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(in-package "ACL2")
; fat32.lisp Mihir Mehta
; Utilities for FAT32.
;; In commit e1d2e33807c4c529d2c7b1eb7d481f466e7a7d67 there's a (successful)
;; attempt to remove the non-local inclusion of this book. Ultimately, though,
;; this commit was rolled back since the redundant redefinition of logapp,
;; loghead, logtail, imod, ifloor and expt2 is deemed excessively
;; cumbersome. This is part of our theory now.
(include-book "ihs/logops-lemmas" :dir :system)
(include-book "centaur/fty/top" :dir :system)
(include-book "file-system-lemmas")
(include-book "utilities/bounded-nat-listp")
(defconst *expt-2-28* (expt 2 28))
;; from page 18 of the FAT specification
(defconst *ms-end-of-cc* (- *expt-2-28* 1))
;; from page 14 of the FAT specification
(defconst *ms-first-data-cluster* 2)
;; from page 18 of the FAT specification
(defconst *ms-bad-cluster* 268435447)
;; from page 15 of the FAT specification
(defconst *ms-min-count-of-clusters* 65525)
;; from page 9 of the FAT specification
(defconst *ms-min-bytes-per-sector* 512)
;; inferred - there can be as few as one sectors in a cluster
(defconst *ms-min-data-region-size* (* *ms-min-bytes-per-sector*
1
*ms-min-count-of-clusters*))
(defconst *ms-max-bytes-per-sector* 4096)
;; inferred - there can be as many as 128 sectors in a cluster
(defconst *ms-max-data-region-size* (* *ms-max-bytes-per-sector*
128
(- *ms-bad-cluster* 2)))
(defconst *ms-d-e-length* 32)
;; observed
(defconst *current-dir-fat32-name* ". ")
(defconst *parent-dir-fat32-name* ".. ")
;; This has not really been observed to be any FAT32 entry's name. In fact, it
;; cannot be, since the FAT specification says that the filesystem's root
;; directory does not have "." and ".." entries, and those entries would have
;; different names even if they existed. Still, this is what our
;; name-to-fat32-name function computes when passed the empty string - and that
;; in turn happens with the empty string preceding the first slash in "/home"
;; or "/vmlinuz".
(defconst *empty-fat32-name* " ")
;; known
(defconst *current-dir-name* ".")
(defconst *parent-dir-name* "..")
;; Page 36 of the FAT specification states that a directory shouldn't have more
;; than 65536 entries. However, *ms-max-d-e-count* below is used for the
;; definition of hifat-bounded-file-alist-p, and since that's applicable to our
;; internal representation of the filesystem, we need to leave room for two
;; entries (dot and dotdot) to be added when we store a directory in the stobj
;; representation. However, *ms-max-dir-size* is applicable to extracting
;; directory contents from the disk, and therefore it needs to be 2097152 as
;; stipulated.
(defconst *ms-max-d-e-count* (- (ash 1 16) 2))
(defconst *ms-max-dir-size* (ash 1 21))
;; from include/uapi/asm-generic/errno-base.h in the linux kernel sources
(defconst *ENOENT* 2) ;; No such file or directory
(defconst *EIO* 5) ;; I/O error
(defconst *EBADF* 9) ;; Bad file number
(defconst *EEXIST* 17) ;; File exists
(defconst *ENOTDIR* 20) ;; Not a directory
(defconst *EISDIR* 21) ;; Is a directory
(defconst *ENOSPC* 28) ;; No space left on device
(defconst *ENAMETOOLONG* 36) ;; File name too long
;; from the stat code in the coreutils sources
(defconst *S_MAGIC_FUSEBLK* #x65735546)
(encapsulate
()
(local
(include-book "arithmetic/top-with-meta" :dir :system))
(local
(defun induction-scheme (bits x)
(if (zp bits)
x
(induction-scheme (- bits 1)
(logcdr x)))))
(defthmd
unsigned-byte-p-alt
(implies (natp bits)
(equal (unsigned-byte-p bits x)
(and (unsigned-byte-p (+ bits 1) x)
(zp (logand (ash 1 bits) x)))))
:hints
(("goal" :in-theory (e/d nil (logand ash logcar logcdr)
(logand* ash*))
:induct (induction-scheme bits x)))))
(encapsulate
()
(local (include-book "arithmetic-5/top" :dir :system))
(defthm
logand-ash-lemma-1
(implies (and (natp c))
(unsigned-byte-p c (logand i (- (ash 1 c) 1))))))
(defund fat32-entry-p (x)
(declare (xargs :guard t))
(unsigned-byte-p 32 x))
(defund fat32-masked-entry-p (x)
(declare (xargs :guard t))
(unsigned-byte-p 28 x))
;; 0 is chosen as the default value based on this comment from Microsoft's FAT
;; overview:
;; The only time that the high 4 bits of FAT32 FAT entries should ever be
;; changed is when the volume is formatted, at which time the whole 32-bit FAT
;; entry should be zeroed, including the high 4 bits.
