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Rw-pair

Rw-pair-equiv

Basic equivalence relation for rw-pair structures.

Definitions and Theorems

Function: rw-pair-equiv$inline

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

Theorem: rw-pair-equiv-is-an-equivalence

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

Theorem: rw-pair-equiv-implies-equal-rw-pair-fix-1

(defthm rw-pair-equiv-implies-equal-rw-pair-fix-1
  (implies (rw-pair-equiv x x-equiv)
           (equal (rw-pair-fix x)
                  (rw-pair-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: rw-pair-fix-under-rw-pair-equiv

(defthm rw-pair-fix-under-rw-pair-equiv
  (rw-pair-equiv (rw-pair-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-rw-pair-fix-1-forward-to-rw-pair-equiv

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

Theorem: equal-of-rw-pair-fix-2-forward-to-rw-pair-equiv

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

Theorem: rw-pair-equiv-of-rw-pair-fix-1-forward

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

Theorem: rw-pair-equiv-of-rw-pair-fix-2-forward

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