Function:
(defun function-option-name-p (option-name) (declare (xargs :guard t)) (let ((acl2::__function__ 'function-option-name-p)) (declare (ignorable acl2::__function__)) (if (member-equal option-name *function-option-names*) t nil)))
Theorem:
(defthm booleanp-of-function-option-name-p (b* ((syntax-good? (function-option-name-p option-name))) (booleanp syntax-good?)) :rule-classes :rewrite)
Function:
(defun function-option-name-fix (option-name) (declare (xargs :guard (function-option-name-p option-name))) (let ((acl2::__function__ 'function-option-name-fix)) (declare (ignorable acl2::__function__)) (mbe :logic (if (function-option-name-p option-name) option-name ':formals) :exec option-name)))
Theorem:
(defthm function-option-name-p-of-function-option-name-fix (b* ((fixed-option-name (function-option-name-fix option-name))) (function-option-name-p fixed-option-name)) :rule-classes :rewrite)
Function:
(defun function-option-name-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (function-option-name-p acl2::x) (function-option-name-p acl2::y)))) (equal (function-option-name-fix acl2::x) (function-option-name-fix acl2::y)))
Theorem:
(defthm function-option-name-equiv-is-an-equivalence (and (booleanp (function-option-name-equiv x y)) (function-option-name-equiv x x) (implies (function-option-name-equiv x y) (function-option-name-equiv y x)) (implies (and (function-option-name-equiv x y) (function-option-name-equiv y z)) (function-option-name-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm function-option-name-equiv-implies-equal-function-option-name-fix-1 (implies (function-option-name-equiv acl2::x x-equiv) (equal (function-option-name-fix acl2::x) (function-option-name-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm function-option-name-fix-under-function-option-name-equiv (function-option-name-equiv (function-option-name-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-function-option-name-fix-1-forward-to-function-option-name-equiv (implies (equal (function-option-name-fix acl2::x) acl2::y) (function-option-name-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-function-option-name-fix-2-forward-to-function-option-name-equiv (implies (equal acl2::x (function-option-name-fix acl2::y)) (function-option-name-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm function-option-name-equiv-of-function-option-name-fix-1-forward (implies (function-option-name-equiv (function-option-name-fix acl2::x) acl2::y) (function-option-name-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm function-option-name-equiv-of-function-option-name-fix-2-forward (implies (function-option-name-equiv acl2::x (function-option-name-fix acl2::y)) (function-option-name-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Function:
(defun function-option-name-lst-p (x) (declare (xargs :guard t)) (let ((acl2::__function__ 'function-option-name-lst-p)) (declare (ignorable acl2::__function__)) (if (atom x) (eq x nil) (and (function-option-name-p (car x)) (function-option-name-lst-p (cdr x))))))
Theorem:
(defthm function-option-name-lst-p-of-cons (equal (function-option-name-lst-p (cons acl2::a acl2::x)) (and (function-option-name-p acl2::a) (function-option-name-lst-p acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-cdr-when-function-option-name-lst-p (implies (function-option-name-lst-p (double-rewrite acl2::x)) (function-option-name-lst-p (cdr acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-when-not-consp (implies (not (consp acl2::x)) (equal (function-option-name-lst-p acl2::x) (not acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-p-of-car-when-function-option-name-lst-p (implies (function-option-name-lst-p acl2::x) (iff (function-option-name-p (car acl2::x)) (or (consp acl2::x) (function-option-name-p nil)))) :rule-classes ((:rewrite)))
Theorem:
(defthm true-listp-when-function-option-name-lst-p-compound-recognizer (implies (function-option-name-lst-p acl2::x) (true-listp acl2::x)) :rule-classes :compound-recognizer)
Theorem:
