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• Svar/4vec-alistlist

Svar/4vec-alistlist-fix

(svar/4vec-alistlist-fix x) is a usual fty list fixing function.

Signature

(svar/4vec-alistlist-fix x) → fty::newx

Arguments

x — Guard (svar/4vec-alistlist-p x).

Returns

fty::newx — Type (svar/4vec-alistlist-p fty::newx).

In the logic, we apply svar/4vec-alist-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: svar/4vec-alistlist-fix$inline

(defun svar/4vec-alistlist-fix$inline (x)
  (declare (xargs :guard (svar/4vec-alistlist-p x)))
  (let ((__function__ 'svar/4vec-alistlist-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (svar/4vec-alist-fix (car x))
                 (svar/4vec-alistlist-fix (cdr x))))
         :exec x)))

Theorem: svar/4vec-alistlist-p-of-svar/4vec-alistlist-fix

(defthm svar/4vec-alistlist-p-of-svar/4vec-alistlist-fix
  (b* ((fty::newx (svar/4vec-alistlist-fix$inline x)))
    (svar/4vec-alistlist-p fty::newx))
  :rule-classes :rewrite)

Theorem: svar/4vec-alistlist-fix-when-svar/4vec-alistlist-p

(defthm svar/4vec-alistlist-fix-when-svar/4vec-alistlist-p
  (implies (svar/4vec-alistlist-p x)
           (equal (svar/4vec-alistlist-fix x) x)))

Function: svar/4vec-alistlist-equiv$inline

(defun svar/4vec-alistlist-equiv$inline (x y)
  (declare (xargs :guard (and (svar/4vec-alistlist-p x)
                              (svar/4vec-alistlist-p y))))
  (equal (svar/4vec-alistlist-fix x)
         (svar/4vec-alistlist-fix y)))

Theorem: svar/4vec-alistlist-equiv-is-an-equivalence

(defthm svar/4vec-alistlist-equiv-is-an-equivalence
  (and (booleanp (svar/4vec-alistlist-equiv x y))
       (svar/4vec-alistlist-equiv x x)
       (implies (svar/4vec-alistlist-equiv x y)
                (svar/4vec-alistlist-equiv y x))
       (implies (and (svar/4vec-alistlist-equiv x y)
                     (svar/4vec-alistlist-equiv y z))
                (svar/4vec-alistlist-equiv x z)))
  :rule-classes (:equivalence))

Theorem: svar/4vec-alistlist-equiv-implies-equal-svar/4vec-alistlist-fix-1

(defthm
  svar/4vec-alistlist-equiv-implies-equal-svar/4vec-alistlist-fix-1
  (implies (svar/4vec-alistlist-equiv x x-equiv)
           (equal (svar/4vec-alistlist-fix x)
                  (svar/4vec-alistlist-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: svar/4vec-alistlist-fix-under-svar/4vec-alistlist-equiv

(defthm svar/4vec-alistlist-fix-under-svar/4vec-alistlist-equiv
  (svar/4vec-alistlist-equiv (svar/4vec-alistlist-fix x)
                             x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-svar/4vec-alistlist-fix-1-forward-to-svar/4vec-alistlist-equiv

(defthm
 equal-of-svar/4vec-alistlist-fix-1-forward-to-svar/4vec-alistlist-equiv
 (implies (equal (svar/4vec-alistlist-fix x) y)
          (svar/4vec-alistlist-equiv x y))
 :rule-classes :forward-chaining)

Theorem: equal-of-svar/4vec-alistlist-fix-2-forward-to-svar/4vec-alistlist-equiv

(defthm
 equal-of-svar/4vec-alistlist-fix-2-forward-to-svar/4vec-alistlist-equiv
 (implies (equal x (svar/4vec-alistlist-fix y))
          (svar/4vec-alistlist-equiv x y))
 :rule-classes :forward-chaining)

Theorem: svar/4vec-alistlist-equiv-of-svar/4vec-alistlist-fix-1-forward

(defthm
     svar/4vec-alistlist-equiv-of-svar/4vec-alistlist-fix-1-forward
  (implies (svar/4vec-alistlist-equiv (svar/4vec-alistlist-fix x)
                                      y)
           (svar/4vec-alistlist-equiv x y))
  :rule-classes :forward-chaining)

Theorem: svar/4vec-alistlist-equiv-of-svar/4vec-alistlist-fix-2-forward

(defthm
     svar/4vec-alistlist-equiv-of-svar/4vec-alistlist-fix-2-forward
  (implies (svar/4vec-alistlist-equiv x (svar/4vec-alistlist-fix y))
           (svar/4vec-alistlist-equiv x y))
  :rule-classes :forward-chaining)

Theorem: car-of-svar/4vec-alistlist-fix-x-under-svar/4vec-alist-equiv

(defthm car-of-svar/4vec-alistlist-fix-x-under-svar/4vec-alist-equiv
  (svar/4vec-alist-equiv (car (svar/4vec-alistlist-fix x))
                         (car x)))

Theorem: car-svar/4vec-alistlist-equiv-congruence-on-x-under-svar/4vec-alist-equiv

(defthm
 car-svar/4vec-alistlist-equiv-congruence-on-x-under-svar/4vec-alist-equiv
 (implies (svar/4vec-alistlist-equiv x x-equiv)
          (svar/4vec-alist-equiv (car x)
                                 (car x-equiv)))
 :rule-classes :congruence)

Theorem: cdr-of-svar/4vec-alistlist-fix-x-under-svar/4vec-alistlist-equiv

(defthm
   cdr-of-svar/4vec-alistlist-fix-x-under-svar/4vec-alistlist-equiv
  (svar/4vec-alistlist-equiv (cdr (svar/4vec-alistlist-fix x))
                             (cdr x)))

