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• Chase-position

Chase-position-equiv

Basic equivalence relation for chase-position structures.

Definitions and Theorems

Function: chase-position-equiv$inline

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

Theorem: chase-position-equiv-is-an-equivalence

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

Theorem: chase-position-equiv-implies-equal-chase-position-fix-1

(defthm chase-position-equiv-implies-equal-chase-position-fix-1
  (implies (chase-position-equiv x x-equiv)
           (equal (chase-position-fix x)
                  (chase-position-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: chase-position-fix-under-chase-position-equiv

(defthm chase-position-fix-under-chase-position-equiv
  (chase-position-equiv (chase-position-fix x)
                        x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-chase-position-fix-1-forward-to-chase-position-equiv

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

Theorem: equal-of-chase-position-fix-2-forward-to-chase-position-equiv

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

Theorem: chase-position-equiv-of-chase-position-fix-1-forward

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

Theorem: chase-position-equiv-of-chase-position-fix-2-forward

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