(defund fat32-entry-fix (x)
(declare (xargs :guard t))
(if (fat32-entry-p x)
x 0))
(defund fat32-masked-entry-fix (x)
(declare (xargs :guard t))
(if (fat32-masked-entry-p x)
x 0))
(defthm natp-of-fat32-masked-entry-fix
(natp (fat32-masked-entry-fix x))
:hints (("goal" :do-not-induct t
:in-theory (enable fat32-masked-entry-fix
fat32-masked-entry-p)))
:rule-classes :type-prescription)
(in-theory (enable fat32-entry-p fat32-entry-fix fat32-masked-entry-p fat32-masked-entry-fix))
(defthm fat32-masked-entry-p-correctness-1
(implies (fat32-masked-entry-p x)
(natp x))
:rule-classes :forward-chaining)
(defthm fat32-entry-fix-when-fat32-entry-p
(implies (fat32-entry-p x)
(equal (fat32-entry-fix x) x))
:hints (("Goal" :in-theory (enable fat32-entry-fix))))
(defthm fat32-masked-entry-fix-when-fat32-masked-entry-p
(implies (fat32-masked-entry-p x)
(equal (fat32-masked-entry-fix x) x))
:hints (("Goal" :in-theory (enable fat32-masked-entry-fix))))
(defthm
fat32-entry-fix-correctness-1
(and (<= 0 (fat32-entry-fix x))
(< (fat32-entry-fix x) (ash 1 32))
(integerp (fat32-entry-fix x)))
:rule-classes
((:linear
:corollary
(and (<= 0 (fat32-entry-fix x))
(< (fat32-entry-fix x) (ash 1 32))))
(:type-prescription
:corollary
(integerp (fat32-entry-fix x))))
:hints
(("goal" :in-theory (enable fat32-entry-fix fat32-entry-p))))
;; Use a mask to take the low 28 bits.
(defund fat32-entry-mask (x)
(declare (xargs :guard (fat32-entry-p x)))
(loghead 28 x))
(defthm
fat32-entry-mask-correctness-1
(fat32-masked-entry-p (fat32-entry-mask x))
:hints (("goal" :in-theory (e/d (fat32-entry-mask fat32-masked-entry-p)
(unsigned-byte-p logand-ash-lemma-1))
:use (:instance logand-ash-lemma-1 (c 28)
(i x)))))
(fty::deffixtype fat32-entry
:pred fat32-entry-p
:fix fat32-entry-fix
:equiv fat32-entry-equiv
:define t
:forward t)
(fty::deffixtype fat32-masked-entry
:pred fat32-masked-entry-p
:fix fat32-masked-entry-fix
:equiv fat32-masked-entry-equiv
:define t
:forward t)
(fty::deflist fat32-entry-list :elt-type fat32-entry-p :true-listp t)
(fty::deflist fat32-masked-entry-list :elt-type fat32-masked-entry-p :true-listp t)
(defthm nat-listp-if-fat32-masked-entry-list-p
(implies (fat32-masked-entry-list-p x)
(nat-listp x))
:rule-classes (:forward-chaining :rewrite))
(in-theory (disable fat32-entry-p fat32-entry-fix fat32-masked-entry-p fat32-masked-entry-fix))
(defthm member-of-fat32-entry-list
(implies (and (member-equal x lst)
(fat32-entry-list-p lst))
(fat32-entry-p x)))
(defthm
fat32-masked-entry-list-p-of-make-list-ac
(implies (and (fat32-masked-entry-p val)
(fat32-masked-entry-list-p ac))
(fat32-masked-entry-list-p (make-list-ac n val ac))))
(defthm fat32-masked-entry-list-p-of-append
(equal (fat32-masked-entry-list-p (binary-append x y))
(and (fat32-masked-entry-list-p (true-list-fix x))
(fat32-masked-entry-list-p y))))
(defthm fat32-masked-entry-list-p-of-true-list-fix
(implies
(fat32-masked-entry-list-p x)
(fat32-masked-entry-list-p (true-list-fix x))))
(defthm fat32-entry-list-p-of-update-nth
(implies
(fat32-entry-list-p l)
(equal (fat32-entry-list-p (update-nth key val l))
(and (<= (nfix key) (len l))
(fat32-entry-p val)))))
(defthm fat32-entry-list-p-of-revappend
(equal (fat32-entry-list-p (revappend x y))
(and (fat32-entry-list-p y)
(fat32-entry-list-p (list-fix x)))))
(defthm fat32-entry-list-of-nthcdr
(implies (fat32-entry-list-p l)
(fat32-entry-list-p (nthcdr n l))))
(defthm
fat32-masked-entry-p-of-nth-when-fat32-masked-entry-list-p
(implies (fat32-masked-entry-list-p l)
(iff (fat32-masked-entry-p (nth n l))
(< (nfix n) (len l))))
:rule-classes
((:rewrite
:corollary (implies (fat32-masked-entry-list-p l)
(equal (fat32-masked-entry-p (nth n l))
(< (nfix n) (len l)))))))
(encapsulate
()
(local
(defthm fat32-entry-list-p-of-first-n-ac-lemma-1
(implies (not (fat32-entry-list-p (true-list-fix ac)))
(not (fat32-entry-list-p (first-n-ac i l ac))))))
(defthm
fat32-entry-list-p-of-first-n-ac
(implies (fat32-entry-list-p l)
(equal (fat32-entry-list-p (first-n-ac n l ac))
(and (fat32-entry-list-p (true-list-fix ac))
(<= (nfix n) (len l)))))))
(defthm
fat32-entry-list-p-of-take
(implies (fat32-entry-list-p l)
(equal (fat32-entry-list-p (take n l))
(<= (nfix n) (len l)))))
(defthm fat32-entry-list-p-of-repeat
(implies (fat32-entry-p x)
(fat32-entry-list-p (repeat n x)))
:hints (("goal" :in-theory (enable repeat))))
(defthm
fat32-entry-list-p-of-resize-list
(implies
(and (fat32-entry-list-p lst)
(fat32-entry-p default-value))
(fat32-entry-list-p (resize-list lst n default-value))))
(defthm natp-when-fat32-entry-p
(implies (fat32-entry-p x) (natp x))
:hints (("goal" :in-theory (enable fat32-entry-p)))
:rule-classes :forward-chaining)
(encapsulate
()
(local (include-book "arithmetic-5/top" :dir :system))
(local
(defthm fat32-entry-p-of-nth-lemma-1
(equal (< (+ -1 n) (len (cdr l)))
(< n (+ 1 (len (cdr l)))))))
(defthm fat32-entry-p-of-nth
(implies (fat32-entry-list-p l)
(equal (fat32-entry-p (nth n l))
(< (nfix n) (len l))))
