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Svex-alist-alist

Svex-alist-alist-fix

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

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

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

Definitions and Theorems

Function: svex-alist-alist-fix$inline

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

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

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

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

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

Function: svex-alist-alist-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: svex-alist-alist-fix-of-acons

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

Theorem: svex-alist-alist-fix-of-append

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

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

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