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Event

Event-equiv

Basic equivalence relation for event structures.

Definitions and Theorems

Function: event-equiv$inline

(defun event-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (eventp acl2::x)
                              (eventp acl2::y))))
  (equal (event-fix acl2::x)
         (event-fix acl2::y)))

Theorem: event-equiv-is-an-equivalence

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

Theorem: event-equiv-implies-equal-event-fix-1

(defthm event-equiv-implies-equal-event-fix-1
  (implies (event-equiv acl2::x x-equiv)
           (equal (event-fix acl2::x)
                  (event-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: event-fix-under-event-equiv

(defthm event-fix-under-event-equiv
  (event-equiv (event-fix acl2::x)
               acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-event-fix-1-forward-to-event-equiv

(defthm equal-of-event-fix-1-forward-to-event-equiv
  (implies (equal (event-fix acl2::x) acl2::y)
           (event-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-event-fix-2-forward-to-event-equiv

(defthm equal-of-event-fix-2-forward-to-event-equiv
  (implies (equal acl2::x (event-fix acl2::y))
           (event-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: event-equiv-of-event-fix-1-forward

(defthm event-equiv-of-event-fix-1-forward
  (implies (event-equiv (event-fix acl2::x)
                        acl2::y)
           (event-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: event-equiv-of-event-fix-2-forward

(defthm event-equiv-of-event-fix-2-forward
  (implies (event-equiv acl2::x (event-fix acl2::y))
           (event-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)