:hints (("Goal" :in-theory (disable nth-when->=-n-len-l)))))
(defund
fat32-update-lower-28
(entry masked-entry)
(declare
(xargs
:guard-hints
(("goal"
:in-theory (enable fat32-entry-p fat32-masked-entry-p)))
:guard (and (fat32-entry-p entry)
(fat32-masked-entry-p masked-entry))))
(logapp 28 masked-entry
(logtail 28
(mbe :exec entry
:logic (fat32-entry-fix entry)))))
(defthm
fat32-update-lower-28-correctness-1
(implies (fat32-masked-entry-p masked-entry)
(fat32-entry-p (fat32-update-lower-28 entry masked-entry)))
:hints (("goal" :in-theory (e/d (fat32-update-lower-28 fat32-entry-p)
(logapp logtail))))
:rule-classes
(:rewrite
(:rewrite
:corollary
(implies (fat32-masked-entry-p masked-entry)
(unsigned-byte-p 32
(fat32-update-lower-28 entry masked-entry)))
:hints (("goal" :in-theory (enable fat32-entry-p))))))
(defthm
fat32-entry-mask-of-fat32-update-lower-28
(implies
(fat32-masked-entry-p masked-entry)
(equal (fat32-entry-mask (fat32-update-lower-28 entry masked-entry))
masked-entry))
:hints
(("goal" :in-theory
(e/d (fat32-update-lower-28 fat32-entry-mask fat32-masked-entry-p)
(unsigned-byte-p loghead logapp)))))
(defthm
fat32-update-lower-28-of-fat32-update-lower-28
(implies
(fat32-masked-entry-p masked-entry1)
(equal (fat32-update-lower-28 (fat32-update-lower-28 entry masked-entry1)
masked-entry2)
(fat32-update-lower-28 entry masked-entry2)))
:hints
(("goal"
:in-theory (e/d (fat32-update-lower-28)
(logapp logtail
(:rewrite fat32-update-lower-28-correctness-1
. 1)))
:use (:instance (:rewrite fat32-update-lower-28-correctness-1 . 1)
(masked-entry masked-entry1)
(entry entry)))))
(defthmd
fat32-update-lower-28-of-fat32-entry-mask
(implies (and (fat32-entry-p entry)
(equal masked-entry (fat32-entry-mask entry)))
(equal (fat32-update-lower-28 entry masked-entry)
entry))
:hints
(("goal"
:in-theory (e/d (fat32-update-lower-28 fat32-entry-mask)
(logapp loghead logtail)))))
;; This is taken from page 18 of the fat specification.
(defund fat32-is-eof (fat-content)
(declare (xargs :guard (fat32-masked-entry-p fat-content)
:guard-hints (("Goal'" :in-theory (enable fat32-masked-entry-p)))))
(>= fat-content #xFFFFFF8))
;; The case-split in the hypothesis is because we can't make a linear rule out
;; of the contrapositive.
(defthm fat32-is-eof-correctness-1
(implies (case-split (< fat-content *ms-bad-cluster*))
(not (fat32-is-eof fat-content)))
:hints (("Goal" :in-theory (enable fat32-is-eof)) ))
(defund
fat32-build-index-list
(fa-table masked-current-cluster
length cluster-size)
(declare
(xargs
:measure (nfix length)
:guard (and (fat32-entry-list-p fa-table)
(fat32-masked-entry-p masked-current-cluster)
(natp length)
(>= masked-current-cluster
*ms-first-data-cluster*)
(< masked-current-cluster (len fa-table))
(integerp cluster-size)
(> cluster-size 0))))
(b*
(((when (or (zp length) (zp cluster-size)))
;; This represents a problem case because masked-current-cluster is a
;; valid non-free cluster, but the length is 0. This loosely
;; corresponds to the infinite loop protection in the function
;; fs/fat/cache.c:fat_get_cluster.
(mv nil (- *eio*)))
(masked-next-cluster
(fat32-entry-mask (nth masked-current-cluster fa-table)))
((when (< masked-next-cluster
*ms-first-data-cluster*))
(mv (list masked-current-cluster)
(- *eio*)))
((when (or (fat32-is-eof masked-next-cluster)
(>= masked-next-cluster (len fa-table))))
(mv (list masked-current-cluster) 0))
((mv tail-index-list tail-error)
(fat32-build-index-list fa-table masked-next-cluster
(nfix (- length cluster-size))
cluster-size)))
(mv (list* masked-current-cluster tail-index-list)
tail-error)))
(defthm bounded-nat-listp-of-fat32-build-index-list
(implies (and (equal b (len fa-table))
(fat32-masked-entry-p masked-current-cluster)
(< masked-current-cluster (len fa-table)))
(b* (((mv index-list &)
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size)))
(bounded-nat-listp index-list b)))
:hints (("Goal" :in-theory (enable fat32-build-index-list))))
(defthm
fat32-masked-entry-list-p-of-fat32-build-index-list
(implies (fat32-masked-entry-p masked-current-cluster)
(b* (((mv index-list &)
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size)))
(fat32-masked-entry-list-p index-list)))
:hints (("goal" :in-theory (enable fat32-build-index-list))))
(defthm
fat32-build-index-list-of-update-nth
(implies
(and (<= *ms-first-data-cluster*
masked-current-cluster)
(fat32-masked-entry-p masked-current-cluster))
(mv-let
(index-list error-code)
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size)
(declare (ignore error-code))
(implies (and (not (member-equal key index-list))
(< (nfix key) (len fa-table)))
(equal (fat32-build-index-list (update-nth key val fa-table)
masked-current-cluster
length cluster-size)
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size)))))
:hints
(("goal" :induct (fat32-build-index-list fa-table masked-current-cluster
length cluster-size)
:in-theory (e/d (fat32-build-index-list)
(nth update-nth))
