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S4veclist

S4veclist-fix

(s4veclist-fix x) is a usual fty list fixing function.

Signature
(s4veclist-fix x) → fty::newx
Arguments: x — Guard (s4veclist-p x).
Returns: fty::newx — Type (s4veclist-p fty::newx).

In the logic, we apply s4vec-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: s4veclist-fix$inline

(defun s4veclist-fix$inline (x)
  (declare (xargs :guard (s4veclist-p x)))
  (let ((__function__ 's4veclist-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (s4vec-fix (car x))
                 (s4veclist-fix (cdr x))))
         :exec x)))

Theorem: s4veclist-p-of-s4veclist-fix

(defthm s4veclist-p-of-s4veclist-fix
  (b* ((fty::newx (s4veclist-fix$inline x)))
    (s4veclist-p fty::newx))
  :rule-classes :rewrite)

Theorem: s4veclist-fix-when-s4veclist-p

(defthm s4veclist-fix-when-s4veclist-p
  (implies (s4veclist-p x)
           (equal (s4veclist-fix x) x)))

Function: s4veclist-equiv$inline

(defun s4veclist-equiv$inline (x y)
  (declare (xargs :guard (and (s4veclist-p x) (s4veclist-p y))))
  (equal (s4veclist-fix x)
         (s4veclist-fix y)))

Theorem: s4veclist-equiv-is-an-equivalence

(defthm s4veclist-equiv-is-an-equivalence
  (and (booleanp (s4veclist-equiv x y))
       (s4veclist-equiv x x)
       (implies (s4veclist-equiv x y)
                (s4veclist-equiv y x))
       (implies (and (s4veclist-equiv x y)
                     (s4veclist-equiv y z))
                (s4veclist-equiv x z)))
  :rule-classes (:equivalence))

Theorem: s4veclist-equiv-implies-equal-s4veclist-fix-1

(defthm s4veclist-equiv-implies-equal-s4veclist-fix-1
  (implies (s4veclist-equiv x x-equiv)
           (equal (s4veclist-fix x)
                  (s4veclist-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: s4veclist-fix-under-s4veclist-equiv

(defthm s4veclist-fix-under-s4veclist-equiv
  (s4veclist-equiv (s4veclist-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-s4veclist-fix-1-forward-to-s4veclist-equiv

(defthm equal-of-s4veclist-fix-1-forward-to-s4veclist-equiv
  (implies (equal (s4veclist-fix x) y)
           (s4veclist-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-s4veclist-fix-2-forward-to-s4veclist-equiv

(defthm equal-of-s4veclist-fix-2-forward-to-s4veclist-equiv
  (implies (equal x (s4veclist-fix y))
           (s4veclist-equiv x y))
  :rule-classes :forward-chaining)

Theorem: s4veclist-equiv-of-s4veclist-fix-1-forward

(defthm s4veclist-equiv-of-s4veclist-fix-1-forward
  (implies (s4veclist-equiv (s4veclist-fix x) y)
           (s4veclist-equiv x y))
  :rule-classes :forward-chaining)

Theorem: s4veclist-equiv-of-s4veclist-fix-2-forward

(defthm s4veclist-equiv-of-s4veclist-fix-2-forward
  (implies (s4veclist-equiv x (s4veclist-fix y))
           (s4veclist-equiv x y))
  :rule-classes :forward-chaining)

Theorem: car-of-s4veclist-fix-x-under-s4vec-equiv

(defthm car-of-s4veclist-fix-x-under-s4vec-equiv
  (s4vec-equiv (car (s4veclist-fix x))
               (car x)))

Theorem: car-s4veclist-equiv-congruence-on-x-under-s4vec-equiv

(defthm car-s4veclist-equiv-congruence-on-x-under-s4vec-equiv
  (implies (s4veclist-equiv x x-equiv)
           (s4vec-equiv (car x) (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-s4veclist-fix-x-under-s4veclist-equiv

(defthm cdr-of-s4veclist-fix-x-under-s4veclist-equiv
  (s4veclist-equiv (cdr (s4veclist-fix x))
                   (cdr x)))

