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Aig-and

Aig-and-pass6

Level 3 Substitution Rules, Both Directions.

Signature
(aig-and-pass6 x y) → (mv status arg1 arg2)

Definitions and Theorems

Function: aig-and-pass6$inline

(defun aig-and-pass6$inline (x y)
  (declare (xargs :guard t))
  (let ((__function__ 'aig-and-pass6))
    (declare (ignorable __function__))
    (b* (((mv status arg1 arg2)
          (aig-and-pass6a y x))
         ((unless (eq status :fail))
          (mv status arg1 arg2)))
      (aig-and-pass6a x y))))

Theorem: aig-and-pass6-correct

(defthm aig-and-pass6-correct
  (b* (((mv ?status ?arg1 ?arg2)
        (aig-and-pass6$inline x y)))
    (equal (and (aig-eval arg1 env)
                (aig-eval arg2 env))
           (and (aig-eval x env)
                (aig-eval y env))))
  :rule-classes nil)

Theorem: aig-and-pass6-reduces-count

(defthm aig-and-pass6-reduces-count
  (b* (((mv ?status ?arg1 ?arg2)
        (aig-and-pass6$inline x y)))
    (implies (eq status :reduced)
             (< (+ (aig-and-count arg1)
                   (aig-and-count arg2))
                (+ (aig-and-count x)
                   (aig-and-count y)))))
  :rule-classes nil)

Theorem: aig-and-pass6-subterm-convention

(defthm aig-and-pass6-subterm-convention
  (b* (((mv ?status ?arg1 ?arg2)
        (aig-and-pass6$inline x y)))
    (implies (equal status :subterm)
             (equal arg2 arg1))))

Theorem: aig-and-pass6-arg2-on-failure

(defthm aig-and-pass6-arg2-on-failure
  (b* (((mv ?status ?arg1 ?arg2)
        (aig-and-pass6$inline x y)))
    (implies (and (equal status :fail) y)
             (iff arg2 t))))

Theorem: aig-and-pass6-when-fail

(defthm aig-and-pass6-when-fail
  (b* (((mv ?status ?arg1 ?arg2)
        (aig-and-pass6$inline x y)))
    (implies (and (not (equal status :subterm))
                  (not (equal status :reduced)))
             (and (equal status :fail)
                  (equal arg1 x)
                  (equal arg2 y)))))