:expand (fat32-build-index-list (update-nth key val fa-table)
masked-current-cluster
length cluster-size))))
(defthm
lower-bounded-integer-listp-of-fat32-build-index-list
(implies
(and (fat32-masked-entry-p masked-current-cluster)
(<= *ms-first-data-cluster*
masked-current-cluster))
(lower-bounded-integer-listp
(mv-nth
0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))
*ms-first-data-cluster*))
:hints
(("goal" :in-theory (enable fat32-build-index-list))))
(defthm
true-listp-of-fat32-build-index-list
(true-listp
(mv-nth
0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size)))
:hints
(("goal" :in-theory (enable fat32-build-index-list)))
:rule-classes (:rewrite :type-prescription))
(defthm
integer-listp-of-fat32-build-index-list
(implies
(integerp masked-current-cluster)
(integer-listp
(mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))))
:hints (("goal" :in-theory (enable fat32-build-index-list))))
(defthm
consp-of-fat32-build-index-list
(equal
(consp (mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size)))
(not (or (zp length) (zp cluster-size))))
:hints (("goal" :in-theory (enable fat32-build-index-list))))
(encapsulate
()
(local (include-book "arithmetic-3/top" :dir :system))
(local (in-theory (enable nfix)))
(local
(set-default-hints
'((nonlinearp-default-hint stable-under-simplificationp
hist pspv))))
(defthm
len-of-fat32-build-index-list-1
(implies
(not (zp cluster-size))
(<= (len (mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size)))
(ceiling (nfix length) cluster-size)))
:hints
(("goal" :induct (fat32-build-index-list fa-table masked-current-cluster
length cluster-size)
:in-theory (enable fat32-build-index-list)))
:rule-classes
(:linear
(:linear
:corollary
(implies
(not (zp cluster-size))
(<= (*
cluster-size
(len (mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))))
(* cluster-size
(ceiling (nfix length) cluster-size)))))
(:linear
:corollary
(implies
(not (zp cluster-size))
(< (*
cluster-size
(len (mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))))
(+ (nfix length) cluster-size)))))))
(defun count-free-clusters-helper (fa-table n)
(declare (xargs :guard (and (fat32-entry-list-p fa-table)
(natp n)
(>= (len fa-table) n))
:guard-hints (("Goal" :in-theory (disable nth)) )))
(if
(zp n)
0
(if
(not (equal (fat32-entry-mask (nth (- n 1) fa-table)) 0))
(count-free-clusters-helper fa-table (- n 1))
(+ 1 (count-free-clusters-helper fa-table (- n 1))))))
(defthm
count-free-clusters-helper-of-update-nth-1
(implies (not (equal (fat32-entry-mask val) 0))
(equal (count-free-clusters-helper (update-nth key val fa-table)
n)
(if (and (< (nfix key) (nfix n))
(equal (fat32-entry-mask (nth key fa-table))
0))
(- (count-free-clusters-helper fa-table n)
1)
(count-free-clusters-helper fa-table n))))
:hints (("goal" :in-theory (disable update-nth))))
(defthm
count-free-clusters-helper-of-update-nth-2
(implies (equal (fat32-entry-mask val) 0)
(equal (count-free-clusters-helper (update-nth key val fa-table)
n)
(if (and (< (nfix key) (nfix n))
(not (equal (fat32-entry-mask (nth key fa-table))
0)))
(+ (count-free-clusters-helper fa-table n)
1)
(count-free-clusters-helper fa-table n))))
:hints (("goal" :in-theory (disable update-nth))))
(defthm count-free-clusters-helper-correctness-1
(<= (count-free-clusters-helper fa-table n)
(nfix n))
:rule-classes :linear)
(defund
count-free-clusters (fa-table)
(declare
(xargs :guard (and (fat32-entry-list-p fa-table)
(>= (len fa-table)
*ms-first-data-cluster*))
:guard-hints (("goal" :in-theory (disable nth)))))
(count-free-clusters-helper
(nthcdr *ms-first-data-cluster* fa-table)
(- (len fa-table)
*ms-first-data-cluster*)))
(defthm count-free-clusters-of-update-nth
(implies (and (<= *ms-first-data-cluster* (nfix key))
(< (nfix key) (len fa-table)))
(equal (count-free-clusters (update-nth key val fa-table))
(cond ((and (equal (fat32-entry-mask (nth key fa-table))
0)
(not (equal (fat32-entry-mask val) 0)))
(- (count-free-clusters fa-table) 1))
((and (equal (fat32-entry-mask val) 0)
(not (equal (fat32-entry-mask (nth key fa-table))
0)))
(+ (count-free-clusters fa-table) 1))
(t (count-free-clusters fa-table)))))
:hints (("goal" :in-theory (enable count-free-clusters))))
(defthm
count-free-clusters-correctness-1
(implies (>= (len fa-table)
*ms-first-data-cluster*)
(<= (count-free-clusters fa-table)
(- (len fa-table)
*ms-first-data-cluster*)))
:rule-classes
((:linear
:trigger-terms
((count-free-clusters fa-table))))
:hints (("goal" :in-theory (enable count-free-clusters))))
(defund
find-n-free-clusters-helper
(fa-table n start)
(declare (xargs :guard (and (fat32-entry-list-p fa-table)
(natp n)
(natp start))))
(if (or (atom fa-table) (zp n))
nil
(if (not (equal (fat32-entry-mask (car fa-table))
0))
(find-n-free-clusters-helper (cdr fa-table)
n (+ start 1))
(cons start
(find-n-free-clusters-helper (cdr fa-table)
(- n 1)
(+ start 1))))))
(defthmd
find-n-free-clusters-helper-correctness-1
(implies (and (fat32-entry-list-p fa-table)
