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• Svarlist

Svarlist-fix

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

Signature

(svarlist-fix x) → fty::newx

Arguments

x — Guard (svarlist-p x).

Returns

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

In the logic, we apply svar-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: svarlist-fix$inline

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

Theorem: svarlist-p-of-svarlist-fix

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

Theorem: svarlist-fix-when-svarlist-p

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

Function: svarlist-equiv$inline

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

Theorem: svarlist-equiv-is-an-equivalence

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

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

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

Theorem: svarlist-fix-under-svarlist-equiv

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

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

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

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

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

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

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

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

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

Theorem: car-of-svarlist-fix-x-under-svar-equiv

(defthm car-of-svarlist-fix-x-under-svar-equiv
        (svar-equiv (car (svarlist-fix x))
                    (car x)))

Theorem: car-svarlist-equiv-congruence-on-x-under-svar-equiv

(defthm car-svarlist-equiv-congruence-on-x-under-svar-equiv
        (implies (svarlist-equiv x x-equiv)
                 (svar-equiv (car x) (car x-equiv)))
        :rule-classes :congruence)

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

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

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

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

Theorem: cons-of-svar-fix-x-under-svarlist-equiv

(defthm cons-of-svar-fix-x-under-svarlist-equiv
        (svarlist-equiv (cons (svar-fix x) y)
                        (cons x y)))

Theorem: cons-svar-equiv-congruence-on-x-under-svarlist-equiv

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

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

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

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

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

Theorem: consp-of-svarlist-fix

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

Theorem: svarlist-fix-under-iff

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

Theorem: svarlist-fix-of-cons

(defthm svarlist-fix-of-cons
        (equal (svarlist-fix (cons a x))
               (cons (svar-fix a) (svarlist-fix x))))

Theorem: len-of-svarlist-fix

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

Theorem: svarlist-fix-of-append

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

Theorem: svarlist-fix-of-repeat

(defthm svarlist-fix-of-repeat
        (equal (svarlist-fix (repeat acl2::n x))
               (repeat acl2::n (svar-fix x))))

Theorem: list-equiv-refines-svarlist-equiv

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

Theorem: nth-of-svarlist-fix

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

Theorem: svarlist-equiv-implies-svarlist-equiv-append-1

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

Theorem: svarlist-equiv-implies-svarlist-equiv-append-2

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

Theorem: svarlist-equiv-implies-svarlist-equiv-nthcdr-2

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

Theorem: svarlist-equiv-implies-svarlist-equiv-take-2

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