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Polarity4

Polarity4-fix

Fixing function for polarity4 bit structures.

Signature
(polarity4-fix x) → fty::fixed
Arguments: x — Guard (polarity4-p x).
Returns: fty::fixed — Type (polarity4-p fty::fixed).

Definitions and Theorems

Function: polarity4-fix

(defun polarity4-fix (x)
  (declare (xargs :guard (polarity4-p x)))
  (let ((__function__ 'polarity4-fix))
    (declare (ignorable __function__))
    (mbe :logic (loghead 4 x) :exec x)))

Theorem: polarity4-p-of-polarity4-fix

(defthm polarity4-p-of-polarity4-fix
  (b* ((fty::fixed (polarity4-fix x)))
    (polarity4-p fty::fixed))
  :rule-classes :rewrite)

Theorem: polarity4-fix-when-polarity4-p

(defthm polarity4-fix-when-polarity4-p
  (implies (polarity4-p x)
           (equal (polarity4-fix x) x)))

Function: polarity4-equiv$inline

(defun polarity4-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (polarity4-p acl2::x)
                              (polarity4-p acl2::y))))
  (equal (polarity4-fix acl2::x)
         (polarity4-fix acl2::y)))

Theorem: polarity4-equiv-is-an-equivalence

(defthm polarity4-equiv-is-an-equivalence
  (and (booleanp (polarity4-equiv x y))
       (polarity4-equiv x x)
       (implies (polarity4-equiv x y)
                (polarity4-equiv y x))
       (implies (and (polarity4-equiv x y)
                     (polarity4-equiv y z))
                (polarity4-equiv x z)))
  :rule-classes (:equivalence))

Theorem: polarity4-equiv-implies-equal-polarity4-fix-1

(defthm polarity4-equiv-implies-equal-polarity4-fix-1
  (implies (polarity4-equiv acl2::x x-equiv)
           (equal (polarity4-fix acl2::x)
                  (polarity4-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: polarity4-fix-under-polarity4-equiv

(defthm polarity4-fix-under-polarity4-equiv
  (polarity4-equiv (polarity4-fix acl2::x)
                   acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: polarity4-fix-of-polarity4-fix-x

(defthm polarity4-fix-of-polarity4-fix-x
  (equal (polarity4-fix (polarity4-fix x))
         (polarity4-fix x)))

Theorem: polarity4-fix-polarity4-equiv-congruence-on-x

(defthm polarity4-fix-polarity4-equiv-congruence-on-x
  (implies (polarity4-equiv x x-equiv)
           (equal (polarity4-fix x)
                  (polarity4-fix x-equiv)))
  :rule-classes :congruence)