(natp n)
(natp start)
(equal b (+ start (len fa-table))))
(bounded-nat-listp
(find-n-free-clusters-helper fa-table n start)
b))
:hints
(("goal'" :in-theory (enable find-n-free-clusters-helper))))
(defthmd
find-n-free-clusters-helper-correctness-2
(implies
(natp start)
(nat-listp (find-n-free-clusters-helper fa-table n start)))
:hints
(("goal" :in-theory (enable find-n-free-clusters-helper))))
(defthmd
find-n-free-clusters-helper-correctness-3
(implies
(and (natp start)
(member-equal
x
(find-n-free-clusters-helper fa-table n start)))
(and (integerp x)
(<= start x)
(< x (+ start (len fa-table)))))
:hints
(("goal" :in-theory (enable find-n-free-clusters-helper))))
(defthm
find-n-free-clusters-helper-correctness-4
(implies
(and
(fat32-entry-list-p fa-table)
(natp n)
(natp start)
(not (equal (fat32-entry-mask (nth (- x start) fa-table))
0)))
(not (member-equal
x
(find-n-free-clusters-helper fa-table n start))))
:hints
(("goal" :in-theory (enable find-n-free-clusters-helper)
:use find-n-free-clusters-helper-correctness-3)
("subgoal *1/2"
:use (:instance find-n-free-clusters-helper-correctness-3
(fa-table (cdr fa-table))
(start (+ 1 start))))))
(defthm
find-n-free-clusters-helper-correctness-6
(implies
(and (fat32-entry-list-p fa-table)
(natp start))
(no-duplicatesp-equal (find-n-free-clusters-helper fa-table n start)))
:hints
(("goal" :in-theory (enable find-n-free-clusters-helper))
("goal'" :induct (find-n-free-clusters-helper fa-table n start))
("subgoal *1/9" :use (:instance find-n-free-clusters-helper-correctness-3
(x start)
(fa-table (cdr fa-table))
(n (- n 1))
(start (+ 1 start))))))
(defthm
find-n-free-clusters-helper-correctness-7
(implies
(and
(natp start)
(< (nfix m)
(len (find-n-free-clusters-helper fa-table n start))))
(and
(integerp
(nth m
(find-n-free-clusters-helper fa-table n start)))
(<= start
(nth m
(find-n-free-clusters-helper fa-table n start)))
(< (nth m
(find-n-free-clusters-helper fa-table n start))
(+ start (len fa-table)))))
:rule-classes
((:linear :corollary
(implies
(and
(natp start)
(< (nfix m)
(len (find-n-free-clusters-helper fa-table n start))))
(and
(<= start
(nth m
(find-n-free-clusters-helper fa-table n start)))
(< (nth m
(find-n-free-clusters-helper fa-table n start))
(+ start (len fa-table))))))
(:rewrite :corollary
(implies
(and
(natp start)
(< (nfix m)
(len (find-n-free-clusters-helper fa-table n start))))
(integerp
(nth m
(find-n-free-clusters-helper fa-table n start))))))
:hints
(("goal"
:use
(:instance
find-n-free-clusters-helper-correctness-3
(x
(nth m
(find-n-free-clusters-helper fa-table n start)))))))
(encapsulate
()
(local
(defthm
len-of-find-n-free-clusters-helper-lemma-1
(implies
(consp fa-table)
(equal
(len (find-n-free-clusters-helper fa-table n start))
(if
(and
(<
(len (find-n-free-clusters-helper (take (- (len fa-table) 1) fa-table)
n start))
(nfix n))
(equal (fat32-entry-mask (nth (- (len fa-table) 1) fa-table))
0))
(+ (len (find-n-free-clusters-helper (take (- (len fa-table) 1) fa-table)
n start))
1)
(len (find-n-free-clusters-helper (take (- (len fa-table) 1) fa-table)
n start)))))
:hints (("goal" :in-theory (enable find-n-free-clusters-helper)
:induct (find-n-free-clusters-helper fa-table n start)
:expand (len (cdr fa-table))))))
(local
(defthm
len-of-find-n-free-clusters-helper-lemma-2
(<= (len (find-n-free-clusters-helper fa-table n start))
(nfix n))
:rule-classes :linear
:hints
(("goal" :in-theory (enable find-n-free-clusters-helper)))))
(defthm
len-of-find-n-free-clusters-helper
(equal (len (find-n-free-clusters-helper (take n2 fa-table)
n1 start))
(min (count-free-clusters-helper fa-table n2)
(nfix n1)))
:hints
(("goal" :in-theory (enable find-n-free-clusters-helper
len-of-find-n-free-clusters-helper-lemma-1)))
:rule-classes
(:rewrite
(:linear
:corollary
(<= (len (find-n-free-clusters-helper fa-table n start))
(nfix n))))))
(in-theory (disable (:linear len-of-find-n-free-clusters-helper)))
(defund
find-n-free-clusters (fa-table n)
(declare (xargs :guard (and (fat32-entry-list-p fa-table)
(natp n))))
;; the first 2 clusters are excluded
(find-n-free-clusters-helper
(nthcdr *ms-first-data-cluster* fa-table)
n *ms-first-data-cluster*))
(defthm
find-n-free-clusters-correctness-1
(implies (and (fat32-entry-list-p fa-table)
(natp n)
(equal b (len fa-table))
(>= (len fa-table)
*ms-first-data-cluster*))
(bounded-nat-listp (find-n-free-clusters fa-table n)
b))
:hints
(("goal"
:in-theory (enable find-n-free-clusters)
:use
((:instance
find-n-free-clusters-helper-correctness-1
(start *ms-first-data-cluster*)
(fa-table (nthcdr *ms-first-data-cluster* fa-table))
(b (len fa-table))))))
:rule-classes
(:rewrite
(:linear
:corollary
(implies (and (fat32-entry-list-p fa-table)
(natp n)
(equal b (len fa-table))
(>= (len fa-table)
*ms-first-data-cluster*)
(consp (find-n-free-clusters fa-table n)))
(< (car (find-n-free-clusters fa-table n))
b)))))
(defthm
find-n-free-clusters-correctness-2
(nat-listp (find-n-free-clusters fa-table n))
