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Chase-stack

Chase-stack-fix

(chase-stack-fix x) is a usual fty list fixing function.

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

In the logic, we apply chase-position-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: chase-stack-fix$inline

(defun chase-stack-fix$inline (x)
  (declare (xargs :guard (chase-stack-p x)))
  (let ((__function__ 'chase-stack-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (chase-position-fix (car x))
                 (chase-stack-fix (cdr x))))
         :exec x)))

Theorem: chase-stack-p-of-chase-stack-fix

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

Theorem: chase-stack-fix-when-chase-stack-p

(defthm chase-stack-fix-when-chase-stack-p
  (implies (chase-stack-p x)
           (equal (chase-stack-fix x) x)))

Function: chase-stack-equiv$inline

(defun chase-stack-equiv$inline (x y)
  (declare (xargs :guard (and (chase-stack-p x)
                              (chase-stack-p y))))
  (equal (chase-stack-fix x)
         (chase-stack-fix y)))

Theorem: chase-stack-equiv-is-an-equivalence

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

Theorem: chase-stack-equiv-implies-equal-chase-stack-fix-1

(defthm chase-stack-equiv-implies-equal-chase-stack-fix-1
  (implies (chase-stack-equiv x x-equiv)
           (equal (chase-stack-fix x)
                  (chase-stack-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: chase-stack-fix-under-chase-stack-equiv

(defthm chase-stack-fix-under-chase-stack-equiv
  (chase-stack-equiv (chase-stack-fix x)
                     x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-chase-stack-fix-1-forward-to-chase-stack-equiv

(defthm equal-of-chase-stack-fix-1-forward-to-chase-stack-equiv
  (implies (equal (chase-stack-fix x) y)
           (chase-stack-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-chase-stack-fix-2-forward-to-chase-stack-equiv

(defthm equal-of-chase-stack-fix-2-forward-to-chase-stack-equiv
  (implies (equal x (chase-stack-fix y))
           (chase-stack-equiv x y))
  :rule-classes :forward-chaining)

Theorem: chase-stack-equiv-of-chase-stack-fix-1-forward

(defthm chase-stack-equiv-of-chase-stack-fix-1-forward
  (implies (chase-stack-equiv (chase-stack-fix x)
                              y)
           (chase-stack-equiv x y))
  :rule-classes :forward-chaining)

Theorem: chase-stack-equiv-of-chase-stack-fix-2-forward

(defthm chase-stack-equiv-of-chase-stack-fix-2-forward
  (implies (chase-stack-equiv x (chase-stack-fix y))
           (chase-stack-equiv x y))
  :rule-classes :forward-chaining)

Theorem: car-of-chase-stack-fix-x-under-chase-position-equiv

(defthm car-of-chase-stack-fix-x-under-chase-position-equiv
  (chase-position-equiv (car (chase-stack-fix x))
                        (car x)))

Theorem: car-chase-stack-equiv-congruence-on-x-under-chase-position-equiv

(defthm
   car-chase-stack-equiv-congruence-on-x-under-chase-position-equiv
  (implies (chase-stack-equiv x x-equiv)
           (chase-position-equiv (car x)
                                 (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-chase-stack-fix-x-under-chase-stack-equiv

(defthm cdr-of-chase-stack-fix-x-under-chase-stack-equiv
  (chase-stack-equiv (cdr (chase-stack-fix x))
                     (cdr x)))

Theorem: cdr-chase-stack-equiv-congruence-on-x-under-chase-stack-equiv

(defthm
      cdr-chase-stack-equiv-congruence-on-x-under-chase-stack-equiv
  (implies (chase-stack-equiv x x-equiv)
           (chase-stack-equiv (cdr x)
                              (cdr x-equiv)))
  :rule-classes :congruence)

Theorem: cons-of-chase-position-fix-x-under-chase-stack-equiv

(defthm cons-of-chase-position-fix-x-under-chase-stack-equiv
  (chase-stack-equiv (cons (chase-position-fix x) y)
                     (cons x y)))

