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(in-package "ACL2")
; find-n-free-blocks.lisp Mihir Mehta
(local (include-book "../file-system-lemmas"))
(include-book "bounded-nat-listp")
(defthm mv-nth-replacement
(equal (mv-nth n (cons a b))
(if (zp n) a (mv-nth (- n 1) b))))
(in-theory (disable mv-nth))
(defun count-free-blocks (alv)
(declare (xargs :guard (and (boolean-listp alv))))
(if (atom alv)
0
(if (car alv)
(count-free-blocks (cdr alv))
(+ (count-free-blocks (cdr alv)) 1))))
(defthm count-free-blocks-correctness-1
(equal (count-free-blocks (binary-append x y))
(+ (count-free-blocks x)
(count-free-blocks y))))
(defthm count-free-blocks-correctness-2
(equal (count-free-blocks (revappend x y))
(+ (count-free-blocks x)
(count-free-blocks y))))
(defthm count-free-blocks-correctness-3
(implies (and (nth key l) (not val))
(equal (count-free-blocks (update-nth key val l))
(+ 1 (count-free-blocks l)))))
(defund find-n-free-blocks-helper (alv n start)
(declare (xargs :guard (and (boolean-listp alv)
(natp n)
(natp start))))
(if (or (atom alv) (zp n))
nil
(if (car alv)
;; this block is taken
(find-n-free-blocks-helper (cdr alv) n (+ start 1))
;; this block isn't taken
(cons start (find-n-free-blocks-helper (cdr alv) (- n 1) (+ start 1))))))
(defthmd find-n-free-blocks-helper-correctness-1
(<= (len (find-n-free-blocks-helper alv n start))
(nfix n))
:rule-classes (:rewrite :linear)
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))))
(defthmd find-n-free-blocks-helper-correctness-2
(equal (len (find-n-free-blocks-helper alv n start))
(min (count-free-blocks alv) (nfix n)))
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))))
(defthmd find-n-free-blocks-helper-correctness-3
(implies (natp start)
(nat-listp (find-n-free-blocks-helper alv n start)))
:rule-classes (:rewrite :type-prescription)
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))))
(defthm
find-n-free-blocks-helper-correctness-4
(implies (> start m)
(not (member-equal m
(find-n-free-blocks-helper alv n start))))
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))))
(defthm
find-n-free-blocks-helper-correctness-5
(implies (and (natp start)
(equal (nth (- m start) alv) t))
(not (member-equal m
(find-n-free-blocks-helper alv n start))))
:hints
(("goal" :in-theory (enable find-n-free-blocks-helper
find-n-free-blocks-helper-correctness-3))))
(defthmd find-n-free-blocks-helper-correctness-6
(no-duplicatesp-equal (find-n-free-blocks-helper alv n start))
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))))
(defthmd find-n-free-blocks-helper-correctness-7
(implies (natp start)
(bounded-nat-listp (find-n-free-blocks-helper alv n start)
(+ start (len alv))))
:hints (("goal" :in-theory (enable find-n-free-blocks-helper))))
(defund find-n-free-blocks (alv n)
(declare (xargs :guard (and (boolean-listp alv)
(natp n))))
(find-n-free-blocks-helper alv n 0))
;; Here are some examples showing how this works.
;; ACL2 !>(find-n-free-blocks (list t nil t) 1)
;; (1)
;; ACL2 !>(find-n-free-blocks (list t nil t) 2)
;; (1)
;; ACL2 !>(find-n-free-blocks (list t nil nil) 2)
;; (1 2)
(defthm
find-n-free-blocks-correctness-1
(<= (len (find-n-free-blocks alv n))
(nfix n))
:rule-classes (:rewrite :linear)
:hints
(("goal" :in-theory (enable find-n-free-blocks
find-n-free-blocks-helper-correctness-1))))
(defthm
find-n-free-blocks-correctness-2
(equal (len (find-n-free-blocks alv n))
(min (count-free-blocks alv) (nfix n)))
:hints
(("goal" :in-theory (enable find-n-free-blocks
find-n-free-blocks-helper-correctness-2))))
(defthm
find-n-free-blocks-correctness-3
(nat-listp (find-n-free-blocks alv n))
:rule-classes (:rewrite :type-prescription)
:hints
(("goal" :in-theory (enable find-n-free-blocks
find-n-free-blocks-helper-correctness-3))))
(defthm
find-n-free-blocks-correctness-5
(implies (equal (nth m alv) t)
(not (member-equal m (find-n-free-blocks alv n))))
:hints
(("goal"
:in-theory (e/d (find-n-free-blocks)
((:rewrite find-n-free-blocks-helper-correctness-5)))
:use (:instance (:rewrite find-n-free-blocks-helper-correctness-5)
(start 0)))))
(defthm
find-n-free-blocks-correctness-6
(no-duplicatesp-equal (find-n-free-blocks alv n))
:hints
(("goal" :in-theory (enable find-n-free-blocks
find-n-free-blocks-helper-correctness-6))))
(defthm
find-n-free-blocks-correctness-7
(implies (equal m (len alv))
(bounded-nat-listp (find-n-free-blocks alv n)
m))
:hints (("goal" :in-theory (enable find-n-free-blocks)
:use (:instance find-n-free-blocks-helper-correctness-7
(start 0)))))
(defun count-free-blocks-alt (alv n)
(declare (xargs :guard (and (natp n) (boolean-listp alv))))
(if (zp n)
0
(+ (if (nth (- n 1) alv) 0 1)
(count-free-blocks-alt alv (- n 1)))))
(defthm
count-free-blocks-alt-correctness-1
(implies (and (boolean-listp alv)
(boolean-listp ac)
(natp n)
(<= n (len alv)))
(equal (count-free-blocks-alt (revappend ac alv)
(+ n (len ac)))
(count-free-blocks (first-n-ac n alv ac))))
:hints (("goal" :induct (first-n-ac n alv ac))))
;; Might be useful at some point.
(defthm
count-free-blocks-alt-correctness-lemma-1
(equal (count-free-blocks (true-list-fix alv))
(count-free-blocks alv))
:hints (("Goal" :in-theory (enable true-list-fix))))
(defthm
count-free-blocks-alt-correctness-2
(implies (and (boolean-listp alv)
(equal n (len alv)))
(equal (count-free-blocks-alt alv n)
(count-free-blocks alv)))
:hints (("goal" :in-theory (disable count-free-blocks-alt-correctness-1)
:use (:instance count-free-blocks-alt-correctness-1
(ac nil)))))
(defun count-before-n (l b)
(declare (xargs :guard (and (natp b) (nat-listp l))))
(if (atom l) 0 (+ (if (< (car l) b) 1 0) (count-before-n (cdr l) b))))
(defthm count-before-n-correctness-1
(<= (count-before-n l b) (len l))
:rule-classes (:linear))
(defthm count-before-n-correctness-2
(implies (nat-listp l)
(iff (equal (count-before-n l b) (len l))
(bounded-nat-listp l b))))
(defthm count-before-n-correctness-3
(implies (nat-listp l)
(equal (count-before-n l 0) 0)))
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