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Oct-escape

Oct-escape-fix

Fixing function for oct-escape structures.

Signature
(oct-escape-fix x) → new-x
Arguments: x — Guard (oct-escapep x).
Returns: new-x — Type (oct-escapep new-x).

Definitions and Theorems

Function: oct-escape-fix$inline

(defun oct-escape-fix$inline (x)
 (declare (xargs :guard (oct-escapep x)))
 (mbe
  :logic
  (case (oct-escape-kind x)
   (:one (b* ((digit (str::oct-digit-char-fix (cdr x))))
           (cons :one digit)))
   (:two (b* ((digit1 (str::oct-digit-char-fix (car (cdr x))))
              (digit2 (str::oct-digit-char-fix (cdr (cdr x)))))
           (cons :two (cons digit1 digit2))))
   (:three
        (b* ((digit1 (str::oct-digit-char-fix (car (cdr x))))
             (digit2 (str::oct-digit-char-fix (car (cdr (cdr x)))))
             (digit3 (str::oct-digit-char-fix (cdr (cdr (cdr x))))))
          (cons :three (cons digit1 (cons digit2 digit3))))))
  :exec x))

Theorem: oct-escapep-of-oct-escape-fix

(defthm oct-escapep-of-oct-escape-fix
  (b* ((new-x (oct-escape-fix$inline x)))
    (oct-escapep new-x))
  :rule-classes :rewrite)

Theorem: oct-escape-fix-when-oct-escapep

(defthm oct-escape-fix-when-oct-escapep
  (implies (oct-escapep x)
           (equal (oct-escape-fix x) x)))

Function: oct-escape-equiv$inline

(defun oct-escape-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (oct-escapep acl2::x)
                              (oct-escapep acl2::y))))
  (equal (oct-escape-fix acl2::x)
         (oct-escape-fix acl2::y)))

Theorem: oct-escape-equiv-is-an-equivalence

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

Theorem: oct-escape-equiv-implies-equal-oct-escape-fix-1

(defthm oct-escape-equiv-implies-equal-oct-escape-fix-1
  (implies (oct-escape-equiv acl2::x x-equiv)
           (equal (oct-escape-fix acl2::x)
                  (oct-escape-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: oct-escape-fix-under-oct-escape-equiv

(defthm oct-escape-fix-under-oct-escape-equiv
  (oct-escape-equiv (oct-escape-fix acl2::x)
                    acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-oct-escape-fix-1-forward-to-oct-escape-equiv

(defthm equal-of-oct-escape-fix-1-forward-to-oct-escape-equiv
  (implies (equal (oct-escape-fix acl2::x) acl2::y)
           (oct-escape-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-oct-escape-fix-2-forward-to-oct-escape-equiv

(defthm equal-of-oct-escape-fix-2-forward-to-oct-escape-equiv
  (implies (equal acl2::x (oct-escape-fix acl2::y))
           (oct-escape-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: oct-escape-equiv-of-oct-escape-fix-1-forward

(defthm oct-escape-equiv-of-oct-escape-fix-1-forward
  (implies (oct-escape-equiv (oct-escape-fix acl2::x)
                             acl2::y)
           (oct-escape-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: oct-escape-equiv-of-oct-escape-fix-2-forward

(defthm oct-escape-equiv-of-oct-escape-fix-2-forward
  (implies (oct-escape-equiv acl2::x (oct-escape-fix acl2::y))
           (oct-escape-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: oct-escape-kind$inline-of-oct-escape-fix-x

(defthm oct-escape-kind$inline-of-oct-escape-fix-x
  (equal (oct-escape-kind$inline (oct-escape-fix x))
         (oct-escape-kind$inline x)))

Theorem: oct-escape-kind$inline-oct-escape-equiv-congruence-on-x

(defthm oct-escape-kind$inline-oct-escape-equiv-congruence-on-x
  (implies (oct-escape-equiv x x-equiv)
           (equal (oct-escape-kind$inline x)
                  (oct-escape-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-oct-escape-fix

(defthm consp-of-oct-escape-fix
  (consp (oct-escape-fix x))
  :rule-classes :type-prescription)

Theorem: oct-escape-fix$inline-of-oct-escape-fix-x

(defthm oct-escape-fix$inline-of-oct-escape-fix-x
  (equal (oct-escape-fix$inline (oct-escape-fix x))
         (oct-escape-fix$inline x)))

Theorem: oct-escape-fix$inline-oct-escape-equiv-congruence-on-x

(defthm oct-escape-fix$inline-oct-escape-equiv-congruence-on-x
  (implies (oct-escape-equiv x x-equiv)
           (equal (oct-escape-fix$inline x)
                  (oct-escape-fix$inline x-equiv)))
  :rule-classes :congruence)