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• Timer

Timer-equiv

Basic equivalence relation for timer structures.

Definitions and Theorems

Function: timer-equiv$inline

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

Theorem: timer-equiv-is-an-equivalence

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

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

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

Theorem: timer-fix-under-timer-equiv

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

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

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

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

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

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

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

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

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