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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))