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String-set

String-sfix

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

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

In the logic, we apply str-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: string-sfix

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

Theorem: string-setp-of-string-sfix

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

Theorem: string-sfix-when-string-setp

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

Theorem: emptyp-string-sfix

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

Theorem: emptyp-of-string-sfix

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

Function: string-sequiv$inline

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

Theorem: string-sequiv-is-an-equivalence

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

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

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

Theorem: string-sfix-under-string-sequiv

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

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

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

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

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

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

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

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

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