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• Node-list

Node-list-fix

(node-list-fix x) is a usual ACL2::fty list fixing function.

Signature

(node-list-fix x) → fty::newx

Arguments

x — Guard (node-listp x).

Returns

fty::newx — Type (node-listp fty::newx).

In the logic, we apply node-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: node-list-fix$inline

(defun node-list-fix$inline (x)
  (declare (xargs :guard (node-listp x)))
  (let ((__function__ 'node-list-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (node-fix (car x))
                 (node-list-fix (cdr x))))
         :exec x)))

Theorem: node-listp-of-node-list-fix

(defthm node-listp-of-node-list-fix
  (b* ((fty::newx (node-list-fix$inline x)))
    (node-listp fty::newx))
  :rule-classes :rewrite)

Theorem: node-list-fix-when-node-listp

(defthm node-list-fix-when-node-listp
  (implies (node-listp x)
           (equal (node-list-fix x) x)))

Function: node-list-equiv$inline

(defun node-list-equiv$inline (x acl2::y)
  (declare (xargs :guard (and (node-listp x)
                              (node-listp acl2::y))))
  (equal (node-list-fix x)
         (node-list-fix acl2::y)))

Theorem: node-list-equiv-is-an-equivalence

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

Theorem: node-list-equiv-implies-equal-node-list-fix-1

(defthm node-list-equiv-implies-equal-node-list-fix-1
  (implies (node-list-equiv x x-equiv)
           (equal (node-list-fix x)
                  (node-list-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: node-list-fix-under-node-list-equiv

(defthm node-list-fix-under-node-list-equiv
  (node-list-equiv (node-list-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-node-list-fix-1-forward-to-node-list-equiv

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

Theorem: equal-of-node-list-fix-2-forward-to-node-list-equiv

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

Theorem: node-list-equiv-of-node-list-fix-1-forward

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

Theorem: node-list-equiv-of-node-list-fix-2-forward

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

Theorem: car-of-node-list-fix-x-under-node-equiv

(defthm car-of-node-list-fix-x-under-node-equiv
  (node-equiv (car (node-list-fix x))
              (car x)))

Theorem: car-node-list-equiv-congruence-on-x-under-node-equiv

(defthm car-node-list-equiv-congruence-on-x-under-node-equiv
  (implies (node-list-equiv x x-equiv)
           (node-equiv (car x) (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-node-list-fix-x-under-node-list-equiv

(defthm cdr-of-node-list-fix-x-under-node-list-equiv
  (node-list-equiv (cdr (node-list-fix x))
                   (cdr x)))

Theorem: cdr-node-list-equiv-congruence-on-x-under-node-list-equiv

(defthm cdr-node-list-equiv-congruence-on-x-under-node-list-equiv
  (implies (node-list-equiv x x-equiv)
           (node-list-equiv (cdr x) (cdr x-equiv)))
  :rule-classes :congruence)

Theorem: cons-of-node-fix-x-under-node-list-equiv

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

Theorem: cons-node-equiv-congruence-on-x-under-node-list-equiv

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

Theorem: cons-of-node-list-fix-y-under-node-list-equiv

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

Theorem: cons-node-list-equiv-congruence-on-y-under-node-list-equiv

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

Theorem: consp-of-node-list-fix

(defthm consp-of-node-list-fix
  (equal (consp (node-list-fix x))
         (consp x)))

Theorem: node-list-fix-under-iff

(defthm node-list-fix-under-iff
  (iff (node-list-fix x) (consp x)))

Theorem: node-list-fix-of-cons

(defthm node-list-fix-of-cons
  (equal (node-list-fix (cons a x))
         (cons (node-fix a) (node-list-fix x))))

Theorem: len-of-node-list-fix

(defthm len-of-node-list-fix
  (equal (len (node-list-fix x)) (len x)))

Theorem: node-list-fix-of-append

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

Theorem: node-list-fix-of-repeat

(defthm node-list-fix-of-repeat
  (equal (node-list-fix (acl2::repeat acl2::n x))
         (acl2::repeat acl2::n (node-fix x))))

Theorem: list-equiv-refines-node-list-equiv

(defthm list-equiv-refines-node-list-equiv
  (implies (list-equiv x acl2::y)
           (node-list-equiv x acl2::y))
  :rule-classes :refinement)

Theorem: nth-of-node-list-fix

(defthm nth-of-node-list-fix
  (equal (nth acl2::n (node-list-fix x))
         (if (< (nfix acl2::n) (len x))
             (node-fix (nth acl2::n x))
           nil)))

Theorem: node-list-equiv-implies-node-list-equiv-append-1

(defthm node-list-equiv-implies-node-list-equiv-append-1
  (implies (node-list-equiv x fty::x-equiv)
           (node-list-equiv (append x acl2::y)
                            (append fty::x-equiv acl2::y)))
  :rule-classes (:congruence))

Theorem: node-list-equiv-implies-node-list-equiv-append-2

(defthm node-list-equiv-implies-node-list-equiv-append-2
  (implies (node-list-equiv acl2::y fty::y-equiv)
           (node-list-equiv (append x acl2::y)
                            (append x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: node-list-equiv-implies-node-list-equiv-nthcdr-2

(defthm node-list-equiv-implies-node-list-equiv-nthcdr-2
  (implies (node-list-equiv acl2::l l-equiv)
           (node-list-equiv (nthcdr acl2::n acl2::l)
                            (nthcdr acl2::n l-equiv)))
  :rule-classes (:congruence))

Theorem: node-list-equiv-implies-node-list-equiv-take-2

(defthm node-list-equiv-implies-node-list-equiv-take-2
  (implies (node-list-equiv acl2::l l-equiv)
           (node-list-equiv (take acl2::n acl2::l)
                            (take acl2::n l-equiv)))
  :rule-classes (:congruence))