:rule-classes
(:rewrite
(:rewrite
:corollary
(implies (consp (find-n-free-clusters fa-table n))
(and (nat-listp (cdr (find-n-free-clusters fa-table n)))
(integerp (car (find-n-free-clusters fa-table n))))))
(:linear
:corollary (implies (consp (find-n-free-clusters fa-table n))
(<= 0
(car (find-n-free-clusters fa-table n))))))
:hints
(("goal"
:in-theory (enable find-n-free-clusters
find-n-free-clusters-helper-correctness-2))))
(defthmd
find-n-free-clusters-correctness-3
(implies (member-equal x (find-n-free-clusters fa-table n))
(and (integerp x)
(<= *ms-first-data-cluster* x)
(< x (len fa-table))))
:hints
(("goal" :in-theory (enable find-n-free-clusters))
("goal'"
:use
(:instance
find-n-free-clusters-helper-correctness-3
(start *ms-first-data-cluster*)
(fa-table (nthcdr *ms-first-data-cluster* fa-table))))))
(defthm
find-n-free-clusters-correctness-4
(implies
(and (fat32-entry-list-p fa-table)
(natp n)
(not (equal (fat32-entry-mask (nth x fa-table))
0)))
(not (member-equal x (find-n-free-clusters fa-table n))))
:hints
(("goal"
:in-theory (e/d (find-n-free-clusters)
(member-of-a-nat-list))
:use
((:instance
member-of-a-nat-list
(lst (find-n-free-clusters-helper
(nthcdr *ms-first-data-cluster* fa-table)
n *ms-first-data-cluster*)))
(:instance
find-n-free-clusters-helper-correctness-3
(fa-table (nthcdr *ms-first-data-cluster* fa-table))
(start *ms-first-data-cluster*))))))
;; Holding off for now on determining what
;; find-n-free-clusters-helper-correctness-5 should look like.
(defthm
find-n-free-clusters-correctness-5
(implies
(and (fat32-entry-list-p fa-table)
(natp n1)
(< (nfix n2)
(len (find-n-free-clusters fa-table n1))))
(equal (fat32-entry-mask
(nth (nth n2 (find-n-free-clusters fa-table n1))
fa-table))
0))
:hints
(("goal"
:do-not-induct t
:in-theory (e/d nil
(find-n-free-clusters-correctness-4))
:use ((:instance
find-n-free-clusters-correctness-4
(n n1)
(x (nth n2
(find-n-free-clusters fa-table n1))))))))
(defthm find-n-free-clusters-correctness-6
(implies (fat32-entry-list-p fa-table)
(no-duplicatesp-equal (find-n-free-clusters fa-table n)))
:hints (("goal" :in-theory (enable find-n-free-clusters))))
(defthm
find-n-free-clusters-correctness-8
(integer-listp (find-n-free-clusters fa-table n)))
(defthm find-n-free-clusters-correctness-7
(implies (case-split (< (nfix m)
(len (find-n-free-clusters fa-table n))))
(and (<= *ms-first-data-cluster*
(nth m (find-n-free-clusters fa-table n)))
(< (nth m (find-n-free-clusters fa-table n))
(len fa-table))))
:rule-classes :linear
:hints (("goal" :in-theory (e/d (find-n-free-clusters) (nth)))))
(defthm
find-n-free-clusters-helper-of-true-list-fix
(equal (find-n-free-clusters-helper (true-list-fix fa-table)
n start)
(find-n-free-clusters-helper fa-table n start))
:hints
(("goal" :in-theory (enable find-n-free-clusters-helper))))
(defthm
len-of-find-n-free-clusters
(equal (len (find-n-free-clusters fa-table n))
(min (count-free-clusters fa-table)
(nfix n)))
:hints (("goal" :in-theory (e/d (count-free-clusters find-n-free-clusters)
(nthcdr len-of-find-n-free-clusters-helper))
:use (:instance len-of-find-n-free-clusters-helper
(n2 (len (nthcdr 2 fa-table)))
(fa-table (nthcdr 2 fa-table))
(n1 n)
(start 2))))
:rule-classes
(:rewrite
(:linear
:corollary
(<= (len (find-n-free-clusters fa-table n))
(nfix n)))))
(in-theory (disable (:linear len-of-find-n-free-clusters)))
(defthm consp-of-find-n-free-clusters
(equal (consp (find-n-free-clusters fa-table n))
(and (not (zp (count-free-clusters fa-table)))
(not (zp n))))
:hints (("goal" :do-not-induct t
:in-theory (e/d (len-when-consp)
(len-of-find-n-free-clusters))
:use len-of-find-n-free-clusters)))
(defthm find-n-free-clusters-when-zp
(implies (zp n)
(equal (find-n-free-clusters fa-table n)
nil))
:hints (("goal" :in-theory (enable find-n-free-clusters
find-n-free-clusters-helper))))
(defcong nat-equiv equal (find-n-free-clusters fa-table n) 2)
(defthmd
fat32-masked-entry-list-p-alt
(equal (fat32-masked-entry-list-p x)
(bounded-nat-listp x *expt-2-28*))
:hints (("goal" :in-theory (enable fat32-masked-entry-p))))
(defthm
fat32-masked-entry-list-p-of-find-n-free-clusters
(implies (and (fat32-entry-list-p fa-table)
(natp n)
(>= (len fa-table) *ms-first-data-cluster*)
(<= (len fa-table) *ms-bad-cluster*))
(fat32-masked-entry-list-p (find-n-free-clusters fa-table n)))
:rule-classes
(:rewrite
(:rewrite
:corollary
(implies
(and (fat32-entry-list-p fa-table)
(natp n)
(>= (len fa-table) *ms-first-data-cluster*)
(<= (len fa-table) *ms-bad-cluster*))
(let ((l (find-n-free-clusters fa-table n)))
(implies (consp l)
(and (fat32-masked-entry-list-p (cdr l))
(fat32-masked-entry-p (car l))))))))
:hints (("goal" :in-theory (disable find-n-free-clusters-correctness-1)
:use ((:instance find-n-free-clusters-correctness-1
(b (len fa-table)))
(:instance fat32-masked-entry-list-p-alt
(x (find-n-free-clusters fa-table n)))
(:instance bounded-nat-listp-correctness-5
(l (find-n-free-clusters fa-table n))
(x (len fa-table))
