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Truth3

Truth3-fix

Fixing function for truth3 bit structures.

Signature
(truth3-fix x) → fty::fixed
Arguments: x — Guard (truth3-p x).
Returns: fty::fixed — Type (truth3-p fty::fixed).

Definitions and Theorems

Function: truth3-fix

(defun truth3-fix (x)
  (declare (xargs :guard (truth3-p x)))
  (let ((__function__ 'truth3-fix))
    (declare (ignorable __function__))
    (mbe :logic (loghead 8 x) :exec x)))

Theorem: truth3-p-of-truth3-fix

(defthm truth3-p-of-truth3-fix
  (b* ((fty::fixed (truth3-fix x)))
    (truth3-p fty::fixed))
  :rule-classes :rewrite)

Theorem: truth3-fix-when-truth3-p

(defthm truth3-fix-when-truth3-p
  (implies (truth3-p x)
           (equal (truth3-fix x) x)))

Function: truth3-equiv$inline

(defun truth3-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (truth3-p acl2::x)
                              (truth3-p acl2::y))))
  (equal (truth3-fix acl2::x)
         (truth3-fix acl2::y)))

Theorem: truth3-equiv-is-an-equivalence

(defthm truth3-equiv-is-an-equivalence
  (and (booleanp (truth3-equiv x y))
       (truth3-equiv x x)
       (implies (truth3-equiv x y)
                (truth3-equiv y x))
       (implies (and (truth3-equiv x y)
                     (truth3-equiv y z))
                (truth3-equiv x z)))
  :rule-classes (:equivalence))

Theorem: truth3-equiv-implies-equal-truth3-fix-1

(defthm truth3-equiv-implies-equal-truth3-fix-1
  (implies (truth3-equiv acl2::x x-equiv)
           (equal (truth3-fix acl2::x)
                  (truth3-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: truth3-fix-under-truth3-equiv

(defthm truth3-fix-under-truth3-equiv
  (truth3-equiv (truth3-fix acl2::x)
                acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: truth3-fix-of-truth3-fix-x

(defthm truth3-fix-of-truth3-fix-x
  (equal (truth3-fix (truth3-fix x))
         (truth3-fix x)))

Theorem: truth3-fix-truth3-equiv-congruence-on-x

(defthm truth3-fix-truth3-equiv-congruence-on-x
  (implies (truth3-equiv x x-equiv)
           (equal (truth3-fix x)
                  (truth3-fix x-equiv)))
  :rule-classes :congruence)