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• Symbol-list

Symbol-list-fix

(symbol-list-fix x) is a usual fty list fixing function.

Signature

(symbol-list-fix x) → fty::newx

Arguments

x — Guard (symbol-listp x).

Returns

fty::newx — Type (symbol-listp 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: symbol-list-fix$inline

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

Theorem: symbol-listp-of-symbol-list-fix

(defthm symbol-listp-of-symbol-list-fix
        (b* ((fty::newx (symbol-list-fix$inline x)))
            (symbol-listp fty::newx))
        :rule-classes :rewrite)

Theorem: symbol-list-fix-when-symbol-listp

(defthm symbol-list-fix-when-symbol-listp
        (implies (symbol-listp x)
                 (equal (symbol-list-fix x) x)))

Function: symbol-list-equiv$inline

(defun symbol-list-equiv$inline (x y)
       (declare (xargs :guard (and (symbol-listp x)
                                   (symbol-listp y))))
       (equal (symbol-list-fix x)
              (symbol-list-fix y)))

Theorem: symbol-list-equiv-is-an-equivalence

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

Theorem: symbol-list-equiv-implies-equal-symbol-list-fix-1

(defthm symbol-list-equiv-implies-equal-symbol-list-fix-1
        (implies (symbol-list-equiv x x-equiv)
                 (equal (symbol-list-fix x)
                        (symbol-list-fix x-equiv)))
        :rule-classes (:congruence))

Theorem: symbol-list-fix-under-symbol-list-equiv

(defthm symbol-list-fix-under-symbol-list-equiv
        (symbol-list-equiv (symbol-list-fix x)
                           x)
        :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-symbol-list-fix-1-forward-to-symbol-list-equiv

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

Theorem: equal-of-symbol-list-fix-2-forward-to-symbol-list-equiv

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

Theorem: symbol-list-equiv-of-symbol-list-fix-1-forward

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

Theorem: symbol-list-equiv-of-symbol-list-fix-2-forward

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

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

(defthm car-of-symbol-list-fix-x-under-symbol-equiv
        (symbol-equiv (car (symbol-list-fix x))
                      (car x)))

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

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

Theorem: cdr-of-symbol-list-fix-x-under-symbol-list-equiv

(defthm cdr-of-symbol-list-fix-x-under-symbol-list-equiv
        (symbol-list-equiv (cdr (symbol-list-fix x))
                           (cdr x)))

Theorem: cdr-symbol-list-equiv-congruence-on-x-under-symbol-list-equiv

(defthm
     cdr-symbol-list-equiv-congruence-on-x-under-symbol-list-equiv
     (implies (symbol-list-equiv x x-equiv)
              (symbol-list-equiv (cdr x)
                                 (cdr x-equiv)))
     :rule-classes :congruence)

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

(defthm cons-of-symbol-fix-x-under-symbol-list-equiv
        (symbol-list-equiv (cons (symbol-fix x) y)
                           (cons x y)))

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

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

Theorem: cons-of-symbol-list-fix-y-under-symbol-list-equiv

(defthm cons-of-symbol-list-fix-y-under-symbol-list-equiv
        (symbol-list-equiv (cons x (symbol-list-fix y))
                           (cons x y)))

Theorem: cons-symbol-list-equiv-congruence-on-y-under-symbol-list-equiv

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

Theorem: consp-of-symbol-list-fix

(defthm consp-of-symbol-list-fix
        (equal (consp (symbol-list-fix x))
               (consp x)))

Theorem: symbol-list-fix-under-iff

(defthm symbol-list-fix-under-iff
        (iff (symbol-list-fix x) (consp x)))

Theorem: symbol-list-fix-of-cons

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

Theorem: len-of-symbol-list-fix

(defthm len-of-symbol-list-fix
        (equal (len (symbol-list-fix x))
               (len x)))

Theorem: symbol-list-fix-of-append

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

Theorem: symbol-list-fix-of-repeat

(defthm symbol-list-fix-of-repeat
        (equal (symbol-list-fix (repeat n x))
               (repeat n (symbol-fix x))))

Theorem: list-equiv-refines-symbol-list-equiv

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

Theorem: nth-of-symbol-list-fix

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

Theorem: symbol-list-equiv-implies-symbol-list-equiv-append-1

(defthm symbol-list-equiv-implies-symbol-list-equiv-append-1
        (implies (symbol-list-equiv x fty::x-equiv)
                 (symbol-list-equiv (append x y)
                                    (append fty::x-equiv y)))
        :rule-classes (:congruence))

Theorem: symbol-list-equiv-implies-symbol-list-equiv-append-2

(defthm symbol-list-equiv-implies-symbol-list-equiv-append-2
        (implies (symbol-list-equiv y fty::y-equiv)
                 (symbol-list-equiv (append x y)
                                    (append x fty::y-equiv)))
        :rule-classes (:congruence))

Theorem: symbol-list-equiv-implies-symbol-list-equiv-nthcdr-2

(defthm symbol-list-equiv-implies-symbol-list-equiv-nthcdr-2
        (implies (symbol-list-equiv l l-equiv)
                 (symbol-list-equiv (nthcdr n l)
                                    (nthcdr n l-equiv)))
        :rule-classes (:congruence))

Theorem: symbol-list-equiv-implies-symbol-list-equiv-take-2

(defthm symbol-list-equiv-implies-symbol-list-equiv-take-2
        (implies (symbol-list-equiv l l-equiv)
                 (symbol-list-equiv (take n l)
                                    (take n l-equiv)))
        :rule-classes (:congruence))