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• Bfr-updates

Bfr-updates-equiv

Basic equivalence relation for bfr-updates structures.

Definitions and Theorems

Function: bfr-updates-equiv$inline

(defun bfr-updates-equiv$inline (x y)
  (declare (xargs :guard (and (bfr-updates-p x)
                              (bfr-updates-p y))))
  (equal (bfr-updates-fix x)
         (bfr-updates-fix y)))

Theorem: bfr-updates-equiv-is-an-equivalence

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

Theorem: bfr-updates-equiv-implies-equal-bfr-updates-fix-1

(defthm bfr-updates-equiv-implies-equal-bfr-updates-fix-1
  (implies (bfr-updates-equiv x x-equiv)
           (equal (bfr-updates-fix x)
                  (bfr-updates-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: bfr-updates-fix-under-bfr-updates-equiv

(defthm bfr-updates-fix-under-bfr-updates-equiv
  (bfr-updates-equiv (bfr-updates-fix x)
                     x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-bfr-updates-fix-1-forward-to-bfr-updates-equiv

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

Theorem: equal-of-bfr-updates-fix-2-forward-to-bfr-updates-equiv

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

Theorem: bfr-updates-equiv-of-bfr-updates-fix-1-forward

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

Theorem: bfr-updates-equiv-of-bfr-updates-fix-2-forward

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