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Obj-alist

Obj-alist-fix

(obj-alist-fix x) is an ACL2::fty alist fixing function that follows the drop-keys strategy.

Signature
(obj-alist-fix x) → fty::newx
Arguments: x — Guard (obj-alist-p x).
Returns: fty::newx — Type (obj-alist-p fty::newx).

Note that in the execution this is just an inline identity function.

Definitions and Theorems

Function: obj-alist-fix$inline

(defun obj-alist-fix$inline (x)
  (declare (xargs :guard (obj-alist-p x)))
  (let ((__function__ 'obj-alist-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (if (consp (car x))
               (cons (cons (caar x) (cdar x))
                     (obj-alist-fix (cdr x)))
             (obj-alist-fix (cdr x))))
         :exec x)))

Theorem: obj-alist-p-of-obj-alist-fix

(defthm obj-alist-p-of-obj-alist-fix
  (b* ((fty::newx (obj-alist-fix$inline x)))
    (obj-alist-p fty::newx))
  :rule-classes :rewrite)

Theorem: obj-alist-fix-when-obj-alist-p

(defthm obj-alist-fix-when-obj-alist-p
  (implies (obj-alist-p x)
           (equal (obj-alist-fix x) x)))

Function: obj-alist-equiv$inline

(defun obj-alist-equiv$inline (x y)
  (declare (xargs :guard (and (obj-alist-p x) (obj-alist-p y))))
  (equal (obj-alist-fix x)
         (obj-alist-fix y)))

Theorem: obj-alist-equiv-is-an-equivalence

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

Theorem: obj-alist-equiv-implies-equal-obj-alist-fix-1

(defthm obj-alist-equiv-implies-equal-obj-alist-fix-1
  (implies (obj-alist-equiv x x-equiv)
           (equal (obj-alist-fix x)
                  (obj-alist-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: obj-alist-fix-under-obj-alist-equiv

(defthm obj-alist-fix-under-obj-alist-equiv
  (obj-alist-equiv (obj-alist-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-obj-alist-fix-1-forward-to-obj-alist-equiv

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

Theorem: equal-of-obj-alist-fix-2-forward-to-obj-alist-equiv

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

Theorem: obj-alist-equiv-of-obj-alist-fix-1-forward

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

Theorem: obj-alist-equiv-of-obj-alist-fix-2-forward

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

Theorem: cons-of-obj-alist-fix-y-under-obj-alist-equiv

(defthm cons-of-obj-alist-fix-y-under-obj-alist-equiv
  (obj-alist-equiv (cons x (obj-alist-fix y))
                   (cons x y)))

Theorem: cons-obj-alist-equiv-congruence-on-y-under-obj-alist-equiv

(defthm cons-obj-alist-equiv-congruence-on-y-under-obj-alist-equiv
  (implies (obj-alist-equiv y y-equiv)
           (obj-alist-equiv (cons x y)
                            (cons x y-equiv)))
  :rule-classes :congruence)

Theorem: obj-alist-fix-of-acons

(defthm obj-alist-fix-of-acons
  (equal (obj-alist-fix (cons (cons a b) x))
         (cons (cons a b) (obj-alist-fix x))))

Theorem: hons-assoc-equal-of-obj-alist-fix

(defthm hons-assoc-equal-of-obj-alist-fix
  (equal (hons-assoc-equal k (obj-alist-fix x))
         (let ((fty::pair (hons-assoc-equal k x)))
           (and fty::pair (cons k (cdr fty::pair))))))

Theorem: obj-alist-fix-of-append

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

Theorem: consp-car-of-obj-alist-fix

(defthm consp-car-of-obj-alist-fix
  (equal (consp (car (obj-alist-fix x)))
         (consp (obj-alist-fix x))))