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Alias-lst

Alias-lst-fix

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

Signature
(alias-lst-fix x) → fty::newx
Arguments: x — Guard (alias-lst-p x).
Returns: fty::newx — Type (alias-lst-p fty::newx).

In the logic, we apply alias-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: alias-lst-fix$inline

(defun alias-lst-fix$inline (x)
  (declare (xargs :guard (alias-lst-p x)))
  (let ((acl2::__function__ 'alias-lst-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (if (atom x)
             x
           (cons (alias-fix (car x))
                 (alias-lst-fix (cdr x))))
         :exec x)))

Theorem: alias-lst-p-of-alias-lst-fix

(defthm alias-lst-p-of-alias-lst-fix
  (b* ((fty::newx (alias-lst-fix$inline x)))
    (alias-lst-p fty::newx))
  :rule-classes :rewrite)

Theorem: alias-lst-fix-when-alias-lst-p

(defthm alias-lst-fix-when-alias-lst-p
  (implies (alias-lst-p x)
           (equal (alias-lst-fix x) x)))

Function: alias-lst-equiv$inline

(defun alias-lst-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (alias-lst-p acl2::x)
                              (alias-lst-p acl2::y))))
  (equal (alias-lst-fix acl2::x)
         (alias-lst-fix acl2::y)))

Theorem: alias-lst-equiv-is-an-equivalence

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

Theorem: alias-lst-equiv-implies-equal-alias-lst-fix-1

(defthm alias-lst-equiv-implies-equal-alias-lst-fix-1
  (implies (alias-lst-equiv acl2::x x-equiv)
           (equal (alias-lst-fix acl2::x)
                  (alias-lst-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: alias-lst-fix-under-alias-lst-equiv

(defthm alias-lst-fix-under-alias-lst-equiv
  (alias-lst-equiv (alias-lst-fix acl2::x)
                   acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-alias-lst-fix-1-forward-to-alias-lst-equiv

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

Theorem: equal-of-alias-lst-fix-2-forward-to-alias-lst-equiv

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

Theorem: alias-lst-equiv-of-alias-lst-fix-1-forward

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

Theorem: alias-lst-equiv-of-alias-lst-fix-2-forward

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

Theorem: car-of-alias-lst-fix-x-under-alias-equiv

(defthm car-of-alias-lst-fix-x-under-alias-equiv
  (alias-equiv (car (alias-lst-fix acl2::x))
               (car acl2::x)))

Theorem: car-alias-lst-equiv-congruence-on-x-under-alias-equiv

(defthm car-alias-lst-equiv-congruence-on-x-under-alias-equiv
  (implies (alias-lst-equiv acl2::x x-equiv)
           (alias-equiv (car acl2::x)
                        (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-alias-lst-fix-x-under-alias-lst-equiv

(defthm cdr-of-alias-lst-fix-x-under-alias-lst-equiv
  (alias-lst-equiv (cdr (alias-lst-fix acl2::x))
                   (cdr acl2::x)))

Theorem: cdr-alias-lst-equiv-congruence-on-x-under-alias-lst-equiv

(defthm cdr-alias-lst-equiv-congruence-on-x-under-alias-lst-equiv
  (implies (alias-lst-equiv acl2::x x-equiv)
           (alias-lst-equiv (cdr acl2::x)
                            (cdr x-equiv)))
  :rule-classes :congruence)

Theorem: cons-of-alias-fix-x-under-alias-lst-equiv

(defthm cons-of-alias-fix-x-under-alias-lst-equiv
  (alias-lst-equiv (cons (alias-fix acl2::x) acl2::y)
                   (cons acl2::x acl2::y)))

Theorem: cons-alias-equiv-congruence-on-x-under-alias-lst-equiv

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

Theorem: cons-of-alias-lst-fix-y-under-alias-lst-equiv

(defthm cons-of-alias-lst-fix-y-under-alias-lst-equiv
  (alias-lst-equiv (cons acl2::x (alias-lst-fix acl2::y))
                   (cons acl2::x acl2::y)))

Theorem: cons-alias-lst-equiv-congruence-on-y-under-alias-lst-equiv

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

Theorem: consp-of-alias-lst-fix

(defthm consp-of-alias-lst-fix
  (equal (consp (alias-lst-fix acl2::x))
         (consp acl2::x)))

Theorem: alias-lst-fix-of-cons

(defthm alias-lst-fix-of-cons
  (equal (alias-lst-fix (cons a x))
         (cons (alias-fix a) (alias-lst-fix x))))

Theorem: len-of-alias-lst-fix

(defthm len-of-alias-lst-fix
  (equal (len (alias-lst-fix acl2::x))
         (len acl2::x)))

Theorem: alias-lst-fix-of-append

(defthm alias-lst-fix-of-append
  (equal (alias-lst-fix (append std::a std::b))
         (append (alias-lst-fix std::a)
                 (alias-lst-fix std::b))))

Theorem: alias-lst-fix-of-repeat

(defthm alias-lst-fix-of-repeat
  (equal (alias-lst-fix (repeat acl2::n acl2::x))
         (repeat acl2::n (alias-fix acl2::x))))

Theorem: nth-of-alias-lst-fix

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

Theorem: alias-lst-equiv-implies-alias-lst-equiv-append-1

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

Theorem: alias-lst-equiv-implies-alias-lst-equiv-append-2

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

Theorem: alias-lst-equiv-implies-alias-lst-equiv-nthcdr-2

(defthm alias-lst-equiv-implies-alias-lst-equiv-nthcdr-2
  (implies (alias-lst-equiv acl2::l l-equiv)
           (alias-lst-equiv (nthcdr acl2::n acl2::l)
                            (nthcdr acl2::n l-equiv)))
  :rule-classes (:congruence))

Theorem: alias-lst-equiv-implies-alias-lst-equiv-take-2

(defthm alias-lst-equiv-implies-alias-lst-equiv-take-2
  (implies (alias-lst-equiv acl2::l l-equiv)
           (alias-lst-equiv (take acl2::n acl2::l)
                            (take acl2::n l-equiv)))
  :rule-classes (:congruence))