		Search-engine friendly clone of the ACL2 documentation.

Fn-indices

Fn-indices-fix

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

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

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

Definitions and Theorems

Function: fn-indices-fix$inline

(defun fn-indices-fix$inline (x)
 (declare (xargs :guard (fn-indices-p x)))
 (let ((__function__ 'fn-indices-fix))
  (declare (ignorable __function__))
  (mbe :logic
       (if (atom x)
           nil
         (let ((rest (fn-indices-fix (cdr x))))
           (if (and (consp (car x))
                    (pseudo-fnsym-p (caar x)))
               (let ((fty::first-key (caar x))
                     (fty::first-val (acl2::nat-list-fix (cdar x))))
                 (cons (cons fty::first-key fty::first-val)
                       rest))
             rest)))
       :exec x)))

Theorem: fn-indices-p-of-fn-indices-fix

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

Theorem: fn-indices-fix-when-fn-indices-p

(defthm fn-indices-fix-when-fn-indices-p
  (implies (fn-indices-p x)
           (equal (fn-indices-fix x) x)))

Function: fn-indices-equiv$inline

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

Theorem: fn-indices-equiv-is-an-equivalence

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

Theorem: fn-indices-equiv-implies-equal-fn-indices-fix-1

(defthm fn-indices-equiv-implies-equal-fn-indices-fix-1
  (implies (fn-indices-equiv x x-equiv)
           (equal (fn-indices-fix x)
                  (fn-indices-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: fn-indices-fix-under-fn-indices-equiv

(defthm fn-indices-fix-under-fn-indices-equiv
  (fn-indices-equiv (fn-indices-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-fn-indices-fix-1-forward-to-fn-indices-equiv

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

Theorem: equal-of-fn-indices-fix-2-forward-to-fn-indices-equiv

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

Theorem: fn-indices-equiv-of-fn-indices-fix-1-forward

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

Theorem: fn-indices-equiv-of-fn-indices-fix-2-forward

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

Theorem: cons-of-nat-list-fix-v-under-fn-indices-equiv

(defthm cons-of-nat-list-fix-v-under-fn-indices-equiv
  (fn-indices-equiv (cons (cons k (acl2::nat-list-fix v)) x)
                    (cons (cons k v) x)))

Theorem: cons-nat-list-equiv-congruence-on-v-under-fn-indices-equiv

(defthm cons-nat-list-equiv-congruence-on-v-under-fn-indices-equiv
  (implies (acl2::nat-list-equiv v v-equiv)
           (fn-indices-equiv (cons (cons k v) x)
                             (cons (cons k v-equiv) x)))
  :rule-classes :congruence)

Theorem: cons-of-fn-indices-fix-y-under-fn-indices-equiv

(defthm cons-of-fn-indices-fix-y-under-fn-indices-equiv
  (fn-indices-equiv (cons x (fn-indices-fix y))
                    (cons x y)))

Theorem: cons-fn-indices-equiv-congruence-on-y-under-fn-indices-equiv

(defthm cons-fn-indices-equiv-congruence-on-y-under-fn-indices-equiv
  (implies (fn-indices-equiv y y-equiv)
           (fn-indices-equiv (cons x y)
                             (cons x y-equiv)))
  :rule-classes :congruence)

Theorem: fn-indices-fix-of-acons

(defthm fn-indices-fix-of-acons
  (equal (fn-indices-fix (cons (cons a b) x))
         (let ((rest (fn-indices-fix x)))
           (if (and (pseudo-fnsym-p a))
               (let ((fty::first-key a)
                     (fty::first-val (acl2::nat-list-fix b)))
                 (cons (cons fty::first-key fty::first-val)
                       rest))
             rest))))

Theorem: hons-assoc-equal-of-fn-indices-fix

(defthm hons-assoc-equal-of-fn-indices-fix
  (equal (hons-assoc-equal k (fn-indices-fix x))
         (let ((fty::pair (hons-assoc-equal k x)))
           (and (pseudo-fnsym-p k)
                fty::pair
                (cons k
                      (acl2::nat-list-fix (cdr fty::pair)))))))

Theorem: fn-indices-fix-of-append

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

Theorem: consp-car-of-fn-indices-fix

(defthm consp-car-of-fn-indices-fix
  (equal (consp (car (fn-indices-fix x)))
         (consp (fn-indices-fix x))))