(rw-pairlist-fix x) is a usual ACL2::fty list fixing function.
(rw-pairlist-fix x) → fty::newx
In the logic, we apply rw-pair-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.
Function:
(defun rw-pairlist-fix$inline (x) (declare (xargs :guard (rw-pairlist-p x))) (let ((__function__ 'rw-pairlist-fix)) (declare (ignorable __function__)) (mbe :logic (if (atom x) nil (cons (rw-pair-fix (car x)) (rw-pairlist-fix (cdr x)))) :exec x)))
Theorem:
(defthm rw-pairlist-p-of-rw-pairlist-fix (b* ((fty::newx (rw-pairlist-fix$inline x))) (rw-pairlist-p fty::newx)) :rule-classes :rewrite)
Theorem:
(defthm rw-pairlist-fix-when-rw-pairlist-p (implies (rw-pairlist-p x) (equal (rw-pairlist-fix x) x)))
Function:
(defun rw-pairlist-equiv$inline (x y) (declare (xargs :guard (and (rw-pairlist-p x) (rw-pairlist-p y)))) (equal (rw-pairlist-fix x) (rw-pairlist-fix y)))
Theorem:
(defthm rw-pairlist-equiv-is-an-equivalence (and (booleanp (rw-pairlist-equiv x y)) (rw-pairlist-equiv x x) (implies (rw-pairlist-equiv x y) (rw-pairlist-equiv y x)) (implies (and (rw-pairlist-equiv x y) (rw-pairlist-equiv y z)) (rw-pairlist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm rw-pairlist-equiv-implies-equal-rw-pairlist-fix-1 (implies (rw-pairlist-equiv x x-equiv) (equal (rw-pairlist-fix x) (rw-pairlist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm rw-pairlist-fix-under-rw-pairlist-equiv (rw-pairlist-equiv (rw-pairlist-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-rw-pairlist-fix-1-forward-to-rw-pairlist-equiv (implies (equal (rw-pairlist-fix x) y) (rw-pairlist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-rw-pairlist-fix-2-forward-to-rw-pairlist-equiv (implies (equal x (rw-pairlist-fix y)) (rw-pairlist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm rw-pairlist-equiv-of-rw-pairlist-fix-1-forward (implies (rw-pairlist-equiv (rw-pairlist-fix x) y) (rw-pairlist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm rw-pairlist-equiv-of-rw-pairlist-fix-2-forward (implies (rw-pairlist-equiv x (rw-pairlist-fix y)) (rw-pairlist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm car-of-rw-pairlist-fix-x-under-rw-pair-equiv (rw-pair-equiv (car (rw-pairlist-fix x)) (car x)))
Theorem:
(defthm car-rw-pairlist-equiv-congruence-on-x-under-rw-pair-equiv (implies (rw-pairlist-equiv x x-equiv) (rw-pair-equiv (car x) (car x-equiv))) :rule-classes :congruence)
Theorem:
(defthm cdr-of-rw-pairlist-fix-x-under-rw-pairlist-equiv (rw-pairlist-equiv (cdr (rw-pairlist-fix x)) (cdr x)))
Theorem:
(defthm cdr-rw-pairlist-equiv-congruence-on-x-under-rw-pairlist-equiv (implies (rw-pairlist-equiv x x-equiv) (rw-pairlist-equiv (cdr x) (cdr x-equiv))) :rule-classes :congruence)
Theorem:
(defthm cons-of-rw-pair-fix-x-under-rw-pairlist-equiv (rw-pairlist-equiv (cons (rw-pair-fix x) y) (cons x y)))
Theorem:
(defthm cons-rw-pair-equiv-congruence-on-x-under-rw-pairlist-equiv (implies (rw-pair-equiv x x-equiv) (rw-pairlist-equiv (cons x y) (cons x-equiv y))) :rule-classes :congruence)
Theorem:
(defthm cons-of-rw-pairlist-fix-y-under-rw-pairlist-equiv (rw-pairlist-equiv (cons x (rw-pairlist-fix y)) (cons x y)))
Theorem:
(defthm cons-rw-pairlist-equiv-congruence-on-y-under-rw-pairlist-equiv (implies (rw-pairlist-equiv y y-equiv) (rw-pairlist-equiv (cons x y) (cons x y-equiv))) :rule-classes :congruence)
Theorem:
(defthm consp-of-rw-pairlist-fix (equal (consp (rw-pairlist-fix x)) (consp x)))
Theorem:
(defthm rw-pairlist-fix-under-iff (iff (rw-pairlist-fix x) (consp x)))
Theorem:
(defthm rw-pairlist-fix-of-cons (equal (rw-pairlist-fix (cons a x)) (cons (rw-pair-fix a) (rw-pairlist-fix x))))
Theorem:
(defthm len-of-rw-pairlist-fix (equal (len (rw-pairlist-fix x)) (len x)))
Theorem:
(defthm rw-pairlist-fix-of-append (equal (rw-pairlist-fix (append std::a std::b)) (append (rw-pairlist-fix std::a) (rw-pairlist-fix std::b))))
Theorem:
(defthm rw-pairlist-fix-of-repeat (equal (rw-pairlist-fix (repeat n x)) (repeat n (rw-pair-fix x))))
Theorem:
(defthm list-equiv-refines-rw-pairlist-equiv (implies (list-equiv x y) (rw-pairlist-equiv x y)) :rule-classes :refinement)
Theorem:
(defthm nth-of-rw-pairlist-fix (equal (nth n (rw-pairlist-fix x)) (if (< (nfix n) (len x)) (rw-pair-fix (nth n x)) nil)))
Theorem:
(defthm rw-pairlist-equiv-implies-rw-pairlist-equiv-append-1 (implies (rw-pairlist-equiv x fty::x-equiv) (rw-pairlist-equiv (append x y) (append fty::x-equiv y))) :rule-classes (:congruence))
Theorem:
(defthm rw-pairlist-equiv-implies-rw-pairlist-equiv-append-2 (implies (rw-pairlist-equiv y fty::y-equiv) (rw-pairlist-equiv (append x y) (append x fty::y-equiv))) :rule-classes (:congruence))
Theorem:
(defthm rw-pairlist-equiv-implies-rw-pairlist-equiv-nthcdr-2 (implies (rw-pairlist-equiv l l-equiv) (rw-pairlist-equiv (nthcdr n l) (nthcdr n l-equiv))) :rule-classes (:congruence))
Theorem:
(defthm rw-pairlist-equiv-implies-rw-pairlist-equiv-take-2 (implies (rw-pairlist-equiv l l-equiv) (rw-pairlist-equiv (take n l) (take n l-equiv))) :rule-classes (:congruence))