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Svex-alist

Svex-alist-fix

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

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

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

Definitions and Theorems

Function: svex-alist-fix$inline

(defun
 svex-alist-fix$inline (x)
 (declare (xargs :guard (svex-alist-p x)))
 (let
   ((__function__ 'svex-alist-fix))
   (declare (ignorable __function__))
   (mbe :logic
        (if (atom x)
            nil
            (let ((rest (svex-alist-fix (cdr x))))
                 (if (and (consp (car x)) (svar-p (caar x)))
                     (let ((fty::first-key (caar x))
                           (fty::first-val (svex-fix (cdar x))))
                          (cons (cons fty::first-key fty::first-val)
                                rest))
                     rest)))
        :exec x)))

Theorem: svex-alist-p-of-svex-alist-fix

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

Theorem: svex-alist-fix-when-svex-alist-p

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

Function: svex-alist-equiv$inline

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

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

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

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

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

Theorem: svex-alist-fix-under-svex-alist-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: svex-alist-fix-of-acons

(defthm
     svex-alist-fix-of-acons
     (equal (svex-alist-fix (cons (cons acl2::a acl2::b) x))
            (let ((rest (svex-alist-fix x)))
                 (if (and (svar-p acl2::a))
                     (let ((fty::first-key acl2::a)
                           (fty::first-val (svex-fix acl2::b)))
                          (cons (cons fty::first-key fty::first-val)
                                rest))
                     rest))))

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

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

Theorem: svex-alist-fix-of-append

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

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

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