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String

String-fix

(string-fix x) is a usual ACL2::fty list fixing function.

Signature
(string-fix x) → fty::newx
Arguments: x — Guard (stringp x).
Returns: fty::newx — Type (stringp fty::newx).

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: string-fix$inline

(defun string-fix$inline (x)
  (declare (xargs :guard (stringp x)))
  (let ((__function__ 'string-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (symbol-fix (car x))
                 (string-fix (cdr x))))
         :exec x)))

Theorem: stringp-of-string-fix

(defthm stringp-of-string-fix
  (b* ((fty::newx (string-fix$inline x)))
    (stringp fty::newx))
  :rule-classes :rewrite)

Theorem: string-fix-when-stringp

(defthm string-fix-when-stringp
  (implies (stringp x)
           (equal (string-fix x) x)))

Function: string-equiv$inline

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

Theorem: string-equiv-is-an-equivalence

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

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

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

Theorem: string-fix-under-string-equiv

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

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

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

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

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

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

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

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

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

Theorem: car-of-string-fix-x-under-symbol-equiv

(defthm car-of-string-fix-x-under-symbol-equiv
  (symbol-equiv (car (string-fix acl2::x))
                (car acl2::x)))

Theorem: car-string-equiv-congruence-on-x-under-symbol-equiv

(defthm car-string-equiv-congruence-on-x-under-symbol-equiv
  (implies (string-equiv acl2::x x-equiv)
           (symbol-equiv (car acl2::x)
                         (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-string-fix-x-under-string-equiv

(defthm cdr-of-string-fix-x-under-string-equiv
  (string-equiv (cdr (string-fix acl2::x))
                (cdr acl2::x)))

Theorem: cdr-string-equiv-congruence-on-x-under-string-equiv

(defthm cdr-string-equiv-congruence-on-x-under-string-equiv
  (implies (string-equiv acl2::x x-equiv)
           (string-equiv (cdr acl2::x)
                         (cdr x-equiv)))
  :rule-classes :congruence)

Theorem: cons-of-symbol-fix-x-under-string-equiv

(defthm cons-of-symbol-fix-x-under-string-equiv
  (string-equiv (cons (symbol-fix acl2::x) acl2::y)
                (cons acl2::x acl2::y)))

Theorem: cons-symbol-equiv-congruence-on-x-under-string-equiv

(defthm cons-symbol-equiv-congruence-on-x-under-string-equiv
  (implies (symbol-equiv acl2::x x-equiv)
           (string-equiv (cons acl2::x acl2::y)
                         (cons x-equiv acl2::y)))
  :rule-classes :congruence)

Theorem: cons-of-string-fix-y-under-string-equiv

(defthm cons-of-string-fix-y-under-string-equiv
  (string-equiv (cons acl2::x (string-fix acl2::y))
                (cons acl2::x acl2::y)))

Theorem: cons-string-equiv-congruence-on-y-under-string-equiv

(defthm cons-string-equiv-congruence-on-y-under-string-equiv
  (implies (string-equiv acl2::y y-equiv)
           (string-equiv (cons acl2::x acl2::y)
                         (cons acl2::x y-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-string-fix

(defthm consp-of-string-fix
  (equal (consp (string-fix acl2::x))
         (consp acl2::x)))

Theorem: string-fix-under-iff

(defthm string-fix-under-iff
  (iff (string-fix acl2::x)
       (consp acl2::x)))

Theorem: string-fix-of-cons

(defthm string-fix-of-cons
  (equal (string-fix (cons a x))
         (cons (symbol-fix a) (string-fix x))))

Theorem: len-of-string-fix

(defthm len-of-string-fix
  (equal (len (string-fix acl2::x))
         (len acl2::x)))

Theorem: string-fix-of-append

(defthm string-fix-of-append
  (equal (string-fix (append std::a std::b))
         (append (string-fix std::a)
                 (string-fix std::b))))

Theorem: string-fix-of-repeat

(defthm string-fix-of-repeat
  (equal (string-fix (repeat acl2::n acl2::x))
         (repeat acl2::n (symbol-fix acl2::x))))

Theorem: list-equiv-refines-string-equiv

(defthm list-equiv-refines-string-equiv
  (implies (list-equiv acl2::x acl2::y)
           (string-equiv acl2::x acl2::y))
  :rule-classes :refinement)

Theorem: nth-of-string-fix

(defthm nth-of-string-fix
  (equal (nth acl2::n (string-fix acl2::x))
         (if (< (nfix acl2::n) (len acl2::x))
             (symbol-fix (nth acl2::n acl2::x))
           nil)))

Theorem: string-equiv-implies-string-equiv-append-1

(defthm string-equiv-implies-string-equiv-append-1
  (implies (string-equiv acl2::x fty::x-equiv)
           (string-equiv (append acl2::x acl2::y)
                         (append fty::x-equiv acl2::y)))
  :rule-classes (:congruence))

Theorem: string-equiv-implies-string-equiv-append-2

(defthm string-equiv-implies-string-equiv-append-2
  (implies (string-equiv acl2::y fty::y-equiv)
           (string-equiv (append acl2::x acl2::y)
                         (append acl2::x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: string-equiv-implies-string-equiv-nthcdr-2

(defthm string-equiv-implies-string-equiv-nthcdr-2
  (implies (string-equiv acl2::l l-equiv)
           (string-equiv (nthcdr acl2::n acl2::l)
                         (nthcdr acl2::n l-equiv)))
  :rule-classes (:congruence))

Theorem: string-equiv-implies-string-equiv-take-2

(defthm string-equiv-implies-string-equiv-take-2
  (implies (string-equiv acl2::l l-equiv)
           (string-equiv (take acl2::n acl2::l)
                         (take acl2::n l-equiv)))
  :rule-classes (:congruence))