(y *expt-2-28*))))))
(defthm
lower-bounded-integer-listp-of-find-n-free-clusters-helper
(implies (integerp start)
(lower-bounded-integer-listp
(find-n-free-clusters-helper fa-table n start)
start))
:hints
(("goal" :in-theory (enable lower-bounded-integer-listp
find-n-free-clusters-helper))
("subgoal *1/5.1'"
:use
(:instance lower-bounded-integer-listp-correctness-5
(l (find-n-free-clusters-helper (cdr fa-table)
(+ -1 n)
(+ 1 start)))
(x (+ 1 start))
(y start)))
("subgoal *1/3''"
:use
(:instance lower-bounded-integer-listp-correctness-5
(l (find-n-free-clusters-helper (cdr fa-table)
n (+ 1 start)))
(x (+ 1 start))
(y start)))))
(defthm
lower-bounded-integer-listp-of-find-n-free-clusters
(lower-bounded-integer-listp (find-n-free-clusters fa-table n)
*ms-first-data-cluster*)
:rule-classes
(:rewrite
(:rewrite
:corollary
(implies
(consp (find-n-free-clusters fa-table n))
(lower-bounded-integer-listp (cdr (find-n-free-clusters fa-table n))
*ms-first-data-cluster*))
:hints (("goal" :in-theory (enable lower-bounded-integer-listp))))
(:linear
:corollary (implies (consp (find-n-free-clusters fa-table n))
(<= *ms-first-data-cluster*
(car (find-n-free-clusters fa-table n))))
:hints (("goal" :in-theory (enable lower-bounded-integer-listp)))))
:hints (("goal" :in-theory (enable find-n-free-clusters))))
(defund
set-indices-in-fa-table
(fa-table index-list value-list)
(declare
(xargs :measure (acl2-count index-list)
:guard (and (fat32-entry-list-p fa-table)
(nat-listp index-list)
(fat32-masked-entry-list-p value-list)
(equal (len index-list)
(len value-list)))))
(if
(atom index-list)
fa-table
(let
((current-index (car index-list)))
(if
(or (not (natp current-index))
(>= current-index (len fa-table)))
fa-table
(set-indices-in-fa-table
(update-nth current-index
(fat32-update-lower-28 (nth current-index fa-table)
(car value-list))
fa-table)
(cdr index-list)
(cdr value-list))))))
(defthm
set-indices-in-fa-table-correctness-1
(implies
(and (fat32-entry-list-p fa-table)
(bounded-nat-listp index-list (len fa-table))
(fat32-masked-entry-list-p value-list)
(equal (len index-list)
(len value-list)))
(fat32-entry-list-p
(set-indices-in-fa-table fa-table index-list value-list)))
:hints (("Goal" :in-theory (enable set-indices-in-fa-table))))
(defthm
set-indices-in-fa-table-correctness-2
(equal (len (set-indices-in-fa-table fa-table index-list value-list))
(len fa-table))
:hints (("goal" :in-theory (enable set-indices-in-fa-table))))
;; Well, it might not be a great idea to borrow a numbering scheme from
;; set-indices.lisp
(defthm
set-indices-in-fa-table-correctness-3
(implies
(not (member-equal (nfix n) index-list))
(equal (nth n
(set-indices-in-fa-table fa-table index-list value-list))
(nth n fa-table)))
:hints (("goal" :in-theory (enable set-indices-in-fa-table))))
(defthm set-indices-in-fa-table-correctness-4
(implies (and (natp key)
(< key (len l))
(nat-listp index-list)
(not (member-equal key index-list)))
(equal (set-indices-in-fa-table (update-nth key val l) index-list value-list)
(update-nth key val (set-indices-in-fa-table l index-list value-list))))
:hints (("Goal" :in-theory (enable set-indices-in-fa-table))))
(defthm
fat32-masked-entry-list-p-when-bounded-nat-listp
(implies
(and (bounded-nat-listp file-index-list (len fa-table))
(<= (len fa-table) *ms-bad-cluster*))
(fat32-masked-entry-list-p file-index-list))
:hints (("goal" :in-theory (enable fat32-masked-entry-p))))
(defthm
fat32-build-index-list-of-set-indices-disjoint
(implies
(and
(natp masked-current-cluster)
(nat-listp index-list)
(not (intersectp-equal
(mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))
index-list)))
(equal (fat32-build-index-list
(set-indices-in-fa-table fa-table index-list value-list)
masked-current-cluster
length cluster-size)
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size)))
:hints
(("goal" :induct (fat32-build-index-list fa-table masked-current-cluster
length cluster-size)
:in-theory (e/d (intersectp-equal fat32-build-index-list)
(intersectp-is-commutative)))))
(defthmd
set-indices-in-fa-table-of-true-list-fix-1
(equal (set-indices-in-fa-table fa-table
index-list (true-list-fix value-list))
(set-indices-in-fa-table fa-table index-list value-list))
:hints (("goal" :in-theory (enable set-indices-in-fa-table))))
(defcong
list-equiv equal
(set-indices-in-fa-table fa-table index-list value-list)
3
:hints
(("goal"
:do-not-induct t
:in-theory (enable list-equiv)
:use ((:instance set-indices-in-fa-table-of-true-list-fix-1
(value-list value-list-equiv))
set-indices-in-fa-table-of-true-list-fix-1))))
(encapsulate
()
(local
(defun
induction-scheme
(file-index-list file-length cluster-size)
(if (or (zp file-length) (zp cluster-size))
file-index-list
(induction-scheme (cdr file-index-list)
(nfix (- file-length cluster-size))
cluster-size))))
(defthm
fat32-build-index-list-of-set-indices-in-fa-table-coincident
(implies
(and (natp file-length)
(no-duplicatesp-equal file-index-list)
(< (* cluster-size