Theorem: cdr-s4veclist-equiv-congruence-on-x-under-s4veclist-equiv

(defthm cdr-s4veclist-equiv-congruence-on-x-under-s4veclist-equiv
  (implies (s4veclist-equiv x x-equiv)
           (s4veclist-equiv (cdr x) (cdr x-equiv)))
  :rule-classes :congruence)

Theorem: cons-of-s4vec-fix-x-under-s4veclist-equiv

(defthm cons-of-s4vec-fix-x-under-s4veclist-equiv
  (s4veclist-equiv (cons (s4vec-fix x) y)
                   (cons x y)))

Theorem: cons-s4vec-equiv-congruence-on-x-under-s4veclist-equiv

(defthm cons-s4vec-equiv-congruence-on-x-under-s4veclist-equiv
  (implies (s4vec-equiv x x-equiv)
           (s4veclist-equiv (cons x y)
                            (cons x-equiv y)))
  :rule-classes :congruence)

Theorem: cons-of-s4veclist-fix-y-under-s4veclist-equiv

(defthm cons-of-s4veclist-fix-y-under-s4veclist-equiv
  (s4veclist-equiv (cons x (s4veclist-fix y))
                   (cons x y)))

Theorem: cons-s4veclist-equiv-congruence-on-y-under-s4veclist-equiv

(defthm cons-s4veclist-equiv-congruence-on-y-under-s4veclist-equiv
  (implies (s4veclist-equiv y y-equiv)
           (s4veclist-equiv (cons x y)
                            (cons x y-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-s4veclist-fix

(defthm consp-of-s4veclist-fix
  (equal (consp (s4veclist-fix x))
         (consp x)))

Theorem: s4veclist-fix-under-iff

(defthm s4veclist-fix-under-iff
  (iff (s4veclist-fix x) (consp x)))

Theorem: s4veclist-fix-of-cons

(defthm s4veclist-fix-of-cons
  (equal (s4veclist-fix (cons a x))
         (cons (s4vec-fix a) (s4veclist-fix x))))

Theorem: len-of-s4veclist-fix

(defthm len-of-s4veclist-fix
  (equal (len (s4veclist-fix x)) (len x)))

Theorem: s4veclist-fix-of-append

(defthm s4veclist-fix-of-append
  (equal (s4veclist-fix (append std::a std::b))
         (append (s4veclist-fix std::a)
                 (s4veclist-fix std::b))))

Theorem: s4veclist-fix-of-repeat

(defthm s4veclist-fix-of-repeat
  (equal (s4veclist-fix (repeat acl2::n x))
         (repeat acl2::n (s4vec-fix x))))

Theorem: list-equiv-refines-s4veclist-equiv

(defthm list-equiv-refines-s4veclist-equiv
  (implies (list-equiv x y)
           (s4veclist-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-s4veclist-fix

(defthm nth-of-s4veclist-fix
  (equal (nth acl2::n (s4veclist-fix x))
         (if (< (nfix acl2::n) (len x))
             (s4vec-fix (nth acl2::n x))
           nil)))

Theorem: s4veclist-equiv-implies-s4veclist-equiv-append-1

(defthm s4veclist-equiv-implies-s4veclist-equiv-append-1
  (implies (s4veclist-equiv x fty::x-equiv)
           (s4veclist-equiv (append x y)
                            (append fty::x-equiv y)))
  :rule-classes (:congruence))

Theorem: s4veclist-equiv-implies-s4veclist-equiv-append-2

(defthm s4veclist-equiv-implies-s4veclist-equiv-append-2
  (implies (s4veclist-equiv y fty::y-equiv)
           (s4veclist-equiv (append x y)
                            (append x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: s4veclist-equiv-implies-s4veclist-equiv-nthcdr-2

(defthm s4veclist-equiv-implies-s4veclist-equiv-nthcdr-2
  (implies (s4veclist-equiv acl2::l l-equiv)
           (s4veclist-equiv (nthcdr acl2::n acl2::l)
                            (nthcdr acl2::n l-equiv)))
  :rule-classes (:congruence))

Theorem: s4veclist-equiv-implies-s4veclist-equiv-take-2

(defthm s4veclist-equiv-implies-s4veclist-equiv-take-2
  (implies (s4veclist-equiv acl2::l l-equiv)
           (s4veclist-equiv (take acl2::n acl2::l)
                            (take acl2::n l-equiv)))
  :rule-classes (:congruence))