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Character-set

Character-sfix

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

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

In the logic, we apply char-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: character-sfix

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

Theorem: character-setp-of-character-sfix

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

Theorem: character-sfix-when-character-setp

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

Theorem: emptyp-character-sfix

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

Theorem: emptyp-of-character-sfix

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

Function: character-sequiv$inline

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

Theorem: character-sequiv-is-an-equivalence

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

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

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

Theorem: character-sfix-under-character-sequiv

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

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

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

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

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

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

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

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

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