseint-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sparseint-equiv-of-sparseint-fix-1-forward

(defthm sparseint-equiv-of-sparseint-fix-1-forward
  (implies (sparseint-equiv (sparseint-fix x) y)
           (sparseint-equiv x y))
  :rule-classes :forward-chaining)

Theorem: sparseint-equiv-of-sparseint-fix-2-forward

(defthm sparseint-equiv-of-sparseint-fix-2-forward
  (implies (sparseint-equiv x (sparseint-fix y))
           (sparseint-equiv x y))
  :rule-classes :forward-chaining)