Theorem: cons-chase-position-equiv-congruence-on-x-under-chase-stack-equiv

(defthm
  cons-chase-position-equiv-congruence-on-x-under-chase-stack-equiv
  (implies (chase-position-equiv x x-equiv)
           (chase-stack-equiv (cons x y)
                              (cons x-equiv y)))
  :rule-classes :congruence)

Theorem: cons-of-chase-stack-fix-y-under-chase-stack-equiv

(defthm cons-of-chase-stack-fix-y-under-chase-stack-equiv
  (chase-stack-equiv (cons x (chase-stack-fix y))
                     (cons x y)))

Theorem: cons-chase-stack-equiv-congruence-on-y-under-chase-stack-equiv

(defthm
     cons-chase-stack-equiv-congruence-on-y-under-chase-stack-equiv
  (implies (chase-stack-equiv y y-equiv)
           (chase-stack-equiv (cons x y)
                              (cons x y-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-chase-stack-fix

(defthm consp-of-chase-stack-fix
  (equal (consp (chase-stack-fix x))
         (consp x)))

Theorem: chase-stack-fix-under-iff

(defthm chase-stack-fix-under-iff
  (iff (chase-stack-fix x) (consp x)))

Theorem: chase-stack-fix-of-cons

(defthm chase-stack-fix-of-cons
  (equal (chase-stack-fix (cons a x))
         (cons (chase-position-fix a)
               (chase-stack-fix x))))

Theorem: len-of-chase-stack-fix

(defthm len-of-chase-stack-fix
  (equal (len (chase-stack-fix x))
         (len x)))

Theorem: chase-stack-fix-of-append

(defthm chase-stack-fix-of-append
  (equal (chase-stack-fix (append std::a std::b))
         (append (chase-stack-fix std::a)
                 (chase-stack-fix std::b))))

Theorem: chase-stack-fix-of-repeat

(defthm chase-stack-fix-of-repeat
  (equal (chase-stack-fix (repeat acl2::n x))
         (repeat acl2::n (chase-position-fix x))))

Theorem: list-equiv-refines-chase-stack-equiv

(defthm list-equiv-refines-chase-stack-equiv
  (implies (list-equiv x y)
           (chase-stack-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-chase-stack-fix

(defthm nth-of-chase-stack-fix
  (equal (nth acl2::n (chase-stack-fix x))
         (if (< (nfix acl2::n) (len x))
             (chase-position-fix (nth acl2::n x))
           nil)))

Theorem: chase-stack-equiv-implies-chase-stack-equiv-append-1

(defthm chase-stack-equiv-implies-chase-stack-equiv-append-1
  (implies (chase-stack-equiv x fty::x-equiv)
           (chase-stack-equiv (append x y)
                              (append fty::x-equiv y)))
  :rule-classes (:congruence))

Theorem: chase-stack-equiv-implies-chase-stack-equiv-append-2

(defthm chase-stack-equiv-implies-chase-stack-equiv-append-2
  (implies (chase-stack-equiv y fty::y-equiv)
           (chase-stack-equiv (append x y)
                              (append x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: chase-stack-equiv-implies-chase-stack-equiv-nthcdr-2

(defthm chase-stack-equiv-implies-chase-stack-equiv-nthcdr-2
  (implies (chase-stack-equiv acl2::l l-equiv)
           (chase-stack-equiv (nthcdr acl2::n acl2::l)
                              (nthcdr acl2::n l-equiv)))
  :rule-classes (:congruence))

Theorem: chase-stack-equiv-implies-chase-stack-equiv-take-2

(defthm chase-stack-equiv-implies-chase-stack-equiv-take-2
  (implies (chase-stack-equiv acl2::l l-equiv)
           (chase-stack-equiv (take acl2::n acl2::l)
                              (take acl2::n l-equiv)))
  :rule-classes (:congruence))