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Aig-and

Aig-and-pass2

Level 2 Contradiction Rule 1 and Idempotence Rule, Both Directions.

Signature
(aig-and-pass2 x y) → (mv status arg1 arg2)

Definitions and Theorems

Function: aig-and-pass2$inline

(defun aig-and-pass2$inline (x y)
  (declare (xargs :guard t))
  (let ((__function__ 'aig-and-pass2))
    (declare (ignorable __function__))
    (b* (((mv status a b) (aig-and-pass2a y x))
         ((unless (eq status :fail))
          (mv status a b)))
      (aig-and-pass2a x y))))

Theorem: aig-and-pass2-correct

(defthm aig-and-pass2-correct
  (b* (((mv ?status ?arg1 ?arg2)
        (aig-and-pass2$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-pass2-never-reduced

(defthm aig-and-pass2-never-reduced
  (b* (((mv ?status ?arg1 ?arg2)
        (aig-and-pass2$inline x y)))
    (not (equal status :reduced))))

Theorem: aig-and-pass2-subterm-convention

(defthm aig-and-pass2-subterm-convention
  (b* (((mv ?status ?arg1 ?arg2)
        (aig-and-pass2$inline x y)))
    (implies (equal status :subterm)
             (equal arg2 arg1))))

Theorem: aig-and-pass2-normalize-status

(defthm aig-and-pass2-normalize-status
  (b* (((mv ?status ?arg1 ?arg2)
        (aig-and-pass2$inline x y)))
    (implies (and (not (equal status :subterm))
                  (not (equal status :reduced)))
             (and (equal status :fail)
                  (equal arg1 x)
                  (equal arg2 y)))))