(calist-fix x) is an ACL2::fty alist fixing function that follows the fix-keys strategy.
Note that in the execution this is just an inline identity function.
Function:
(defun calist-fix$inline (x) (declare (xargs :guard (calistp x))) (let ((__function__ 'calist-fix)) (declare (ignorable __function__)) (mbe :logic (if (atom x) nil (let ((rest (calist-fix (cdr x)))) (if (and (consp (car x))) (let ((fty::first-key (caar x)) (fty::first-val (bfix (cdar x)))) (if (hons-assoc-equal fty::first-key rest) rest (cons (cons fty::first-key fty::first-val) rest))) rest))) :exec x)))
Theorem:
(defthm calistp-of-calist-fix (b* ((fty::newx (calist-fix$inline x))) (calistp fty::newx)) :rule-classes :rewrite)
Theorem:
(defthm calist-fix-when-calistp (implies (calistp x) (equal (calist-fix x) x)))
Function:
(defun calist-equiv$inline (x y) (declare (xargs :guard (and (calistp x) (calistp y)))) (equal (calist-fix x) (calist-fix y)))
Theorem:
(defthm calist-equiv-is-an-equivalence (and (booleanp (calist-equiv x y)) (calist-equiv x x) (implies (calist-equiv x y) (calist-equiv y x)) (implies (and (calist-equiv x y) (calist-equiv y z)) (calist-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm calist-equiv-implies-equal-calist-fix-1 (implies (calist-equiv x x-equiv) (equal (calist-fix x) (calist-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm calist-fix-under-calist-equiv (calist-equiv (calist-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-calist-fix-1-forward-to-calist-equiv (implies (equal (calist-fix x) y) (calist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-calist-fix-2-forward-to-calist-equiv (implies (equal x (calist-fix y)) (calist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm calist-equiv-of-calist-fix-1-forward (implies (calist-equiv (calist-fix x) y) (calist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm calist-equiv-of-calist-fix-2-forward (implies (calist-equiv x (calist-fix y)) (calist-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm cons-of-bfix-v-under-calist-equiv (calist-equiv (cons (cons k (bfix v)) x) (cons (cons k v) x)))
Theorem:
(defthm cons-bit-equiv-congruence-on-v-under-calist-equiv (implies (acl2::bit-equiv v v-equiv) (calist-equiv (cons (cons k v) x) (cons (cons k v-equiv) x))) :rule-classes :congruence)
Theorem:
(defthm cons-of-calist-fix-y-under-calist-equiv (calist-equiv (cons x (calist-fix y)) (cons x y)))
Theorem:
(defthm cons-calist-equiv-congruence-on-y-under-calist-equiv (implies (calist-equiv y y-equiv) (calist-equiv (cons x y) (cons x y-equiv))) :rule-classes :congruence)
Theorem:
(defthm calist-fix-of-acons (equal (calist-fix (cons (cons a b) x)) (let ((rest (calist-fix x))) (if (and) (let ((fty::first-key a) (fty::first-val (bfix b))) (if (hons-assoc-equal fty::first-key rest) rest (cons (cons fty::first-key fty::first-val) rest))) rest))))
Theorem:
(defthm consp-car-of-calist-fix (equal (consp (car (calist-fix x))) (consp (calist-fix x))))