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(in-package "ACL2")
; hifat-entry-count.lisp Mihir Mehta
; hifat-entry-count is related to the problem of transforming a potentially loopy
; FAT32 disk image into a tree in a bounded amount of time. Some lemmas for
; reasoning about it are placed here.
(include-book "hifat-equiv")
;; We're not counting this very directory, because the root does not have a
;; directory entry for itself.
(defund
hifat-entry-count (fs)
(declare (xargs :guard (and (m1-file-alist-p fs)
(hifat-no-dups-p fs))
:guard-hints
(("Goal" :in-theory (disable m1-directory-file-p-of-m1-file-fix)))))
(if
(atom fs)
0
(+
(hifat-entry-count (cdr fs))
(if
(consp
(assoc-equal (mbe :logic (fat32-filename-fix (caar fs))
:exec (caar fs))
(mbe :logic (hifat-file-alist-fix (cdr fs))
:exec (cdr fs))))
0
(if
(m1-directory-file-p (mbe :logic (m1-file-fix (cdar fs))
:exec (cdar fs)))
(+ 1
(hifat-entry-count (m1-file->contents (cdar fs))))
1)))))
(defthm hifat-entry-count-of-hifat-file-alist-fix
(equal (hifat-entry-count (hifat-file-alist-fix fs))
(hifat-entry-count fs))
:hints (("Goal" :in-theory (enable hifat-entry-count hifat-file-alist-fix))))
(defthm
m1-file-alist-p-of-remove1-assoc-equal
(implies (m1-file-alist-p m1-file-alist)
(m1-file-alist-p (remove1-assoc-equal key m1-file-alist))))
(defthmd
hifat-entry-count-when-hifat-no-dups-p
(implies
(and (m1-file-alist-p m1-file-alist)
(hifat-no-dups-p m1-file-alist)
(consp (assoc-equal x m1-file-alist)))
(equal
(hifat-entry-count m1-file-alist)
(+
(hifat-entry-count (remove1-assoc x m1-file-alist))
1
(if
(m1-directory-file-p (cdr (assoc-equal x m1-file-alist)))
(hifat-entry-count
(m1-file->contents (cdr (assoc-equal x m1-file-alist))))
0))))
:hints
(("goal"
:in-theory (enable hifat-entry-count hifat-no-dups-p))))
(defthm
hifat-entry-count-of-cdr-1
(implies (and (consp fs)
(m1-file-alist-p fs)
(hifat-no-dups-p fs))
(< (hifat-entry-count (cdr fs))
(hifat-entry-count fs)))
:rule-classes :linear
:hints
(("goal"
:in-theory (enable hifat-no-dups-p hifat-entry-count))))
(defthm
hifat-entry-count-of-cdr-2
(implies
(and (consp fs)
(hifat-no-dups-p fs)
(m1-file-alist-p fs)
(not (m1-regular-file-p (cdr (car fs)))))
(< (+ (hifat-entry-count (m1-file->contents (cdr (car fs))))
(hifat-entry-count (cdr fs)))
(hifat-entry-count fs)))
:rule-classes :linear
:hints (("goal" :in-theory (enable hifat-no-dups-p)
:expand (hifat-entry-count fs))))
(encapsulate
()
(local
(defun
induction-scheme
(m1-file-alist1 m1-file-alist2)
(declare
(xargs
:ruler-extenders nil ; Matt K. change for ruler-extenders mod 2/2021
:guard (and (m1-file-alist-p m1-file-alist1)
(m1-file-alist-p m1-file-alist2))
:hints (("goal" :in-theory (enable m1-file->contents
m1-directory-file-p)))))
(b* (((when (atom m1-file-alist1)) t)
((when (or (atom (car m1-file-alist1))
(not (stringp (car (car m1-file-alist1))))))
(and (member-equal (car m1-file-alist1)
m1-file-alist2)
(induction-scheme (cdr m1-file-alist1)
(remove1-assoc-equal
(caar m1-file-alist1)
m1-file-alist2))))
(name (caar m1-file-alist1))
(file1 (cdar m1-file-alist1))
((unless (consp (assoc-equal name m1-file-alist2)))
nil)
(file2 (cdr (assoc-equal name m1-file-alist2))))
(if (not (m1-directory-file-p file1))
(and (not (m1-directory-file-p file2))
(induction-scheme (cdr m1-file-alist1)
(remove1-assoc-equal
name
m1-file-alist2))
(equal (m1-file->contents file1)
(m1-file->contents file2)))
(and (m1-directory-file-p file2)
(induction-scheme (cdr m1-file-alist1)
(remove1-assoc-equal
name
m1-file-alist2))
(induction-scheme (m1-file->contents file1)
(m1-file->contents file2)))))))
(local
(defthm
induction-scheme-correctness
(implies (and (hifat-no-dups-p m1-file-alist1)
(m1-file-alist-p m1-file-alist1))
(iff (induction-scheme m1-file-alist1 m1-file-alist2)
(hifat-subsetp m1-file-alist1 m1-file-alist2)))
:hints (("goal" :induct (induction-scheme m1-file-alist1 m1-file-alist2)
:in-theory (enable hifat-no-dups-p hifat-subsetp)))))
(defthm
hifat-entry-count-when-hifat-subsetp
(implies (and (hifat-no-dups-p m1-file-alist1)
(m1-file-alist-p m1-file-alist1)
(hifat-no-dups-p m1-file-alist2)
(m1-file-alist-p m1-file-alist2)
(hifat-subsetp m1-file-alist1 m1-file-alist2))
(<= (hifat-entry-count m1-file-alist1)
(hifat-entry-count m1-file-alist2)))
:rule-classes :linear
:hints
(("goal" :induct (induction-scheme m1-file-alist1 m1-file-alist2)
:in-theory (enable hifat-no-dups-p hifat-entry-count hifat-subsetp))
("subgoal *1/7"
:use (:instance (:rewrite hifat-entry-count-when-hifat-no-dups-p)
(m1-file-alist m1-file-alist2)
(x (car (car m1-file-alist1)))))
("subgoal *1/4"
:use (:instance (:rewrite hifat-entry-count-when-hifat-no-dups-p)
(m1-file-alist m1-file-alist2)
(x (car (car m1-file-alist1))))))))
;; This rule is kinda problematic because it has caused an infinite rewrite at
;; least once in hifat-to-lofat-inversion-big-induction, which was only
;; resolved by disabling it. It would be nice to make this a plain congruence
;; rule - but that would require the m1-file-alist and hifat-no-dups-p
;; hypotheses to be removed, which in turn would require the definition of
;; hifat-entry-count to be changed.
