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Symbol-set

Symbol-sfix

(symbol-sfix x) is a usual fty set fixing function.

Signature
(symbol-sfix x) → *
Arguments: x — Guard (symbol-setp x).

In the logic, we apply symbol-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: symbol-sfix

(defun symbol-sfix (x)
  (declare (xargs :guard (symbol-setp x)))
  (mbe :logic (if (symbol-setp x) x nil)
       :exec x))

Theorem: symbol-setp-of-symbol-sfix

(defthm symbol-setp-of-symbol-sfix
  (symbol-setp (symbol-sfix x)))

Theorem: symbol-sfix-when-symbol-setp

(defthm symbol-sfix-when-symbol-setp
  (implies (symbol-setp x)
           (equal (symbol-sfix x) x)))

Theorem: emptyp-symbol-sfix

(defthm emptyp-symbol-sfix
  (implies (or (set::emptyp x)
               (not (symbol-setp x)))
           (set::emptyp (symbol-sfix x))))

Theorem: emptyp-of-symbol-sfix

(defthm emptyp-of-symbol-sfix
  (equal (set::emptyp (symbol-sfix x))
         (or (not (symbol-setp x))
             (set::emptyp x))))

Function: symbol-sequiv$inline

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

Theorem: symbol-sequiv-is-an-equivalence

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

Theorem: symbol-sequiv-implies-equal-symbol-sfix-1

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

Theorem: symbol-sfix-under-symbol-sequiv

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

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

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

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

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

Theorem: symbol-sequiv-of-symbol-sfix-1-forward

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

Theorem: symbol-sequiv-of-symbol-sfix-2-forward

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