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• Lit-list

Lit-list-fix

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

Signature

(lit-list-fix x) → fty::newx

Arguments

x — Guard (lit-listp x).

Returns

fty::newx — Type (lit-listp fty::newx).

In the logic, we apply lit-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: lit-list-fix$inline

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

Theorem: lit-listp-of-lit-list-fix

(defthm lit-listp-of-lit-list-fix
        (b* ((fty::newx (lit-list-fix$inline x)))
            (lit-listp fty::newx))
        :rule-classes :rewrite)

Theorem: lit-list-fix-when-lit-listp

(defthm lit-list-fix-when-lit-listp
        (implies (lit-listp x)
                 (equal (lit-list-fix x) x)))

Function: lit-list-equiv$inline

(defun lit-list-equiv$inline (acl2::x acl2::y)
       (declare (xargs :guard (and (lit-listp acl2::x)
                                   (lit-listp acl2::y))))
       (equal (lit-list-fix acl2::x)
              (lit-list-fix acl2::y)))

Theorem: lit-list-equiv-is-an-equivalence

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

Theorem: lit-list-equiv-implies-equal-lit-list-fix-1

(defthm lit-list-equiv-implies-equal-lit-list-fix-1
        (implies (lit-list-equiv acl2::x x-equiv)
                 (equal (lit-list-fix acl2::x)
                        (lit-list-fix x-equiv)))
        :rule-classes (:congruence))

Theorem: lit-list-fix-under-lit-list-equiv

(defthm lit-list-fix-under-lit-list-equiv
        (lit-list-equiv (lit-list-fix acl2::x)
                        acl2::x)
        :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-lit-list-fix-1-forward-to-lit-list-equiv

(defthm equal-of-lit-list-fix-1-forward-to-lit-list-equiv
        (implies (equal (lit-list-fix acl2::x) acl2::y)
                 (lit-list-equiv acl2::x acl2::y))
        :rule-classes :forward-chaining)

Theorem: equal-of-lit-list-fix-2-forward-to-lit-list-equiv

(defthm equal-of-lit-list-fix-2-forward-to-lit-list-equiv
        (implies (equal acl2::x (lit-list-fix acl2::y))
                 (lit-list-equiv acl2::x acl2::y))
        :rule-classes :forward-chaining)

Theorem: lit-list-equiv-of-lit-list-fix-1-forward

(defthm lit-list-equiv-of-lit-list-fix-1-forward
        (implies (lit-list-equiv (lit-list-fix acl2::x)
                                 acl2::y)
                 (lit-list-equiv acl2::x acl2::y))
        :rule-classes :forward-chaining)

Theorem: lit-list-equiv-of-lit-list-fix-2-forward

(defthm lit-list-equiv-of-lit-list-fix-2-forward
        (implies (lit-list-equiv acl2::x (lit-list-fix acl2::y))
                 (lit-list-equiv acl2::x acl2::y))
        :rule-classes :forward-chaining)

Theorem: car-of-lit-list-fix-x-under-lit-equiv

(defthm car-of-lit-list-fix-x-under-lit-equiv
        (lit-equiv (car (lit-list-fix acl2::x))
                   (car acl2::x)))

Theorem: car-lit-list-equiv-congruence-on-x-under-lit-equiv

(defthm car-lit-list-equiv-congruence-on-x-under-lit-equiv
        (implies (lit-list-equiv acl2::x x-equiv)
                 (lit-equiv (car acl2::x) (car x-equiv)))
        :rule-classes :congruence)

Theorem: cdr-of-lit-list-fix-x-under-lit-list-equiv

(defthm cdr-of-lit-list-fix-x-under-lit-list-equiv
        (lit-list-equiv (cdr (lit-list-fix acl2::x))
                        (cdr acl2::x)))

Theorem: cdr-lit-list-equiv-congruence-on-x-under-lit-list-equiv

(defthm cdr-lit-list-equiv-congruence-on-x-under-lit-list-equiv
        (implies (lit-list-equiv acl2::x x-equiv)
                 (lit-list-equiv (cdr acl2::x)
                                 (cdr x-equiv)))
        :rule-classes :congruence)

Theorem: cons-of-lit-fix-x-under-lit-list-equiv

(defthm cons-of-lit-fix-x-under-lit-list-equiv
        (lit-list-equiv (cons (lit-fix acl2::x) acl2::y)
                        (cons acl2::x acl2::y)))

Theorem: cons-lit-equiv-congruence-on-x-under-lit-list-equiv

(defthm cons-lit-equiv-congruence-on-x-under-lit-list-equiv
        (implies (lit-equiv acl2::x x-equiv)
                 (lit-list-equiv (cons acl2::x acl2::y)
                                 (cons x-equiv acl2::y)))
        :rule-classes :congruence)

Theorem: cons-of-lit-list-fix-y-under-lit-list-equiv

(defthm cons-of-lit-list-fix-y-under-lit-list-equiv
        (lit-list-equiv (cons acl2::x (lit-list-fix acl2::y))
                        (cons acl2::x acl2::y)))

Theorem: cons-lit-list-equiv-congruence-on-y-under-lit-list-equiv

(defthm cons-lit-list-equiv-congruence-on-y-under-lit-list-equiv
        (implies (lit-list-equiv acl2::y y-equiv)
                 (lit-list-equiv (cons acl2::x acl2::y)
                                 (cons acl2::x y-equiv)))
        :rule-classes :congruence)

Theorem: consp-of-lit-list-fix

(defthm consp-of-lit-list-fix
        (equal (consp (lit-list-fix acl2::x))
               (consp acl2::x)))

Theorem: lit-list-fix-under-iff

(defthm lit-list-fix-under-iff
        (iff (lit-list-fix acl2::x)
             (consp acl2::x)))

Theorem: lit-list-fix-of-cons

(defthm lit-list-fix-of-cons
        (equal (lit-list-fix (cons a x))
               (cons (lit-fix a) (lit-list-fix x))))

Theorem: len-of-lit-list-fix

(defthm len-of-lit-list-fix
        (equal (len (lit-list-fix acl2::x))
               (len acl2::x)))

Theorem: lit-list-fix-of-append

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

Theorem: lit-list-fix-of-repeat

(defthm lit-list-fix-of-repeat
        (equal (lit-list-fix (acl2::repeat acl2::n acl2::x))
               (acl2::repeat acl2::n (lit-fix acl2::x))))

Theorem: list-equiv-refines-lit-list-equiv

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

Theorem: nth-of-lit-list-fix

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

Theorem: lit-list-equiv-implies-lit-list-equiv-append-1

(defthm lit-list-equiv-implies-lit-list-equiv-append-1
        (implies (lit-list-equiv acl2::x fty::x-equiv)
                 (lit-list-equiv (append acl2::x acl2::y)
                                 (append fty::x-equiv acl2::y)))
        :rule-classes (:congruence))

Theorem: lit-list-equiv-implies-lit-list-equiv-append-2

(defthm lit-list-equiv-implies-lit-list-equiv-append-2
        (implies (lit-list-equiv acl2::y fty::y-equiv)
                 (lit-list-equiv (append acl2::x acl2::y)
                                 (append acl2::x fty::y-equiv)))
        :rule-classes (:congruence))

Theorem: lit-list-equiv-implies-lit-list-equiv-nthcdr-2

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

Theorem: lit-list-equiv-implies-lit-list-equiv-take-2

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