Theorem: cdr-svar/4vec-alistlist-equiv-congruence-on-x-under-svar/4vec-alistlist-equiv

(defthm
 cdr-svar/4vec-alistlist-equiv-congruence-on-x-under-svar/4vec-alistlist-equiv
 (implies (svar/4vec-alistlist-equiv x x-equiv)
          (svar/4vec-alistlist-equiv (cdr x)
                                     (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-svar/4vec-alist-fix-x-under-svar/4vec-alistlist-equiv

(defthm
      cons-of-svar/4vec-alist-fix-x-under-svar/4vec-alistlist-equiv
  (svar/4vec-alistlist-equiv (cons (svar/4vec-alist-fix x) y)
                             (cons x y)))

Theorem: cons-svar/4vec-alist-equiv-congruence-on-x-under-svar/4vec-alistlist-equiv

(defthm
 cons-svar/4vec-alist-equiv-congruence-on-x-under-svar/4vec-alistlist-equiv
 (implies (svar/4vec-alist-equiv x x-equiv)
          (svar/4vec-alistlist-equiv (cons x y)
                                     (cons x-equiv y)))
 :rule-classes :congruence)

Theorem: cons-of-svar/4vec-alistlist-fix-y-under-svar/4vec-alistlist-equiv

(defthm
  cons-of-svar/4vec-alistlist-fix-y-under-svar/4vec-alistlist-equiv
  (svar/4vec-alistlist-equiv (cons x (svar/4vec-alistlist-fix y))
                             (cons x y)))

Theorem: cons-svar/4vec-alistlist-equiv-congruence-on-y-under-svar/4vec-alistlist-equiv

(defthm
 cons-svar/4vec-alistlist-equiv-congruence-on-y-under-svar/4vec-alistlist-equiv
 (implies (svar/4vec-alistlist-equiv y y-equiv)
          (svar/4vec-alistlist-equiv (cons x y)
                                     (cons x y-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-svar/4vec-alistlist-fix

(defthm consp-of-svar/4vec-alistlist-fix
  (equal (consp (svar/4vec-alistlist-fix x))
         (consp x)))

Theorem: svar/4vec-alistlist-fix-under-iff

(defthm svar/4vec-alistlist-fix-under-iff
  (iff (svar/4vec-alistlist-fix x)
       (consp x)))

Theorem: svar/4vec-alistlist-fix-of-cons

(defthm svar/4vec-alistlist-fix-of-cons
  (equal (svar/4vec-alistlist-fix (cons a x))
         (cons (svar/4vec-alist-fix a)
               (svar/4vec-alistlist-fix x))))

Theorem: len-of-svar/4vec-alistlist-fix

(defthm len-of-svar/4vec-alistlist-fix
  (equal (len (svar/4vec-alistlist-fix x))
         (len x)))

Theorem: svar/4vec-alistlist-fix-of-append

(defthm svar/4vec-alistlist-fix-of-append
  (equal (svar/4vec-alistlist-fix (append std::a std::b))
         (append (svar/4vec-alistlist-fix std::a)
                 (svar/4vec-alistlist-fix std::b))))

Theorem: svar/4vec-alistlist-fix-of-repeat

(defthm svar/4vec-alistlist-fix-of-repeat
  (equal (svar/4vec-alistlist-fix (repeat acl2::n x))
         (repeat acl2::n (svar/4vec-alist-fix x))))

Theorem: list-equiv-refines-svar/4vec-alistlist-equiv

(defthm list-equiv-refines-svar/4vec-alistlist-equiv
  (implies (list-equiv x y)
           (svar/4vec-alistlist-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-svar/4vec-alistlist-fix

(defthm nth-of-svar/4vec-alistlist-fix
  (equal (nth acl2::n (svar/4vec-alistlist-fix x))
         (if (< (nfix acl2::n) (len x))
             (svar/4vec-alist-fix (nth acl2::n x))
           nil)))

Theorem: svar/4vec-alistlist-equiv-implies-svar/4vec-alistlist-equiv-append-1

(defthm
 svar/4vec-alistlist-equiv-implies-svar/4vec-alistlist-equiv-append-1
 (implies (svar/4vec-alistlist-equiv x fty::x-equiv)
          (svar/4vec-alistlist-equiv (append x y)
                                     (append fty::x-equiv y)))
 :rule-classes (:congruence))

Theorem: svar/4vec-alistlist-equiv-implies-svar/4vec-alistlist-equiv-append-2

(defthm
 svar/4vec-alistlist-equiv-implies-svar/4vec-alistlist-equiv-append-2
 (implies (svar/4vec-alistlist-equiv y fty::y-equiv)
          (svar/4vec-alistlist-equiv (append x y)
                                     (append x fty::y-equiv)))
 :rule-classes (:congruence))

Theorem: svar/4vec-alistlist-equiv-implies-svar/4vec-alistlist-equiv-nthcdr-2

(defthm
 svar/4vec-alistlist-equiv-implies-svar/4vec-alistlist-equiv-nthcdr-2
 (implies (svar/4vec-alistlist-equiv acl2::l l-equiv)
          (svar/4vec-alistlist-equiv (nthcdr acl2::n acl2::l)
                                     (nthcdr acl2::n l-equiv)))
 :rule-classes (:congruence))

Theorem: svar/4vec-alistlist-equiv-implies-svar/4vec-alistlist-equiv-take-2

(defthm
 svar/4vec-alistlist-equiv-implies-svar/4vec-alistlist-equiv-take-2
 (implies (svar/4vec-alistlist-equiv acl2::l l-equiv)
          (svar/4vec-alistlist-equiv (take acl2::n acl2::l)
                                     (take acl2::n l-equiv)))
 :rule-classes (:congruence))