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• Stack

Stack-fix

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

Signature

(stack-fix x) → fty::newx

Arguments

x — Guard (stackp x).

Returns

fty::newx — Type (stackp fty::newx).

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

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

Theorem: stackp-of-stack-fix

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

Theorem: stack-fix-when-stackp

(defthm stack-fix-when-stackp
  (implies (stackp x)
           (equal (stack-fix x) x)))

Function: stack-equiv$inline

(defun stack-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (stackp acl2::x)
                              (stackp acl2::y))))
  (equal (stack-fix acl2::x)
         (stack-fix acl2::y)))

Theorem: stack-equiv-is-an-equivalence

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

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

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

Theorem: stack-fix-under-stack-equiv

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

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

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

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

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

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

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

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

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

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

(defthm car-of-stack-fix-x-under-frame-equiv
  (frame-equiv (car (stack-fix acl2::x))
               (car acl2::x)))

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

(defthm car-stack-equiv-congruence-on-x-under-frame-equiv
  (implies (stack-equiv acl2::x x-equiv)
           (frame-equiv (car acl2::x)
                        (car x-equiv)))
  :rule-classes :congruence)

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

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

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

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

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

(defthm cons-of-frame-fix-x-under-stack-equiv
  (stack-equiv (cons (frame-fix acl2::x) acl2::y)
               (cons acl2::x acl2::y)))

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

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

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

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

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

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

Theorem: consp-of-stack-fix

(defthm consp-of-stack-fix
  (equal (consp (stack-fix acl2::x))
         (consp acl2::x)))

Theorem: stack-fix-under-iff

(defthm stack-fix-under-iff
  (iff (stack-fix acl2::x)
       (consp acl2::x)))

Theorem: stack-fix-of-cons

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

Theorem: len-of-stack-fix

(defthm len-of-stack-fix
  (equal (len (stack-fix acl2::x))
         (len acl2::x)))

Theorem: stack-fix-of-append

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

Theorem: stack-fix-of-repeat

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

Theorem: list-equiv-refines-stack-equiv

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

Theorem: nth-of-stack-fix

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

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

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

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

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

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

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

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

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