		Search-engine friendly clone of the ACL2 documentation.

Truth4

Truth4-fix

Fixing function for truth4 bit structures.

Signature
(truth4-fix x) → fty::fixed
Arguments: x — Guard (truth4-p x).
Returns: fty::fixed — Type (truth4-p fty::fixed).

Definitions and Theorems

Function: truth4-fix

(defun truth4-fix (x)
  (declare (xargs :guard (truth4-p x)))
  (let ((__function__ 'truth4-fix))
    (declare (ignorable __function__))
    (mbe :logic (loghead 16 x) :exec x)))

Theorem: truth4-p-of-truth4-fix

(defthm truth4-p-of-truth4-fix
  (b* ((fty::fixed (truth4-fix x)))
    (truth4-p fty::fixed))
  :rule-classes :rewrite)

Theorem: truth4-fix-when-truth4-p

(defthm truth4-fix-when-truth4-p
  (implies (truth4-p x)
           (equal (truth4-fix x) x)))

Function: truth4-equiv$inline

(defun truth4-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (truth4-p acl2::x)
                              (truth4-p acl2::y))))
  (equal (truth4-fix acl2::x)
         (truth4-fix acl2::y)))

Theorem: truth4-equiv-is-an-equivalence

(defthm truth4-equiv-is-an-equivalence
  (and (booleanp (truth4-equiv x y))
       (truth4-equiv x x)
       (implies (truth4-equiv x y)
                (truth4-equiv y x))
       (implies (and (truth4-equiv x y)
                     (truth4-equiv y z))
                (truth4-equiv x z)))
  :rule-classes (:equivalence))

Theorem: truth4-equiv-implies-equal-truth4-fix-1

(defthm truth4-equiv-implies-equal-truth4-fix-1
  (implies (truth4-equiv acl2::x x-equiv)
           (equal (truth4-fix acl2::x)
                  (truth4-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: truth4-fix-under-truth4-equiv

(defthm truth4-fix-under-truth4-equiv
  (truth4-equiv (truth4-fix acl2::x)
                acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: truth4-fix-of-truth4-fix-x

(defthm truth4-fix-of-truth4-fix-x
  (equal (truth4-fix (truth4-fix x))
         (truth4-fix x)))

Theorem: truth4-fix-truth4-equiv-congruence-on-x

(defthm truth4-fix-truth4-equiv-congruence-on-x
  (implies (truth4-equiv x x-equiv)
           (equal (truth4-fix x)
                  (truth4-fix x-equiv)))
  :rule-classes :congruence)