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• Scratchlist

Scratchlist-fix

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

Signature

(scratchlist-fix x) → fty::newx

Arguments

x — Guard (scratchlist-p x).

Returns

fty::newx — Type (scratchlist-p fty::newx).

In the logic, we apply scratchobj-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: scratchlist-fix$inline

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

Theorem: scratchlist-p-of-scratchlist-fix

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

Theorem: scratchlist-fix-when-scratchlist-p

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

Function: scratchlist-equiv$inline

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

Theorem: scratchlist-equiv-is-an-equivalence

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

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

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

Theorem: scratchlist-fix-under-scratchlist-equiv

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

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

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

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

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

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

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

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

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

Theorem: car-of-scratchlist-fix-x-under-scratchobj-equiv

(defthm car-of-scratchlist-fix-x-under-scratchobj-equiv
  (scratchobj-equiv (car (scratchlist-fix x))
                    (car x)))

Theorem: car-scratchlist-equiv-congruence-on-x-under-scratchobj-equiv

(defthm car-scratchlist-equiv-congruence-on-x-under-scratchobj-equiv
  (implies (scratchlist-equiv x x-equiv)
           (scratchobj-equiv (car x)
                             (car x-equiv)))
  :rule-classes :congruence)

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

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

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

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

Theorem: cons-of-scratchobj-fix-x-under-scratchlist-equiv

(defthm cons-of-scratchobj-fix-x-under-scratchlist-equiv
  (scratchlist-equiv (cons (scratchobj-fix x) y)
                     (cons x y)))

Theorem: cons-scratchobj-equiv-congruence-on-x-under-scratchlist-equiv

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

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

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

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

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

Theorem: consp-of-scratchlist-fix

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

Theorem: scratchlist-fix-under-iff

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

Theorem: scratchlist-fix-of-cons

(defthm scratchlist-fix-of-cons
  (equal (scratchlist-fix (cons a x))
         (cons (scratchobj-fix a)
               (scratchlist-fix x))))

Theorem: len-of-scratchlist-fix

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

Theorem: scratchlist-fix-of-append

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

Theorem: scratchlist-fix-of-repeat

(defthm scratchlist-fix-of-repeat
  (equal (scratchlist-fix (acl2::repeat n x))
         (acl2::repeat n (scratchobj-fix x))))

Theorem: list-equiv-refines-scratchlist-equiv

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

Theorem: nth-of-scratchlist-fix

(defthm nth-of-scratchlist-fix
  (equal (nth n (scratchlist-fix x))
         (if (< (nfix n) (len x))
             (scratchobj-fix (nth n x))
           nil)))

Theorem: scratchlist-equiv-implies-scratchlist-equiv-append-1

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

Theorem: scratchlist-equiv-implies-scratchlist-equiv-append-2

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

Theorem: scratchlist-equiv-implies-scratchlist-equiv-nthcdr-2

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

Theorem: scratchlist-equiv-implies-scratchlist-equiv-take-2

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