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