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(in-package "ACL2")
(defun bounded-nat-listp (l b)
(declare (xargs :guard (natp b)))
(if (atom l)
(equal l nil)
(and (natp (car l))
(< (car l) b)
(bounded-nat-listp (cdr l) b))))
(defthm bounded-nat-listp-correctness-1
(implies (bounded-nat-listp l b)
(nat-listp l))
:rule-classes (:rewrite :forward-chaining))
(defthm bounded-nat-listp-correctness-2
(implies (true-listp x)
(equal (bounded-nat-listp (binary-append x y)
b)
(and (bounded-nat-listp x b)
(bounded-nat-listp y b)))))
(defthm bounded-nat-listp-correctness-3
(implies (and (bounded-nat-listp l (+ b 1))
(natp b)
(not (bounded-nat-listp l b)))
(member-equal b l))
:rule-classes :forward-chaining)
(defthm bounded-nat-listp-correctness-4
(implies (bounded-nat-listp l b)
(not (member-equal b l)))
:rule-classes :forward-chaining)
(defthmd bounded-nat-listp-correctness-5
(implies (and (<= x y) (bounded-nat-listp l x))
(bounded-nat-listp l y)))
(defthm bounded-nat-listp-of-make-list-ac
(implies (and (bounded-nat-listp ac b) (natp val) (< val b))
(bounded-nat-listp (make-list-ac n val ac) b)))
(defthm car-of-last-when-bounded-nat-listp
(implies (and (< 0 b) (bounded-nat-listp l b))
(< (car (last l)) b))
:hints (("goal" :induct (bounded-nat-listp l b)))
:rule-classes :linear)
(defun lower-bounded-integer-listp (l b)
(declare (xargs :guard (integerp b)))
(if (atom l)
(equal l nil)
(and (integerp (car l))
(>= (car l) b)
(lower-bounded-integer-listp (cdr l)
b))))
(defthm lower-bounded-integer-listp-correctness-2
(implies (true-listp x)
(equal (lower-bounded-integer-listp (binary-append x y)
b)
(and (lower-bounded-integer-listp x b)
(lower-bounded-integer-listp y b)))))
(defthmd lower-bounded-integer-listp-correctness-5
(implies (and (<= y x) (lower-bounded-integer-listp l x))
(lower-bounded-integer-listp l y)))
(defthm lower-bounded-integer-listp-of-remove
(implies (lower-bounded-integer-listp l b)
(lower-bounded-integer-listp (remove-equal x l) b)))
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