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• Aig-semantics

Aig-env-equiv

We say the environments X and Y are equivalent when they give equivalent values to variables looked up with aig-env-lookup.

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

Function: aig-env-equiv

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

Definitions and Theorems

Theorem: aig-env-equiv-necc

(defthm aig-env-equiv-necc
  (implies (not (and (iff (aig-env-lookup key x)
                          (aig-env-lookup key y))))
           (not (aig-env-equiv x y))))

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

(defthm aig-env-equiv-witnessing-witness-rule-correct
  (implies (not ((lambda (key y x)
                   (not (iff (aig-env-lookup key x)
                             (aig-env-lookup key y))))
                 (aig-env-equiv-witness x y)
                 y x))
           (aig-env-equiv x y))
  :rule-classes nil)

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

(defthm aig-env-equiv-instancing-instance-rule-correct
  (implies (not (iff (aig-env-lookup key x)
                     (aig-env-lookup key y)))
           (not (aig-env-equiv x y)))
  :rule-classes nil)

Theorem: aig-env-equiv-is-an-equivalence

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

Theorem: aig-env-equiv-implies-equal-aig-env-lookup-2

(defthm aig-env-equiv-implies-equal-aig-env-lookup-2
  (implies (aig-env-equiv x x-equiv)
           (equal (aig-env-lookup key x)
                  (aig-env-lookup key x-equiv)))
  :rule-classes (:congruence))

Theorem: alist-equiv-refines-aig-env-equiv

(defthm alist-equiv-refines-aig-env-equiv
  (implies (alist-equiv x y)
           (aig-env-equiv x y))
  :rule-classes (:refinement))