(+ -1 (len file-index-list)))
file-length)
(lower-bounded-integer-listp file-index-list *ms-first-data-cluster*)
(bounded-nat-listp file-index-list (len fa-table))
(consp file-index-list)
(<= (len fa-table) *ms-bad-cluster*)
(not (zp cluster-size))
(fat32-masked-entry-p fat-content)
(or
(fat32-is-eof fat-content)
(<= (len fa-table) fat-content)))
(equal (fat32-build-index-list
(set-indices-in-fa-table fa-table file-index-list
(append (cdr file-index-list)
(list fat-content)))
(car file-index-list)
file-length cluster-size)
(mv file-index-list 0)))
:hints
(("goal"
:in-theory (e/d (set-indices-in-fa-table fat32-build-index-list)
(fat32-masked-entry-list-p-when-bounded-nat-listp))
:induct (induction-scheme file-index-list
file-length cluster-size)
:expand
((:free (fa-table value-list)
(set-indices-in-fa-table fa-table file-index-list value-list))
(fat32-build-index-list
(update-nth (car file-index-list)
(fat32-update-lower-28 (nth (car file-index-list) fa-table)
(cadr file-index-list))
(set-indices-in-fa-table fa-table (cdr file-index-list)
(append (cddr file-index-list)
'(268435455))))
(car file-index-list)
file-length cluster-size)
(fat32-build-index-list
(update-nth (car file-index-list)
(fat32-update-lower-28 (nth (car file-index-list) fa-table)
fat-content)
fa-table)
(car file-index-list)
file-length cluster-size)))
("subgoal *1/2" :use fat32-masked-entry-list-p-when-bounded-nat-listp))))
(defthm
set-indices-in-fa-table-when-atom
(implies (atom index-list)
(equal (set-indices-in-fa-table fa-table index-list value-list)
fa-table))
:hints (("goal" :in-theory (enable set-indices-in-fa-table))))
(defthm
member-of-fat32-build-index-list
(implies
(and (equal (mv-nth 1
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))
0)
(equal (fat32-entry-mask (nth x fa-table))
0))
(not (member-equal
x
(mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size)))))
:hints (("goal" :in-theory (enable fat32-build-index-list))))
(defthm
integerp-of-fat32-build-index-list
(implies
(not (equal (mv-nth 1
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))
0))
(equal (mv-nth 1
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))
(- *eio*)))
:hints (("goal" :in-theory (enable fat32-build-index-list)))
:rule-classes
(:rewrite
(:type-prescription
:corollary
(and
(integerp (mv-nth 1
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size)))
(<= (mv-nth 1
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))
0)))))
(defthm
fat32-build-index-list-correctness-1
(implies
(and (fat32-entry-list-p fa-table)
(natp n)
(equal (mv-nth 1
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))
0))
(not (intersectp-equal
(mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))
(find-n-free-clusters fa-table n))))
:hints (("goal" :in-theory (e/d (intersectp-equal fat32-build-index-list)
(intersectp-is-commutative))))
:rule-classes
(:rewrite
(:rewrite
:corollary
(implies
(and
(fat32-entry-list-p fa-table)
(natp n)
(equal (mv-nth 1
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))
0))
(not (intersectp-equal
(find-n-free-clusters fa-table n)
(mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))))))))
(encapsulate ()
(local
(defthmd
lemma
(implies (and (bounded-nat-listp index-list (len fa-table))
(fat32-masked-entry-p val)
(member-equal n index-list))
(equal (nth n
(set-indices-in-fa-table fa-table index-list
(make-list-ac (len index-list)
val nil)))
(fat32-update-lower-28 (nth n fa-table)
val)))
:hints (("goal" :in-theory (enable set-indices-in-fa-table)))))
(defthm
nth-of-set-indices-in-fa-table-when-member
(implies (and (bounded-nat-listp index-list (len fa-table))
(fat32-masked-entry-p val))
(equal (nth n
(set-indices-in-fa-table fa-table index-list
(make-list-ac (len index-list)
val nil)))
(if (member-equal (nfix n) index-list)
(fat32-update-lower-28 (nth n fa-table)
val)
(nth n fa-table))))
:hints
(("goal"
:in-theory (e/d (set-indices-in-fa-table)
((:rewrite set-indices-in-fa-table-correctness-3)))
:use ((:instance (:rewrite set-indices-in-fa-table-correctness-3)
(value-list (make-list-ac (len index-list) val nil))
(index-list index-list)
(fa-table fa-table)
(n n))
(:instance lemma (n (nfix n))))))))
(defthm
fat32-build-index-list-correctness-2
(implies
(equal (mv-nth 1
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))
0)
(member-equal
masked-current-cluster
(mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))))
:hints
(("goal" :in-theory (enable fat32-build-index-list))))
(defthm
car-of-last-of-fat32-build-index-list
(implies
(and (fat32-masked-entry-p masked-current-cluster)
(< masked-current-cluster (len fa-table)))
(<
(car (last (mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size))))
(len fa-table)))
:hints
(("goal"
:in-theory (disable car-of-last-when-bounded-nat-listp)
:use
(:instance
car-of-last-when-bounded-nat-listp
(l (mv-nth 0
(fat32-build-index-list fa-table masked-current-cluster
length cluster-size)))
(b (len fa-table)))))
:rule-classes :linear)
|