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Backtrace

Backtrace-fix

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

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

In the logic, we apply backtrace-frame-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: backtrace-fix$inline

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

Theorem: backtrace-p-of-backtrace-fix

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

Theorem: backtrace-fix-when-backtrace-p

(defthm backtrace-fix-when-backtrace-p
  (implies (backtrace-p x)
           (equal (backtrace-fix x) x)))

Function: backtrace-equiv$inline

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

Theorem: backtrace-equiv-is-an-equivalence

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

Theorem: backtrace-equiv-implies-equal-backtrace-fix-1

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

Theorem: backtrace-fix-under-backtrace-equiv

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

Theorem: equal-of-backtrace-fix-1-forward-to-backtrace-equiv

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

Theorem: equal-of-backtrace-fix-2-forward-to-backtrace-equiv

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

Theorem: backtrace-equiv-of-backtrace-fix-1-forward

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

Theorem: backtrace-equiv-of-backtrace-fix-2-forward

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

Theorem: car-of-backtrace-fix-x-under-backtrace-frame-equiv

(defthm car-of-backtrace-fix-x-under-backtrace-frame-equiv
  (backtrace-frame-equiv (car (backtrace-fix x))
                         (car x)))

Theorem: car-backtrace-equiv-congruence-on-x-under-backtrace-frame-equiv

(defthm
    car-backtrace-equiv-congruence-on-x-under-backtrace-frame-equiv
  (implies (backtrace-equiv x x-equiv)
           (backtrace-frame-equiv (car x)
                                  (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-backtrace-fix-x-under-backtrace-equiv

(defthm cdr-of-backtrace-fix-x-under-backtrace-equiv
  (backtrace-equiv (cdr (backtrace-fix x))
                   (cdr x)))

Theorem: cdr-backtrace-equiv-congruence-on-x-under-backtrace-equiv

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

Theorem: cons-of-backtrace-frame-fix-x-under-backtrace-equiv

(defthm cons-of-backtrace-frame-fix-x-under-backtrace-equiv
  (backtrace-equiv (cons (backtrace-frame-fix x) y)
                   (cons x y)))

Theorem: cons-backtrace-frame-equiv-congruence-on-x-under-backtrace-equiv

(defthm
   cons-backtrace-frame-equiv-congruence-on-x-under-backtrace-equiv
  (implies (backtrace-frame-equiv x x-equiv)
           (backtrace-equiv (cons x y)
                            (cons x-equiv y)))
  :rule-classes :congruence)

Theorem: cons-of-backtrace-fix-y-under-backtrace-equiv

(defthm cons-of-backtrace-fix-y-under-backtrace-equiv
  (backtrace-equiv (cons x (backtrace-fix y))
                   (cons x y)))

Theorem: cons-backtrace-equiv-congruence-on-y-under-backtrace-equiv

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

Theorem: consp-of-backtrace-fix

(defthm consp-of-backtrace-fix
  (equal (consp (backtrace-fix x))
         (consp x)))

Theorem: backtrace-fix-under-iff

(defthm backtrace-fix-under-iff
  (iff (backtrace-fix x) (consp x)))

Theorem: backtrace-fix-of-cons

(defthm backtrace-fix-of-cons
  (equal (backtrace-fix (cons a x))
         (cons (backtrace-frame-fix a)
               (backtrace-fix x))))

Theorem: len-of-backtrace-fix

(defthm len-of-backtrace-fix
  (equal (len (backtrace-fix x)) (len x)))

Theorem: backtrace-fix-of-append

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

Theorem: backtrace-fix-of-repeat

(defthm backtrace-fix-of-repeat
  (equal (backtrace-fix (acl2::repeat n x))
         (acl2::repeat n (backtrace-frame-fix x))))

Theorem: list-equiv-refines-backtrace-equiv

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

Theorem: nth-of-backtrace-fix

(defthm nth-of-backtrace-fix
  (equal (nth n (backtrace-fix x))
         (if (< (nfix n) (len x))
             (backtrace-frame-fix (nth n x))
           nil)))

Theorem: backtrace-equiv-implies-backtrace-equiv-append-1

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

Theorem: backtrace-equiv-implies-backtrace-equiv-append-2

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

Theorem: backtrace-equiv-implies-backtrace-equiv-nthcdr-2

(defthm backtrace-equiv-implies-backtrace-equiv-nthcdr-2
  (implies (backtrace-equiv l l-equiv)
           (backtrace-equiv (nthcdr n l)
                            (nthcdr n l-equiv)))
  :rule-classes (:congruence))

Theorem: backtrace-equiv-implies-backtrace-equiv-take-2

(defthm backtrace-equiv-implies-backtrace-equiv-take-2
  (implies (backtrace-equiv l l-equiv)
           (backtrace-equiv (take n l)
                            (take n l-equiv)))
  :rule-classes (:congruence))