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