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Backref-alist

Backref-alist-fix

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

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

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

Definitions and Theorems

Function: backref-alist-fix$inline

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

Theorem: backref-alist-p-of-backref-alist-fix

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

Theorem: backref-alist-fix-when-backref-alist-p

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

Function: backref-alist-equiv$inline

(defun backref-alist-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (backref-alist-p acl2::x)
                              (backref-alist-p acl2::y))))
  (equal (backref-alist-fix acl2::x)
         (backref-alist-fix acl2::y)))

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

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

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

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

Theorem: backref-alist-fix-under-backref-alist-equiv

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

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

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

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

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

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

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

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

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

Theorem: cons-of-backref-fix-v-under-backref-alist-equiv

(defthm cons-of-backref-fix-v-under-backref-alist-equiv
  (backref-alist-equiv (cons (cons acl2::k (backref-fix acl2::v))
                             acl2::x)
                       (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-backref-equiv-congruence-on-v-under-backref-alist-equiv

(defthm cons-backref-equiv-congruence-on-v-under-backref-alist-equiv
  (implies
       (backref-equiv acl2::v v-equiv)
       (backref-alist-equiv (cons (cons acl2::k acl2::v) acl2::x)
                            (cons (cons acl2::k v-equiv) acl2::x)))
  :rule-classes :congruence)

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

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

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

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

Theorem: backref-alist-fix-of-acons

(defthm backref-alist-fix-of-acons
  (equal (backref-alist-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons acl2::a (backref-fix acl2::b))
               (backref-alist-fix x))))

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

(defthm hons-assoc-equal-of-backref-alist-fix
  (equal (hons-assoc-equal acl2::k (backref-alist-fix acl2::x))
         (let ((fty::pair (hons-assoc-equal acl2::k acl2::x)))
           (and fty::pair
                (cons acl2::k
                      (backref-fix (cdr fty::pair)))))))

Theorem: backref-alist-fix-of-append

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

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

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