(defthm function-option-name-lst-p-of-list-fix (implies (function-option-name-lst-p acl2::x) (function-option-name-lst-p (acl2::list-fix acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-sfix (iff (function-option-name-lst-p (set::sfix acl2::x)) (or (function-option-name-lst-p acl2::x) (not (set::setp acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-insert (iff (function-option-name-lst-p (set::insert acl2::a acl2::x)) (and (function-option-name-lst-p (set::sfix acl2::x)) (function-option-name-p acl2::a))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-delete (implies (function-option-name-lst-p acl2::x) (function-option-name-lst-p (set::delete acl2::k acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-mergesort (iff (function-option-name-lst-p (set::mergesort acl2::x)) (function-option-name-lst-p (acl2::list-fix acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-union (iff (function-option-name-lst-p (set::union acl2::x acl2::y)) (and (function-option-name-lst-p (set::sfix acl2::x)) (function-option-name-lst-p (set::sfix acl2::y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-intersect-1 (implies (function-option-name-lst-p acl2::x) (function-option-name-lst-p (set::intersect acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-intersect-2 (implies (function-option-name-lst-p acl2::y) (function-option-name-lst-p (set::intersect acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-difference (implies (function-option-name-lst-p acl2::x) (function-option-name-lst-p (set::difference acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-duplicated-members (implies (function-option-name-lst-p acl2::x) (function-option-name-lst-p (acl2::duplicated-members acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-rev (equal (function-option-name-lst-p (acl2::rev acl2::x)) (function-option-name-lst-p (acl2::list-fix acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-append (equal (function-option-name-lst-p (append acl2::a acl2::b)) (and (function-option-name-lst-p (acl2::list-fix acl2::a)) (function-option-name-lst-p acl2::b))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-rcons (iff (function-option-name-lst-p (acl2::rcons acl2::a acl2::x)) (and (function-option-name-p acl2::a) (function-option-name-lst-p (acl2::list-fix acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-p-when-member-equal-of-function-option-name-lst-p (and (implies (and (member-equal acl2::a acl2::x) (function-option-name-lst-p acl2::x)) (function-option-name-p acl2::a)) (implies (and (function-option-name-lst-p acl2::x) (member-equal acl2::a acl2::x)) (function-option-name-p acl2::a))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-when-subsetp-equal (and (implies (and (subsetp-equal acl2::x acl2::y) (function-option-name-lst-p acl2::y)) (equal (function-option-name-lst-p acl2::x) (true-listp acl2::x))) (implies (and (function-option-name-lst-p acl2::y) (subsetp-equal acl2::x acl2::y)) (equal (function-option-name-lst-p acl2::x) (true-listp acl2::x)))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-set-difference-equal (implies (function-option-name-lst-p acl2::x) (function-option-name-lst-p (set-difference-equal acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-intersection-equal-1 (implies (function-option-name-lst-p (double-rewrite acl2::x)) (function-option-name-lst-p (intersection-equal acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-intersection-equal-2 (implies (function-option-name-lst-p (double-rewrite acl2::y)) (function-option-name-lst-p (intersection-equal acl2::x acl2::y))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-union-equal (equal (function-option-name-lst-p (union-equal acl2::x acl2::y)) (and (function-option-name-lst-p (acl2::list-fix acl2::x)) (function-option-name-lst-p (double-rewrite acl2::y)))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-take (implies (function-option-name-lst-p (double-rewrite acl2::x)) (iff (function-option-name-lst-p (take acl2::n acl2::x)) (or (function-option-name-p nil) (<= (nfix acl2::n) (len acl2::x))))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-repeat (iff (function-option-name-lst-p (acl2::repeat acl2::n acl2::x)) (or (function-option-name-p acl2::x) (zp acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-p-of-nth-when-function-option-name-lst-p (implies (and (function-option-name-lst-p acl2::x) (< (nfix acl2::n) (len acl2::x))) (function-option-name-p (nth acl2::n acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-update-nth (implies (function-option-name-lst-p (double-rewrite acl2::x)) (iff (function-option-name-lst-p (update-nth acl2::n acl2::y acl2::x)) (and (function-option-name-p acl2::y) (or (<= (nfix acl2::n) (len acl2::x)) (function-option-name-p nil))))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-butlast (implies (function-option-name-lst-p (double-rewrite acl2::x)) (function-option-name-lst-p (butlast acl2::x acl2::n))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-nthcdr (implies (function-option-name-lst-p (double-rewrite acl2::x)) (function-option-name-lst-p (nthcdr acl2::n acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-last (implies (function-option-name-lst-p (double-rewrite acl2::x)) (function-option-name-lst-p (last acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-remove (implies (function-option-name-lst-p acl2::x) (function-option-name-lst-p (remove acl2::a acl2::x))) :rule-classes ((:rewrite)))
