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String-element

String-element-fix

Fixing function for string-element structures.

Signature
(string-element-fix x) → new-x
Arguments: x — Guard (string-elementp x).
Returns: new-x — Type (string-elementp new-x).

Definitions and Theorems

Function: string-element-fix$inline

(defun string-element-fix$inline (x)
 (declare (xargs :guard (string-elementp x)))
 (let ((__function__ 'string-element-fix))
  (declare (ignorable __function__))
  (mbe :logic
       (case (string-element-kind x)
         (:char (b* ((get (acl2::char-fix (std::da-nth 0 (cdr x)))))
                  (cons :char (list get))))
         (:escape (b* ((get (escape-fix (std::da-nth 0 (cdr x)))))
                    (cons :escape (list get)))))
       :exec x)))

Theorem: string-elementp-of-string-element-fix

(defthm string-elementp-of-string-element-fix
  (b* ((new-x (string-element-fix$inline x)))
    (string-elementp new-x))
  :rule-classes :rewrite)

Theorem: string-element-fix-when-string-elementp

(defthm string-element-fix-when-string-elementp
  (implies (string-elementp x)
           (equal (string-element-fix x) x)))

Function: string-element-equiv$inline

(defun string-element-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (string-elementp acl2::x)
                              (string-elementp acl2::y))))
  (equal (string-element-fix acl2::x)
         (string-element-fix acl2::y)))

Theorem: string-element-equiv-is-an-equivalence

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

Theorem: string-element-equiv-implies-equal-string-element-fix-1

(defthm string-element-equiv-implies-equal-string-element-fix-1
  (implies (string-element-equiv acl2::x x-equiv)
           (equal (string-element-fix acl2::x)
                  (string-element-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: string-element-fix-under-string-element-equiv

(defthm string-element-fix-under-string-element-equiv
  (string-element-equiv (string-element-fix acl2::x)
                        acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-string-element-fix-1-forward-to-string-element-equiv

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

Theorem: equal-of-string-element-fix-2-forward-to-string-element-equiv

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

Theorem: string-element-equiv-of-string-element-fix-1-forward

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

Theorem: string-element-equiv-of-string-element-fix-2-forward

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

Theorem: string-element-kind$inline-of-string-element-fix-x

(defthm string-element-kind$inline-of-string-element-fix-x
  (equal (string-element-kind$inline (string-element-fix x))
         (string-element-kind$inline x)))

Theorem: string-element-kind$inline-string-element-equiv-congruence-on-x

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

Theorem: consp-of-string-element-fix

(defthm consp-of-string-element-fix
  (consp (string-element-fix x))
  :rule-classes :type-prescription)