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Svexl-node-array

Svexl-node-array-fix

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

Signature
(svexl-node-array-fix x) → fty::newx
Arguments: x — Guard (svexl-node-array-p x).
Returns: fty::newx — Type (svexl-node-array-p fty::newx).

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

Definitions and Theorems

Function: svexl-node-array-fix$inline

(defun svexl-node-array-fix$inline (x)
  (declare (xargs :guard (svexl-node-array-p x)))
  (let ((acl2::__function__ 'svexl-node-array-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (if (atom x)
             nil
           (if (consp (car x))
               (cons (cons (nfix (caar x))
                           (svexl-node-fix (cdar x)))
                     (svexl-node-array-fix (cdr x)))
             (svexl-node-array-fix (cdr x))))
         :exec x)))

Theorem: svexl-node-array-p-of-svexl-node-array-fix

(defthm svexl-node-array-p-of-svexl-node-array-fix
  (b* ((fty::newx (svexl-node-array-fix$inline x)))
    (svexl-node-array-p fty::newx))
  :rule-classes :rewrite)

Theorem: svexl-node-array-fix-when-svexl-node-array-p

(defthm svexl-node-array-fix-when-svexl-node-array-p
  (implies (svexl-node-array-p x)
           (equal (svexl-node-array-fix x) x)))

Function: svexl-node-array-equiv$inline

(defun svexl-node-array-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (svexl-node-array-p acl2::x)
                              (svexl-node-array-p acl2::y))))
  (equal (svexl-node-array-fix acl2::x)
         (svexl-node-array-fix acl2::y)))

Theorem: svexl-node-array-equiv-is-an-equivalence

(defthm svexl-node-array-equiv-is-an-equivalence
  (and (booleanp (svexl-node-array-equiv x y))
       (svexl-node-array-equiv x x)
       (implies (svexl-node-array-equiv x y)
                (svexl-node-array-equiv y x))
       (implies (and (svexl-node-array-equiv x y)
                     (svexl-node-array-equiv y z))
                (svexl-node-array-equiv x z)))
  :rule-classes (:equivalence))

Theorem: svexl-node-array-equiv-implies-equal-svexl-node-array-fix-1

(defthm svexl-node-array-equiv-implies-equal-svexl-node-array-fix-1
  (implies (svexl-node-array-equiv acl2::x x-equiv)
           (equal (svexl-node-array-fix acl2::x)
                  (svexl-node-array-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: svexl-node-array-fix-under-svexl-node-array-equiv

(defthm svexl-node-array-fix-under-svexl-node-array-equiv
  (svexl-node-array-equiv (svexl-node-array-fix acl2::x)
                          acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-svexl-node-array-fix-1-forward-to-svexl-node-array-equiv

(defthm
  equal-of-svexl-node-array-fix-1-forward-to-svexl-node-array-equiv
  (implies (equal (svexl-node-array-fix acl2::x)
                  acl2::y)
           (svexl-node-array-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-svexl-node-array-fix-2-forward-to-svexl-node-array-equiv

(defthm
  equal-of-svexl-node-array-fix-2-forward-to-svexl-node-array-equiv
  (implies (equal acl2::x (svexl-node-array-fix acl2::y))
           (svexl-node-array-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: svexl-node-array-equiv-of-svexl-node-array-fix-1-forward

(defthm svexl-node-array-equiv-of-svexl-node-array-fix-1-forward
  (implies (svexl-node-array-equiv (svexl-node-array-fix acl2::x)
                                   acl2::y)
           (svexl-node-array-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: svexl-node-array-equiv-of-svexl-node-array-fix-2-forward

(defthm svexl-node-array-equiv-of-svexl-node-array-fix-2-forward
 (implies
     (svexl-node-array-equiv acl2::x (svexl-node-array-fix acl2::y))
     (svexl-node-array-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: cons-of-nfix-k-under-svexl-node-array-equiv

(defthm cons-of-nfix-k-under-svexl-node-array-equiv
  (svexl-node-array-equiv (cons (cons (nfix acl2::k) acl2::v)
                                acl2::x)
                          (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-nat-equiv-congruence-on-k-under-svexl-node-array-equiv

(defthm cons-nat-equiv-congruence-on-k-under-svexl-node-array-equiv
 (implies
     (acl2::nat-equiv acl2::k k-equiv)
     (svexl-node-array-equiv (cons (cons acl2::k acl2::v) acl2::x)
                             (cons (cons k-equiv acl2::v) acl2::x)))
 :rule-classes :congruence)

Theorem: cons-of-svexl-node-fix-v-under-svexl-node-array-equiv

(defthm cons-of-svexl-node-fix-v-under-svexl-node-array-equiv
  (svexl-node-array-equiv
       (cons (cons acl2::k (svexl-node-fix acl2::v))
             acl2::x)
       (cons (cons acl2::k acl2::v) acl2::x)))

Theorem: cons-svexl-node-equiv-congruence-on-v-under-svexl-node-array-equiv

(defthm
 cons-svexl-node-equiv-congruence-on-v-under-svexl-node-array-equiv
 (implies
     (svexl-node-equiv acl2::v v-equiv)
     (svexl-node-array-equiv (cons (cons acl2::k acl2::v) acl2::x)
                             (cons (cons acl2::k v-equiv) acl2::x)))
 :rule-classes :congruence)

Theorem: cons-of-svexl-node-array-fix-y-under-svexl-node-array-equiv

(defthm cons-of-svexl-node-array-fix-y-under-svexl-node-array-equiv
  (svexl-node-array-equiv
       (cons acl2::x (svexl-node-array-fix acl2::y))
       (cons acl2::x acl2::y)))

Theorem: cons-svexl-node-array-equiv-congruence-on-y-under-svexl-node-array-equiv

(defthm
 cons-svexl-node-array-equiv-congruence-on-y-under-svexl-node-array-equiv
 (implies (svexl-node-array-equiv acl2::y y-equiv)
          (svexl-node-array-equiv (cons acl2::x acl2::y)
                                  (cons acl2::x y-equiv)))
 :rule-classes :congruence)

Theorem: svexl-node-array-fix-of-acons

(defthm svexl-node-array-fix-of-acons
  (equal (svexl-node-array-fix (cons (cons acl2::a acl2::b) x))
         (cons (cons (nfix acl2::a)
                     (svexl-node-fix acl2::b))
               (svexl-node-array-fix x))))

Theorem: svexl-node-array-fix-of-append

(defthm svexl-node-array-fix-of-append
  (equal (svexl-node-array-fix (append std::a std::b))
         (append (svexl-node-array-fix std::a)
                 (svexl-node-array-fix std::b))))

Theorem: consp-car-of-svexl-node-array-fix

(defthm consp-car-of-svexl-node-array-fix
  (equal (consp (car (svexl-node-array-fix x)))
         (consp (svexl-node-array-fix x))))