Theorem:
(defthm function-option-name-lst-p-of-revappend (equal (function-option-name-lst-p (revappend acl2::x acl2::y)) (and (function-option-name-lst-p (acl2::list-fix acl2::x)) (function-option-name-lst-p acl2::y))) :rule-classes ((:rewrite)))
Function:
(defun function-option-name-lst-fix$inline (x) (declare (xargs :guard (function-option-name-lst-p x))) (let ((acl2::__function__ 'function-option-name-lst-fix)) (declare (ignorable acl2::__function__)) (mbe :logic (if (atom x) nil (cons (function-option-name-fix (car x)) (function-option-name-lst-fix (cdr x)))) :exec x)))
Theorem:
(defthm function-option-name-lst-p-of-function-option-name-lst-fix (b* ((fty::newx (function-option-name-lst-fix$inline x))) (function-option-name-lst-p fty::newx)) :rule-classes :rewrite)
Theorem:
(defthm function-option-name-lst-fix-when-function-option-name-lst-p (implies (function-option-name-lst-p x) (equal (function-option-name-lst-fix x) x)))
Function:
(defun function-option-name-lst-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (function-option-name-lst-p acl2::x) (function-option-name-lst-p acl2::y)))) (equal (function-option-name-lst-fix acl2::x) (function-option-name-lst-fix acl2::y)))
Theorem:
(defthm function-option-name-lst-equiv-is-an-equivalence (and (booleanp (function-option-name-lst-equiv x y)) (function-option-name-lst-equiv x x) (implies (function-option-name-lst-equiv x y) (function-option-name-lst-equiv y x)) (implies (and (function-option-name-lst-equiv x y) (function-option-name-lst-equiv y z)) (function-option-name-lst-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm function-option-name-lst-equiv-implies-equal-function-option-name-lst-fix-1 (implies (function-option-name-lst-equiv acl2::x x-equiv) (equal (function-option-name-lst-fix acl2::x) (function-option-name-lst-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm function-option-name-lst-fix-under-function-option-name-lst-equiv (function-option-name-lst-equiv (function-option-name-lst-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-function-option-name-lst-fix-1-forward-to-function-option-name-lst-equiv (implies (equal (function-option-name-lst-fix acl2::x) acl2::y) (function-option-name-lst-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-function-option-name-lst-fix-2-forward-to-function-option-name-lst-equiv (implies (equal acl2::x (function-option-name-lst-fix acl2::y)) (function-option-name-lst-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm function-option-name-lst-equiv-of-function-option-name-lst-fix-1-forward (implies (function-option-name-lst-equiv (function-option-name-lst-fix acl2::x) acl2::y) (function-option-name-lst-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm function-option-name-lst-equiv-of-function-option-name-lst-fix-2-forward (implies (function-option-name-lst-equiv acl2::x (function-option-name-lst-fix acl2::y)) (function-option-name-lst-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm car-of-function-option-name-lst-fix-x-under-function-option-name-equiv (function-option-name-equiv (car (function-option-name-lst-fix acl2::x)) (car acl2::x)))
Theorem:
(defthm car-function-option-name-lst-equiv-congruence-on-x-under-function-option-name-equiv (implies (function-option-name-lst-equiv acl2::x x-equiv) (function-option-name-equiv (car acl2::x) (car x-equiv))) :rule-classes :congruence)
Theorem:
(defthm cdr-of-function-option-name-lst-fix-x-under-function-option-name-lst-equiv (function-option-name-lst-equiv (cdr (function-option-name-lst-fix acl2::x)) (cdr acl2::x)))
Theorem:
(defthm cdr-function-option-name-lst-equiv-congruence-on-x-under-function-option-name-lst-equiv (implies (function-option-name-lst-equiv acl2::x x-equiv) (function-option-name-lst-equiv (cdr acl2::x) (cdr x-equiv))) :rule-classes :congruence)
Theorem:
