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Aig-semantics

Aig-equiv

We say the AIGs X and Y are equivalent when they always evaluate to the same value, per aig-eval.

This is a universal equivalence, introduced using def-universal-equiv.

Function: aig-equiv

(defun aig-equiv (x y)
       (declare (xargs :non-executable t))
       (declare (xargs :guard t))
       (prog2$ (throw-nonexec-error 'aig-equiv
                                    (list x y))
               (let ((env (aig-equiv-witness x y)))
                    (and (equal (aig-eval x env)
                                (aig-eval y env))))))

Definitions and Theorems

Theorem: aig-equiv-necc

(defthm aig-equiv-necc
        (implies (not (and (equal (aig-eval x env)
                                  (aig-eval y env))))
                 (not (aig-equiv x y))))

Theorem: aig-equiv-witnessing-witness-rule-correct

(defthm aig-equiv-witnessing-witness-rule-correct
        (implies (not ((lambda (env y x)
                               (not (equal (aig-eval x env)
                                           (aig-eval y env))))
                       (aig-equiv-witness x y)
                       y x))
                 (aig-equiv x y))
        :rule-classes nil)

Theorem: aig-equiv-instancing-instance-rule-correct

(defthm aig-equiv-instancing-instance-rule-correct
        (implies (not (equal (aig-eval x env)
                             (aig-eval y env)))
                 (not (aig-equiv x y)))
        :rule-classes nil)

Theorem: aig-equiv-is-an-equivalence

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