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Alternative-list

Alternative-list-fix

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

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

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

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

Theorem: alternative-listp-of-alternative-list-fix

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

Theorem: alternative-list-fix-when-alternative-listp

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

Function: alternative-list-equiv$inline

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

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

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

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

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

Theorem: alternative-list-fix-under-alternative-list-equiv

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: consp-of-alternative-list-fix

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

Theorem: alternative-list-fix-under-iff

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

Theorem: alternative-list-fix-of-cons

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

Theorem: len-of-alternative-list-fix

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

Theorem: alternative-list-fix-of-append

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

Theorem: alternative-list-fix-of-repeat

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

Theorem: list-equiv-refines-alternative-list-equiv

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

Theorem: nth-of-alternative-list-fix

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

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

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

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

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

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

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

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

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