(defthm cons-of-function-option-name-fix-x-under-function-option-name-lst-equiv (function-option-name-lst-equiv (cons (function-option-name-fix acl2::x) acl2::y) (cons acl2::x acl2::y)))
Theorem:
(defthm cons-function-option-name-equiv-congruence-on-x-under-function-option-name-lst-equiv (implies (function-option-name-equiv acl2::x x-equiv) (function-option-name-lst-equiv (cons acl2::x acl2::y) (cons x-equiv acl2::y))) :rule-classes :congruence)
Theorem:
(defthm cons-of-function-option-name-lst-fix-y-under-function-option-name-lst-equiv (function-option-name-lst-equiv (cons acl2::x (function-option-name-lst-fix acl2::y)) (cons acl2::x acl2::y)))
Theorem:
(defthm cons-function-option-name-lst-equiv-congruence-on-y-under-function-option-name-lst-equiv (implies (function-option-name-lst-equiv acl2::y y-equiv) (function-option-name-lst-equiv (cons acl2::x acl2::y) (cons acl2::x y-equiv))) :rule-classes :congruence)
Theorem:
(defthm consp-of-function-option-name-lst-fix (equal (consp (function-option-name-lst-fix acl2::x)) (consp acl2::x)))
Theorem:
(defthm function-option-name-lst-fix-under-iff (iff (function-option-name-lst-fix acl2::x) (consp acl2::x)))
Theorem:
(defthm function-option-name-lst-fix-of-cons (equal (function-option-name-lst-fix (cons a x)) (cons (function-option-name-fix a) (function-option-name-lst-fix x))))
Theorem:
(defthm len-of-function-option-name-lst-fix (equal (len (function-option-name-lst-fix acl2::x)) (len acl2::x)))
Theorem:
(defthm function-option-name-lst-fix-of-append (equal (function-option-name-lst-fix (append std::a std::b)) (append (function-option-name-lst-fix std::a) (function-option-name-lst-fix std::b))))
Theorem:
(defthm function-option-name-lst-fix-of-repeat (equal (function-option-name-lst-fix (acl2::repeat acl2::n acl2::x)) (acl2::repeat acl2::n (function-option-name-fix acl2::x))))
Theorem:
(defthm list-equiv-refines-function-option-name-lst-equiv (implies (acl2::list-equiv acl2::x acl2::y) (function-option-name-lst-equiv acl2::x acl2::y)) :rule-classes :refinement)
Theorem:
(defthm nth-of-function-option-name-lst-fix (equal (nth acl2::n (function-option-name-lst-fix acl2::x)) (if (< (nfix acl2::n) (len acl2::x)) (function-option-name-fix (nth acl2::n acl2::x)) nil)))
Theorem:
(defthm function-option-name-lst-equiv-implies-function-option-name-lst-equiv-append-1 (implies (function-option-name-lst-equiv acl2::x fty::x-equiv) (function-option-name-lst-equiv (append acl2::x acl2::y) (append fty::x-equiv acl2::y))) :rule-classes (:congruence))
Theorem:
(defthm function-option-name-lst-equiv-implies-function-option-name-lst-equiv-append-2 (implies (function-option-name-lst-equiv acl2::y fty::y-equiv) (function-option-name-lst-equiv (append acl2::x acl2::y) (append acl2::x fty::y-equiv))) :rule-classes (:congruence))
Theorem:
(defthm function-option-name-lst-equiv-implies-function-option-name-lst-equiv-nthcdr-2 (implies (function-option-name-lst-equiv acl2::l l-equiv) (function-option-name-lst-equiv (nthcdr acl2::n acl2::l) (nthcdr acl2::n l-equiv))) :rule-classes (:congruence))
Theorem:
(defthm function-option-name-lst-equiv-implies-function-option-name-lst-equiv-take-2 (implies (function-option-name-lst-equiv acl2::l l-equiv) (function-option-name-lst-equiv (take acl2::n acl2::l) (take acl2::n l-equiv))) :rule-classes (:congruence))
Theorem:
(defthm function-option-name-fix-preserves-member (implies (member x used :test 'equal) (member (function-option-name-fix x) (function-option-name-lst-fix used) :test 'equal)))
Theorem:
(defthm function-option-name-lst-fix-preserves-subsetp (implies (subsetp used-1 used-2 :test 'equal) (subsetp (function-option-name-lst-fix used-1) (function-option-name-lst-fix used-2) :test 'equal)))
Theorem:
(defthm function-option-name-lst-fix-preserves-set-equiv (implies (acl2::set-equiv used-1 used-2) (acl2::set-equiv (function-option-name-lst-fix used-1) (function-option-name-lst-fix used-2))) :rule-classes (:congruence))
Theorem:
(defthm function-option-name-lst-p-and-member (implies (and (member x used) (not (function-option-name-p x))) (not (function-option-name-lst-p used))))
Theorem:
(defthm function-option-name-lst-p--monotonicity (implies (and (equal (true-listp used-1) (true-listp used-2)) (subsetp used-1 used-2 :test 'equal) (function-option-name-lst-p used-2)) (function-option-name-lst-p used-1)))
Theorem:
(defthm function-option-name-lst-p--congruence (implies (true-set-equiv used-1 used-2) (equal (function-option-name-lst-p used-1) (function-option-name-lst-p used-2))) :rule-classes (:congruence))