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Set

The fixing function used here is sfix.

The name sequiv of the equivalence relation introduced here is ``structurally similar'' to the name sfix of the fixing function.

Definitions and Theorems

Function: sequiv$inline

(defun sequiv$inline (x y)
  (declare (xargs :guard (and (setp x) (setp y))))
  (equal (sfix x) (sfix y)))

Theorem: sequiv-is-an-equivalence

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

Theorem: sequiv-implies-equal-sfix-1

(defthm sequiv-implies-equal-sfix-1
  (implies (sequiv x x-equiv)
           (equal (sfix x) (sfix x-equiv)))
  :rule-classes (:congruence))

Theorem: sfix-under-sequiv

(defthm sfix-under-sequiv
  (sequiv (sfix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sfix-1-forward-to-sequiv

(defthm equal-of-sfix-1-forward-to-sequiv
  (implies (equal (sfix x) y)
           (sequiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-sfix-2-forward-to-sequiv

(defthm equal-of-sfix-2-forward-to-sequiv
  (implies (equal x (sfix y))
           (sequiv x y))
  :rule-classes :forward-chaining)

Theorem: sequiv-of-sfix-1-forward

(defthm sequiv-of-sfix-1-forward
  (implies (sequiv (sfix x) y)
           (sequiv x y))
  :rule-classes :forward-chaining)

Theorem: sequiv-of-sfix-2-forward

(defthm sequiv-of-sfix-2-forward
  (implies (sequiv x (sfix y))
           (sequiv x y))
  :rule-classes :forward-chaining)