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Pathlist

Pathlist-fix

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

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

In the logic, we apply path-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: pathlist-fix$inline

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

Theorem: pathlist-p-of-pathlist-fix

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

Theorem: pathlist-fix-when-pathlist-p

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

Function: pathlist-equiv$inline

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

Theorem: pathlist-equiv-is-an-equivalence

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

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

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

Theorem: pathlist-fix-under-pathlist-equiv

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

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

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

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

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

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

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

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

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

Theorem: car-of-pathlist-fix-x-under-path-equiv

(defthm car-of-pathlist-fix-x-under-path-equiv
  (path-equiv (car (pathlist-fix x))
              (car x)))

Theorem: car-pathlist-equiv-congruence-on-x-under-path-equiv

(defthm car-pathlist-equiv-congruence-on-x-under-path-equiv
  (implies (pathlist-equiv x x-equiv)
           (path-equiv (car x) (car x-equiv)))
  :rule-classes :congruence)

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

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

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

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

Theorem: cons-of-path-fix-x-under-pathlist-equiv

(defthm cons-of-path-fix-x-under-pathlist-equiv
  (pathlist-equiv (cons (path-fix x) y)
                  (cons x y)))

Theorem: cons-path-equiv-congruence-on-x-under-pathlist-equiv

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

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

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

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

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

Theorem: consp-of-pathlist-fix

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

Theorem: pathlist-fix-under-iff

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

Theorem: pathlist-fix-of-cons

(defthm pathlist-fix-of-cons
  (equal (pathlist-fix (cons a x))
         (cons (path-fix a) (pathlist-fix x))))

Theorem: len-of-pathlist-fix

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

Theorem: pathlist-fix-of-append

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

Theorem: pathlist-fix-of-repeat

(defthm pathlist-fix-of-repeat
  (equal (pathlist-fix (repeat acl2::n x))
         (repeat acl2::n (path-fix x))))

Theorem: list-equiv-refines-pathlist-equiv

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

Theorem: nth-of-pathlist-fix

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

Theorem: pathlist-equiv-implies-pathlist-equiv-append-1

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

Theorem: pathlist-equiv-implies-pathlist-equiv-append-2

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

Theorem: pathlist-equiv-implies-pathlist-equiv-nthcdr-2

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

Theorem: pathlist-equiv-implies-pathlist-equiv-take-2

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