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Binding-list

Binding-list-fix

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

Signature
(binding-list-fix x) → fty::newx
Arguments: x — Guard (binding-listp x).
Returns: fty::newx — Type (binding-listp fty::newx).

In the logic, we apply binding-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: binding-list-fix$inline

(defun binding-list-fix$inline (x)
  (declare (xargs :guard (binding-listp x)))
  (let ((acl2::__function__ 'binding-list-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (binding-fix (car x))
                 (binding-list-fix (cdr x))))
         :exec x)))

Theorem: binding-listp-of-binding-list-fix

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

Theorem: binding-list-fix-when-binding-listp

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

Function: binding-list-equiv$inline

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

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

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

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

(defthm binding-list-equiv-implies-equal-binding-list-fix-1
  (implies (binding-list-equiv acl2::x x-equiv)
           (equal (binding-list-fix acl2::x)
                  (binding-list-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: binding-list-fix-under-binding-list-equiv

(defthm binding-list-fix-under-binding-list-equiv
  (binding-list-equiv (binding-list-fix acl2::x)
                      acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

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

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

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

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

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

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

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

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

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

(defthm car-of-binding-list-fix-x-under-binding-equiv
  (binding-equiv (car (binding-list-fix acl2::x))
                 (car acl2::x)))

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

(defthm car-binding-list-equiv-congruence-on-x-under-binding-equiv
  (implies (binding-list-equiv acl2::x x-equiv)
           (binding-equiv (car acl2::x)
                          (car x-equiv)))
  :rule-classes :congruence)

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

(defthm cdr-of-binding-list-fix-x-under-binding-list-equiv
  (binding-list-equiv (cdr (binding-list-fix acl2::x))
                      (cdr acl2::x)))

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

(defthm
    cdr-binding-list-equiv-congruence-on-x-under-binding-list-equiv
  (implies (binding-list-equiv acl2::x x-equiv)
           (binding-list-equiv (cdr acl2::x)
                               (cdr x-equiv)))
  :rule-classes :congruence)

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

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

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

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

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

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

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

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

Theorem: consp-of-binding-list-fix

(defthm consp-of-binding-list-fix
  (equal (consp (binding-list-fix acl2::x))
         (consp acl2::x)))

Theorem: binding-list-fix-under-iff

(defthm binding-list-fix-under-iff
  (iff (binding-list-fix acl2::x)
       (consp acl2::x)))

Theorem: binding-list-fix-of-cons

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

Theorem: len-of-binding-list-fix

(defthm len-of-binding-list-fix
  (equal (len (binding-list-fix acl2::x))
         (len acl2::x)))

Theorem: binding-list-fix-of-append

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

Theorem: binding-list-fix-of-repeat

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

Theorem: list-equiv-refines-binding-list-equiv

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

Theorem: nth-of-binding-list-fix

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

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

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

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

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

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

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

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

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