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