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4v-operations

4v-and

Four-valued semantics for and gates.

(4v-and a b) returns:

F when either input is F, or
X when either input is X/Z, or
T when both inputs are T.

See 4v-unfloat for an explanation of the Z case.

Definitions and Theorems

Function: 4v-and$inline

(defun 4v-and$inline (a b)
  (declare (xargs :guard t))
  (mbe :logic (4vcases a (t (4v-unfloat b))
                       (f (4vf))
                       (& (4vcases b (f (4vf)) (& (4vx)))))
       :exec (cond ((or (eq a (4vf)) (eq b (4vf))) (4vf))
                   ((and (eq a (4vt)) (eq b (4vt))) (4vt))
                   (t (4vx)))))

Theorem: 4v-equiv-implies-equal-4v-and-2

(defthm 4v-equiv-implies-equal-4v-and-2
  (implies (4v-equiv b b-equiv)
           (equal (4v-and a b) (4v-and a b-equiv)))
  :rule-classes (:congruence))

Theorem: 4v-equiv-implies-equal-4v-and-1

(defthm 4v-equiv-implies-equal-4v-and-1
  (implies (4v-equiv a a-equiv)
           (equal (4v-and a b) (4v-and a-equiv b)))
  :rule-classes (:congruence))