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A4vec-operations

A3vec-bit?

Symbolic version of a3vec-bit?.

Signature
(a3vec-bit? x y y3p z z3p) → res
Arguments: x — Guard (a4vec-p x).; y — Guard (a4vec-p y).; z — Guard (a4vec-p z).
Returns: res — Type (a4vec-p res).

Definitions and Theorems

Function: a3vec-bit?

(defun a3vec-bit? (x y y3p z z3p)
 (declare (xargs :guard (and (a4vec-p x)
                             (a4vec-p y)
                             (a4vec-p z))))
 (let ((__function__ 'a3vec-bit?))
   (declare (ignorable __function__))
   (b* (((a4vec a) x)
        ((a4vec b) y)
        ((a4vec c) z)
        (a=x (aig-logandc2-ss a.upper a.lower))
        (boolcase (aig-logior-ss (aig-logand-ss a.lower b.upper)
                                 (aig-logandc1-ss a.upper c.upper)))
        (xcase (aig-logior-ss (if z3p c.upper
                                (aig-logior-ss c.upper c.lower))
                              (if y3p b.upper
                                (aig-logior-ss b.upper b.lower))))
        (upper (aig-logior-ss boolcase (aig-logand-ss a=x xcase)))
        (boolcase (aig-logior-ss (aig-logand-ss a.lower b.lower)
                                 (aig-logandc1-ss a.upper c.lower)))
        (xcase (aig-logand-ss (if z3p c.lower
                                (aig-logand-ss c.upper c.lower))
                              (if y3p b.lower
                                (aig-logand-ss b.upper b.lower))))
        (lower (aig-logior-ss boolcase (aig-logand-ss a=x xcase))))
     (a4vec upper lower))))

Theorem: a4vec-p-of-a3vec-bit?

(defthm a4vec-p-of-a3vec-bit?
  (b* ((res (a3vec-bit? x y y3p z z3p)))
    (a4vec-p res))
  :rule-classes :rewrite)

Theorem: a3vec-bit?-correct

(defthm a3vec-bit?-correct
  (implies
       (and (case-split (implies y3p (3vec-p (a4vec-eval y env))))
            (case-split (implies z3p (3vec-p (a4vec-eval z env))))
            (3vec-p (a4vec-eval x env)))
       (equal (a4vec-eval (a3vec-bit? x y y3p z z3p)
                          env)
              (3vec-bit? (a4vec-eval x env)
                         (a4vec-eval y env)
                         (a4vec-eval z env)))))

Theorem: a3vec-bit?-of-a4vec-fix-x

(defthm a3vec-bit?-of-a4vec-fix-x
  (equal (a3vec-bit? (a4vec-fix x) y y3p z z3p)
         (a3vec-bit? x y y3p z z3p)))

Theorem: a3vec-bit?-a4vec-equiv-congruence-on-x

(defthm a3vec-bit?-a4vec-equiv-congruence-on-x
  (implies (a4vec-equiv x x-equiv)
           (equal (a3vec-bit? x y y3p z z3p)
                  (a3vec-bit? x-equiv y y3p z z3p)))
  :rule-classes :congruence)

Theorem: a3vec-bit?-of-a4vec-fix-y

(defthm a3vec-bit?-of-a4vec-fix-y
  (equal (a3vec-bit? x (a4vec-fix y) y3p z z3p)
         (a3vec-bit? x y y3p z z3p)))

Theorem: a3vec-bit?-a4vec-equiv-congruence-on-y

(defthm a3vec-bit?-a4vec-equiv-congruence-on-y
  (implies (a4vec-equiv y y-equiv)
           (equal (a3vec-bit? x y y3p z z3p)
                  (a3vec-bit? x y-equiv y3p z z3p)))
  :rule-classes :congruence)

Theorem: a3vec-bit?-of-a4vec-fix-z

(defthm a3vec-bit?-of-a4vec-fix-z
  (equal (a3vec-bit? x y y3p (a4vec-fix z) z3p)
         (a3vec-bit? x y y3p z z3p)))

Theorem: a3vec-bit?-a4vec-equiv-congruence-on-z

(defthm a3vec-bit?-a4vec-equiv-congruence-on-z
  (implies (a4vec-equiv z z-equiv)
           (equal (a3vec-bit? x y y3p z z3p)
                  (a3vec-bit? x y y3p z-equiv z3p)))
  :rule-classes :congruence)