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Func-alist

Func-alist-fix

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

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

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

Definitions and Theorems

Function: func-alist-fix$inline

(defun func-alist-fix$inline (x)
  (declare (xargs :guard (func-alistp x)))
  (let ((acl2::__function__ 'func-alist-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (if (atom x)
             nil
           (if (consp (car x))
               (cons (cons (symbol-fix (caar x))
                           (func-fix (cdar x)))
                     (func-alist-fix (cdr x)))
             (func-alist-fix (cdr x))))
         :exec x)))

Theorem: func-alistp-of-func-alist-fix

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

Theorem: func-alist-fix-when-func-alistp

(defthm func-alist-fix-when-func-alistp
  (implies (func-alistp x)
           (equal (func-alist-fix x) x)))

Function: func-alist-equiv$inline

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

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

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

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

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

Theorem: func-alist-fix-under-func-alist-equiv

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

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

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

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

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

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

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

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

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

Theorem: cons-of-symbol-fix-k-under-func-alist-equiv

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

Theorem: cons-symbol-equiv-congruence-on-k-under-func-alist-equiv

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

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

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

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

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

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

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

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

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

Theorem: func-alist-fix-of-acons

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

Theorem: func-alist-fix-of-append

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

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

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