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