(defthm
hifat-entry-count-when-hifat-equiv
(implies (hifat-equiv m1-file-alist1 m1-file-alist2)
(equal (hifat-entry-count m1-file-alist1)
(hifat-entry-count m1-file-alist2)))
:rule-classes :congruence
:hints
(("goal"
:in-theory (e/d (hifat-equiv)
(hifat-entry-count-when-hifat-subsetp))
:use
((:instance hifat-entry-count-when-hifat-subsetp
(m1-file-alist1 (hifat-file-alist-fix m1-file-alist2))
(m1-file-alist2 (hifat-file-alist-fix m1-file-alist1)))
(:instance hifat-entry-count-when-hifat-subsetp
(m1-file-alist1 (hifat-file-alist-fix m1-file-alist1))
(m1-file-alist2 (hifat-file-alist-fix m1-file-alist2)))))))
(defthm hifat-entry-count-of-remove-assoc-equal
(implies (and (m1-file-alist-p fs)
(hifat-no-dups-p fs)
(consp (assoc-equal x fs)))
(< (hifat-entry-count (remove-assoc-equal x fs))
(hifat-entry-count fs)))
:hints (("goal" :in-theory (e/d (hifat-entry-count hifat-no-dups-p))
:induct (assoc-equal x fs)))
:rule-classes :linear)
(local
(defthm hifat-entry-count-of-put-assoc-equal-lemma-1
(implies (and (m1-file-alist-p fs)
(not (fat32-filename-p x)))
(not (consp (assoc-equal x fs))))
:hints (("goal" :in-theory (enable m1-file-alist-p)))))
(defthm
consp-of-assoc-of-hifat-file-alist-fix
(implies (m1-file-alist-p hifat-file-alist)
(iff (consp (assoc-equal name
(hifat-file-alist-fix hifat-file-alist)))
(consp (assoc-equal name hifat-file-alist))))
:hints (("goal" :in-theory (enable hifat-file-alist-fix)
:induct (assoc-equal name hifat-file-alist))))
(defthm
hifat-entry-count-of-put-assoc-equal
(implies
(and (m1-file-alist-p fs)
(hifat-no-dups-p fs)
(m1-file-p val)
(fat32-filename-p name))
(equal
(hifat-entry-count (put-assoc-equal name val fs))
(+
(hifat-entry-count fs)
(if
(atom (assoc-equal name fs))
1
(if
(m1-directory-file-p (cdr (assoc-equal name fs)))
(- (hifat-entry-count (m1-file->contents (cdr (assoc-equal name fs)))))
0))
(if (m1-directory-file-p val)
(hifat-entry-count (m1-file->contents val))
0))))
:hints (("goal" :in-theory (enable hifat-entry-count hifat-no-dups-p)
:induct (assoc-equal name fs))))
(defthm
hifat-entry-count-of-hifat-remove-file
(implies (equal (mv-nth 1 (hifat-remove-file fs path))
0)
(< (hifat-entry-count (mv-nth 0 (hifat-remove-file fs path)))
(hifat-entry-count fs)))
:hints (("goal" :in-theory (enable hifat-remove-file)))
:rule-classes :linear)
(defthm
hifat-entry-count-of-hifat-place-file
(implies
(m1-file-p file)
(equal
(hifat-entry-count (mv-nth 0 (hifat-place-file fs path file)))
(if
(zp (mv-nth 1 (hifat-place-file fs path file)))
(+
(hifat-entry-count fs)
(if (m1-regular-file-p file)
0
(hifat-entry-count (m1-file->contents file)))
(cond
((not (zp (mv-nth 1 (hifat-find-file fs path))))
1)
((m1-regular-file-p (mv-nth 0 (hifat-find-file fs path)))
0)
(t
(-
(hifat-entry-count
(m1-file->contents (mv-nth 0 (hifat-find-file fs path))))))))
(hifat-entry-count fs))))
:hints (("goal" :induct (hifat-place-file fs path file)
:in-theory (enable hifat-place-file hifat-find-file))))
|
